text:
Scouting/BSA/Citizenship in the World Merit Badge

Requirement 1

 * " in a particular country. The revised 2006 merit badge booklet requires input from the individual scout as this is a subjective question.  An example for the answer is, "I believe a citizen in the world is one who contributes to society in one way or another."

 * ...and what you think it takes to be a good world citizen.

This is also a subjective question that requires the individual scout put in his opinions. An example for this part is, "I believe a good world citizen is one who contributes effort to strive for the good of his/her fellow citizens."

Requirement 2

 * 2A) Explain how one becomes a citizen in the United States...

One can become an American citizen in various ways. The 2005 BSA "Citizenship in the World" handbook points these out: 1) "By blood if born to parents who are already citizens of [The United States]" "By soil if born within the legal borders of a nation" -"Through naturalization" (in which the person must meet certain requirements which can be found here) -"Through naturalization of parents"

 * 2B) ...and explain the rights, duties, and obligations of U.S. citizenship.

American citizens have a number of rights, such as freedom of speech, the right to bear arms, due process, religion, against quartering soldiers, and against double jeopardy (most of these rights also apply to non-citizens) and are also taxed in order to provide money for essential services. Other duties of American citizens might include obeying laws and voting. Obligations are those elements of citizenship which are required under the law, for example, the obligation to obey the law.

 * 2C) Discuss the SIMILARITIES and DIFFERENCES between the rights, duties, and obligations of U.S. citizens and of two other countries.

A U.S. citizens' rights, duties, and obligations are briefly explained in subsection 2B. Here are examples of how they are similar from those of Chinese and Irish citizens' rights, duties, and obligations in the following manner: -Both Chinese citizens and U.S. citizens have the obligation to receive education. -Both Chinese citizens and U.S. citizens have the duty to get a Visa to travel to other countries. -Both Irish citizens and U.S. citizens have the right to vote. -Both Irish citizens and U.S. citizens have the duty to serve on a jury when called upon.

Requirement 3
Do the following:

 *  A) pick a current world event. In relation to this current event, discuss with your counselor how a country's national interest and its relationship with other countries might affect areas such as its security, its economy, its values, and the health of its citizens.

 * B) Select a foreign country and discuss with your counselor how its geography, natural resources, and climate influence its economy and its global partnerships with other countries.

After World War 1, Germany's economy was hard struck by the effects of the Treaty of Versailles, leading to a national sense of bitterness towards countries like France and the United Kingdom, which help set the stage for World War 2, when Germany invaded many of its neighbors. After World War 2, Japan adopted a constitution that does not allow the creation of a military force, hampering the country's ability to help in military operations with the United Nations and allies like the United States. In the run-up to the 2003 occupation of Iraq, many American commentators expressed bitterness over France's refusal to support the invasion. Many perceived a lack of gratitude for American aid during World War 2, while France's defenders pointed out that country's aid during the American War for Independence. Culture includes a society's language, religion, lifestyle, clothing, decor and many other aspects. Cultural splits within countries sometimes lead to strife because different parts of the country have different interests. The American Civil War is an example of a country torn apart by the opposing interests of culturally distinct regions.

More recently, religious animosity has led to violence and chaos in many Middle-Eastern countries, especially Israel. The creation of Israel was prompted by fears of anti-Semitism leading to a repeat of the attempted genocide of the Jews during the Holocaust, and Israel's foreign relations have long been affected by fears of anti-Semitism harming Jews abroad. Conflicts over religion, geography, ethnicity and other issues have spread in different countries has caused strife and war. For example, the Kurds live in several countries, including Iraq and Turkey, the latter a key US ally in the region. With Kurdish leaders calling for an independent Kurdistan, the US was torn between supporting its Kurdish allies and the Turkish government, which believed that any independence for Iraqi Kurds could spur a revival of similar hopes among Turkish Kurds. Nationalism plays an important role in this phenomenon, which has also included calls for a Greater Somalia, a view that has twice led to war with Ethiopia over the Somali Ogaden region, and Greater Serbia|, a motivation for the Kosovo War. Demographics are the make-up of a country. Some countries have a very young population, such as Zambia, while others have a large elderly population, such as Sweden. Some countries are made up nearly entirely of one ethnic group speaking a single language, such as Iceland, while others are a melting pot of dozens or hundreds of ethnic groups, such as many of the equatorial African countries like the Republic of the Congo.

Demographic conflict has long been a source of strife on the diverse continent of Africa. In South Africa, for example, a white minority long controlled a huge black majority (Apartheid), eventually straining relations with countries like the United States, which imposed an embargo as a result. Ethnically diverse states such as most of the Sub-Saharan African countries were created from European colonies in the region, which had no basis in rational geographic or cultural boundaries. As a result, many of the earliest leaders in these countries pursued as a high priority the formation of a national identity; this occurred in nations across Africa, including Zambia and Sudan. In the rush to create a distinctive culture, minorities and dissenters were often marginalized or oppressed, leading to revolutions and civil strife. Politics can occur within a single country or among countries. Activities involving the internal political landscape of a country can have an affect on national interests and foreign relations through the election or installment of leaders who enact foreign policy, while laws passed by a particular country may have an affect on the economy of a different country.

Internal conflicts between political blocs of capitalists and socialists led to a polarization of international relations. Countries like the United States, France and Italy jockeyed for power with the Soviet Union and satellites in newly-independent countries across Africa, South America and Asia (see The Cold War). Following the internationally divisive US-led occupation of Iraq in 2003, foreign countries around the world were forced to pay close attention to the American political landscape. Some, like Israel, were supportive of the conservative right, while others, like France, supported the liberal left.

The national interests of a country are often very complex, and hinge on multiple factors involving geography, natural resources and climate, as well as social values, security issues and concerns regarding foreign countries.

Geography can affect a country's national resources through several ways. A country's shape can dictate its national interest.

Nearly all countries have some kind of natural resources. It is always in the best interest of the producers of a product (for example oil) to keep prices and demand high.

Because all countries need food and water, climate is extremely important. Without adequate rain, for example, agriculture can not occur and food will have to be imported and this can make a country's economy fall. Weather can also have debilitating effects on trade, both within and outside the boundaries of country.

Below are two links to additional information. They will provide informative graphs that had answered the previous question 4, which is worded exactly as the current question 3b.

Graphed Countries 1: Saudi Arabia

Graphed Countries 2: United States, Tuvalu, Botswana, and Bolivia

Requirement 4
Do TWO of the following:
 * A)Explain international law and how it differs from national law. Explain the role of international law and how international law can be used as a tool for conflict resolution

A state is ordinarily said to be sovereign - i.e. capable of enforcing its own laws on its own territory, and only on its own territory (for example, American laws do not apply in France, even if the people involved are all American). International law is a relatively recent invention, and concerns agreements between countries in pacts called treaties. International law can enforce standards (such as standardized weights and measures for the purposes of trade) and proscribe conduct according to treaties concerning human rights. With international organizations enforcing these treaties, often ineffectively, but also often with great success, countries whose governments ignore human rights treaties can be dealt with by multi-national alliances, which are perceived as having more legitimacy if a broken human rights treaty can be blamed.

 * B) Using resources such as major daily newspapers, the Internet (with your parent's permission), and news magazines, observe a current issue that involves international trade, foreign exchange, balance of payments, tariffs, and free trade. Explain what you have learned.  Include in your discussion an explanation of why countries must cooperate in order for world trade and global competition to thrive.This option requires the scout to do independent research.

 * C) Select TWO of the following organizations and describe their role in the world.

shaping the research agenda and stimulating the generation, translation and dissemination of valuable knowledge; setting norms and standards and promoting and monitoring their implementation; articulating ethical and evidence-based policy options; providing technical support, catalysing change, and building sustainable institutional capacity; and monitoring the health situation and assessing health trends.
 * The United Nations- is an international organization whose stated aims are facilitating cooperation in international law, international security, economic development, social progress, human rights, and achievement of world peace.
 * The World Court- is the primary judicial organ of the United Nations. It is based in the Peace Palace in The Hague, Netherlands. Its main functions are to settle legal disputes submitted to it by states and to provide advisory opinions on legal questions submitted to it by duly authorized international organs, agencies, and the UN General Assembly.
 * World Organization of the Scout Movement-The mission of WOSM is to contribute to the education of young people, through a value system based on the Scout Promise and Scout Law, to help build a better world where people are self-fulfilled as individuals and play a constructive role in society. World Organization of the Scout Movement is organized into regions and operates with a conference, committee and bureau.
 * The World Health Organization-providing leadership on matters critical to health and engaging in partnerships where joint action is needed;
 * Amnesty International- is a non-governmental organisation focused on human rights with over 3 million members and supporters around the world. The objective of the organisation is "to conduct research and generate action to prevent and end grave abuses of human rights, and to demand justice for those whose rights have been violated."
 * The International Red Cross- is an international humanitarian movement with approximately 97 million volunteers, members and staff worldwide which was founded to protect human life and health, to ensure respect for all human beings, and to prevent and alleviate human suffering, without any discrimination based on nationality, race, sex, religious beliefs, class or political opinions.
 * CARE-It helps with health with almost all of the countries.

Requirement 5
Do the following:
 * A) Discuss the differences between constitutional and nonconstitutional governments.

Constitutional governments have a written document stating their laws, rights and responsibilities. It must also be limited in power, follow a higher law, and have constitutional stability. Nonconstitutional governments are dictatorships and absolute monarchies. These are hard on the citizens because they don't have many rights. Totalitarian and authoritarian are the two types of nonconstitutional governments. In an authoritarian government, the power to make decisions is in the hands of one person or a small group. Totalitarian government is another form of authoritarian in which the government controls every aspect of society.
 * B) Name at least five different types of governments currently in power in the world.

United States- Federal republic, Representative democracy(democratic republic), Canada- Federal parliamentary democracy and constitutional monarchy, Spain- Parliamentary monarchy, Poland- Republic, Libya- People's Congress, North Korea- Single-party state, Brazil- Democratic federal republic, Niger- Parliamentary democracy.
 * C) Show on a world map countries that use each of these five different forms of government.

Requirement 6
Do the Following:

 * A) Explain how a government is represented abroad and how the United States government is accredited to international organizations.

A government is represented abroad by an ambassador or a consul. The United States government is accredited to international organizations based on the requirements of each individual organization. For example, here are the requirements for the United Nations

 * B) Describe the roles of the following in the conduct of foreign relations:

Ambassador: a diplomatic official of the highest rank, sent by one sovereign or state to another as its resident representative (ambassador extraordinary and plenipotentiary), usually for a specific length of time. He or she is a diplomatic official heading his or her country's permanent mission to certain international organizations; such as the United Nations; and is an authorized representative of a nation to deliver their nation's messages and protect the national interest of the nation they represent through diplomatic means.

Consul: an official appointed by the government of one country to look after its commercial interests and the welfare of its citizens in another country, as well as providing assistance to Americans traveling abroad.

Bureau of International Information Programs: The Bureau of International Information Programs (IIP) is the principal international strategic communications service for the foreign affairs community. IIP designs, develops, and implements a variety of information initiatives and strategic communications programs, including Internet and print publications, traveling and electronically transmitted speaker programs, and information resource services. These reach--and are created strictly for--key international audiences, such as the media, government officials, opinion leaders, and the general public in more than 140 countries around the world.

U.S Agency for International Development: The United States Agency for International Development (USAID) is the United States federal government agency primarily responsible for administering civilian foreign aid. President John F. Kennedy created USAID in 1961 by executive order to implement development assistance programs in the areas authorized by the Congress in the Foreign Assistance Act of 1961. An independent federal agency, USAID receives overall foreign policy guidance from the United States Secretary of State and seeks to "extend a helping hand to those people overseas struggling to make a better life, recover from a disaster or striving to live in a free and democratic country.

United States and Foreign Commercial Service: The U.S. Commercial Service (CS), the trade promotion arm of the U.S. Department of Commerce's International Trade Administration, helps U.S. companies succeed in markets around the world. Led by Assistant Secretary of Commerce and Director General Suresh Kumar and located across the United States and in U.S. Embassies and Consulates in nearly 80 countries, the CS global network of trade professionals helps thousands of U.S. companies to export goods and services worth billions of dollars every year.

 * C) Explain the purpose of a passport and visa for international travel.

A passport is an official document issued to a citizen by a government affirming the individual’s nationality. The passport permits the citizen to travel or live abroad and entitles the citizen to privileges guaranteed by international custom and treaties: safe passage, lawful aid, and protection while under the foreign government’s jurisdiction. A visa is a permit to enter another country and must be issued by the country one wishes to enter. It is a special seal stamped on a valid passport that indicates an official of that country has approved the visit. Prior to leaving, the traveler should apply for a visa-if required by that particular country-at the appropriate foreign embassy or nearest consulate. A foreign national wishing to enter the United States must apply for a visa outside the United States at embassies and consulates worldwide. An immigrant visa permits a person to stay in the United States permanently. A non-immigrant visa allows a foreign national to enter the country for a temporary stay. Tourists, temporary workers, missionaries, business professionals, artists, and athletes travel on temporary visas.

Requirement 7
Do two of the following (with your parent's permission) and share with your counselor what you have learned:

 * A) Visit the Web site of the U.S. State Department. Learn more about an issue you find interesting that is discussed on this Web site.

Web Site: http://www.state.gov/

Issues Vary.

 * B) Visit the Web site of an international news organization or foreign government...

Example Websites (news organizations): Al Jazeera

 * ...OR examine a foreign newspaper available at your local library, bookstore, or newsstand. Find a news story about a human right realized in the United States that is not recognized in another country.

 * C) Visit with a student or Scout from another country and discuss the typical values, holidays, ethnic foods, and traditions practiced or enjoyed there.

 * D) Attend a world Scout jamboree.

 * E) Participate in or attend an international event in your area, such as an ethnic festival, concert, or play.

Author's Note
Writing this article has been a pleasurable learning experience for me and I am happy to have been able to give back to the scouting community.

I hope that my efforts serve as a reminder to the reader that it is important to both give and take information in order to be a good world citizen.


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Hedonic modelling

Hedonic modelling
Related phrases: hedonic pricing approach, hedonic tone, hedonic model, hedonic modeling, hedonic scale, hedonic calculus, hedonic demand theory, hedonic regression, hedonic pricing, "Measure the pleasure"

For more contenders for the million dollar view, see []

Definition of Hedonic
Relating to like or dislike. See also hedonics and Hedonic motivation

Background
This was a page for collaborative original research on the use of hedonic models for estimating the value of a view from a building. This was planning for work in progress in 2007, published in 2008 and 2009. Two papers on the value of a (sea and veld) view were presented. The primary author was based at the School of Construction Economics and Management at the University of the Witwatersrand, Johannesburg, South Africa.

Viewpoint
The authors believe that hedonic modelling is a useful tool in assessing the value of buildings where view is an asset. This includes hotels, apartment blocks (flats), residential homes. Data on property transactions and selling points must be readily available, making this the least expensive and least complicated approach.

Preproposal planning
Beginning of a research proposal for co-authored work.

Research Question
What is the value of a view? Alternatively: What is a view worth? +/-ve

Dependent variable
Property price (ideally deeds office); rental / purchase price

Independent variables that might be available
Adjacent property prices, plot size zoned, number of bedrooms, floor area of rooms, number of bathrooms, degrees of seaview, lakes, wetlands, forest, scrub, open fields/space, parks, city scapes; water pollution, air pollution, farmland flies, car and aeroplane noise levels, mine dumps, slime dams, power lines, cellphone towers etc.

Confounding variables
Inflation of property prices, noise, access to transport and amenities, snob value, etc.

Key words
Hedonic regression modelling, property purchase prices

Importance of study
Aid to property valuation; Model can be used to reduce visual impact of new developments; Model can be used recursively by developers to optimize returns on multi-unit developments or estimate for planning and decision-making the potential impacts of construction projects.

Method

 * 1) Capture valuations.
 * 2) Multiple Linear Regression Analysis

Reading List

 * [http://www.harc.edu/Projects/Nature/Projects/Methods Valuing Nature in

Texas]

Method] Ecosystem Valuation Section 3
 * [http://www.ecosystemvaluation.org/hedonic_pricing.htm Hedonic Pricing

What’s in a View?] Steven C. Bourassa, Martin Hoesli, and Jian Sun
 * [http://www.hec.unige.ch/recherches_publications/cahiers/2003/2003.13.pdf

The Effect of Environmental Amenities on House Values] by Quentin Gillard in The Professional Geographer (1981) vol. 33 p. 216
 * [http://www.doi.org/10.1111/j.0033-0124.1981.00216.x

source: wikimedia

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Hedonic modelling


text:
Wikipedia:Sockpuppet investigations/DolphinOfJustice/Archive

Suspected sockpuppets

 * User compare report Auto-generated every hour.
 * Editor interaction utility

All have very low edits and have exclusively edited the page The Line Madder created by the sockmaster about a non notable self-published book by a non notable author, and have all contributed to Articles for deletion/The Line Madder - all voting keep. Kudpung กุดผึ้ง (talk) 06:39, 15 September 2017 (UTC)

Comments by other users

Clerk, CheckUser, and/or patrolling admin comments

 * - Please compare these accounts. Thanks, GABgab 15:30, 15 September 2017 (UTC)

. All those listed above and the master. CU please check for sleepers. Kudpung กุดผึ้ง (talk) 04:06, 16 September 2017 (UTC)
 * No sleepers, but it looks like the actual master is User:DolphinOfJustice, if someone wants to move the case. Courcelles (talk) 20:04, 18 September 2017 (UTC)

 * Moved, tagged, closing. GABgab 22:40, 18 September 2017 (UTC)


text:
User:Vishalghule

Name:Vishal Ghule DOB:19-07-2000 Place of Birth:Vizag City:kalyan State:Maharastra Country:India

title: User:Vishalghule


text:
User:Theresa knott/sandbox

Here is an excel file for you to download [[Media:Year 5 prep school experiments.xls |Excel spreadsheet]]

title: User:Theresa knott/sandbox


text:
Anthony Vaz

Anthony Querobino Exaltacao Vaz (14 September 1932 – 7 August 1982) was a Kenyan field hockey player. He was Kenya's first flag bearer and captain at the Summer Olympic Games. He competed at the 1956 Summer Olympics, the 1960 Summer Olympics and the 1964 Summer Olympics.

Early life
Anthony Querobino Exaltacao Vaz was born on 14 September 1932 and was a Goan. He attended and graduated from King Alfred's College (now known as the University of Winchester) in Winchester, England.

Sporting career
Vaz was a member of the Goan Institute in Nairobi, Kenya, described as "most celebrated social and sports club within the East African Goan community." He was a member of their field hockey team as a right full back as early as 1955. He also played cricket and football, being described as an "all-around sportsman."

Vaz was called up to the Kenya national field hockey team for the first time in 1956–to represent his country at the 1956 Summer Olympics in Melbourne, Australia. 1956 was Kenya's inaugural appearance at the games, and Vaz was named both the overall team captain and the country's flag bearer. He helped the Kenya team place 10th out of 12 teams in field hockey with a record of 2–2–2.

Vaz continued playing for the national team and in 1959 represented them at the East and Central African Championships, being an important piece of their tournament-winning squad. The following year, he was selected to participate in the 1960 Summer Olympics in Rome, Italy, his second Olympic appearance. Out of 16 teams, Vaz helped his country place eighth with a 2–2–2 record.

At the 1960 Olympics, Vaz was interviewed by American journalist Al Abrams of the Pittsburgh Post-Gazette, who noted that he was a "dark, handsome lad of 27 who comes from Kenya, an English province in East Africa, some thousands of miles away from Rome" and that he "spoke perfect English." Vaz described his past Olympic experience and noted that "Soccer football, field hockey, cricket and tennis are our major sports in Kenya, Uganda and Tanganyika."

In 1961, Vaz toured Rhodesia with the national field hockey team and the following year played matches against the Pakistan national team. He was noted as one of the players who always gave opposing teams "a good run for their money." He was selected for his third and final Olympic appearance at the 1964 Summer Olympics in Tokyo, Japan, helping Kenya place sixth with a 4–1–3 record. A modern source about Kenyan field hockey noted that Vaz was one of the players who "easily" could have been considered the best Goan at the Olympics at the time.

During his sporting career, Vaz's height was 156 cm and his weight was 64 kg.

Personal life and death
Outside of sports, Vaz worked as a school teacher and physical fitness instructor. He trained Olympic gold medalist Kipchoge Keino at the start of his track and field career. He later spent time in England and died in Lambeth on 7 August 1982, at the age of 49.

title: Anthony Vaz


text:
Talk:Now That's What I Call Music! 43 (American series)

Track Listing now is not on the Now Album!!!!!!!
Are y'all crazy!!!!!! This current track listing is NOT the actual track listing!!!!!! — Preceding unsigned comment added by JordanKyser22 (talk • contribs) 21:26, 4 August 2012 (UTC)

Excuse me, you have is a wrong track list for Now, Vol. 43. Thank you. --<IP_ADDRESS> (talk) 21:54, 4 August 2012 (UTC)

Change it back please. --<IP_ADDRESS> (talk) 21:56, 4 August 2012 (UTC)


text:
Talk:Antonino Lo Surdo

Translated by?
It seems that this page was created as a translation from the corresponding one on it:wiki --<IP_ADDRESS> (talk) 22:46, 2 October 2014 (UTC)

title: Talk:Antonino Lo Surdo


text:
Wikipedia:Main Page history/2017 September 18

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text:
Antigonia

Antigonia may refer to:

Places

 * Antigonia (Chaonia), near Saraqinisht, southern Albania
 * Archaeological National park of Antigonea
 * Antigonia (Paeonia), in Paeonia, Macedon
 * Antigonia Psaphara, in Crucis, Chalcidice, Greece
 * Antigonia (Syria), in Seleucis, Syria, now near Antakya, Hatay Province, Turkey
 * Alexandria Troas or Antigonia, in Troas, Asia Minor, now Eski Stambul, Çanakkale Province, Turkey
 * Diocese of Antigonia, a former see in the Roman province Hellespontus
 * Nicaea or Antigonia, Bithynia, now İznik, Bursa Province, Turkey
 * Mantinea or Antigonia, Arcadia, Greece

Other uses

 * Antigonia (fish), a genus of fish

title: Antigonia


text:
School talk:Philosophy

Wekk I would be prepared to write something about the history of continental philosphy, from Nietzsche to Badiou.== Updates ==

Suggestions
Let's face it, nothing like a functioning wikiversity will exist for a long, long time. The whole project was premature. Perhaps however we can offer something to the interested layman. I am thinking we should make a reading list.

 * I strongly disagree. I believe that the Wikiversity classes might have been premature (but they still might have been an interesting experiment). However, these and other experiments led to a discussion about whether Wikiversity belonged in Wikibooks which, in turn, was quite a put-off for people working here at Wikiversity, I believe. As far as I have seen, these discussions have brought the content of Wikiversity to a stand-still. However, much of the material in wikiversity was within the wikibooks policy (textbooks, notes and reading lists). (Alex beta 18:30, 30 April 2006 (UTC))

Metaphysics, epistemology, the philosophy of science, normative ethics, the philosophy of language, the philosophy of religion and the philosophy of mind are missing. I would suggest that these things are an absolute minimum for this to be taken seriously as an introduction to philosophy.

Moving the Introduction to Philosophy
Perhaps the Introduction to Philosophy at the top of the School of Philosophy should be moved to the contents list. (Alex beta 15:31, 23 December 2006 (UTC))

Errors (Typos, Factual Mistakes, Etc)
why is there no spell checker built in editor? Asaashok 18:58, 28 September 2008 (UTC)

Voting
I am not sure if this is on topic, but I think it is, as it concerns the future of this department. Please vote yes on the Wikiversity proposal, there's no point doing anything unless it's passed.

 * I am not sure that I agree, although it is true that I have not contributed (except for minor edits) since I read that the Wikiversity was on Votes for Deletion. It currently seems improbable that we will reach two thirds of the votes by the first of November. If this is the case, and Wikiversity does not get its own Wiki, you might want to consider voting "Keep" on the Votes for deletion/Wikiversity page. (Alex beta 12:13, 21 October 2005 (UTC))

Alex's Awaiting
I have been awaiting the birth of the school of philosophy in wikiversity for some time. So far, no one has dared to set down a few stepping stones for the rest of us. I suspect that part of the problem is that "What is philosophy?" is itself a philosophical question, and it is perhaps difficult to find the right words to give someone a sense of the kind of investigation we are involved in. (Alex beta 22:05, 15 Sep 2004 (UTC))

Alex's Looking
I have checked the main pages of a couple of schools and found that the structure of the page is usually a list of courses on offer, some links to books and external resources, and some way to ask for help. In our case, it will be useful to create a list of resources freely available (For example, Kant's Critique of Pure Reason or Wittgenstein's Tractatus are in the public domain and can be found on the internet). I would also suggest the creation of some introductory courses: The History of Ancient Philosophy, Introduction to Philosophical Reasoning, Logic and so on... I would love to start to work on the Logic course if it is possible to coordinate a small group of volunteers in order to save this school from a speedy deletion. Some suggestions on how to organise some way to ask for help would be appreciated. (Is there anyone else out there?) (Alex beta 22:37, 15 Sep 2004 (UTC))

 * I would start with the Pre-Socratics moving up to Plato then Aristotle. Before Plato, philosophy and science were one and the same, where the basic drive was to understand the natural world and life itself. Thales, was one, Heraclitus another. Heraclitus was the one who observed that "you can never step into the same river twice." Plato (I believe) represented the first systematic or "dialectic" (and he literally used dialogs) approach to reaching a state of truth about something. The field of philosphy today (like many other disciplines) is riven with specialization. There are branches, for example, that deal exclusively with areas of biomedical ethics. The best approach would be an historical one for starters, to lay the foundation. -CL

 * Thanks for answering. I agree. The history of ancient philosophy is probably a good introduction to philosophy. I will add it to the list of courses and try to write something in the next few days.

I find your comment "philosophy and science were one and the same" very interesting. It is true that there was no clear distinction between the two in Pre-Socratic thought. However, their method of enquiry could hardly be identified with what we consider science nowadays, but their search for the ultimate origin (Arché) of the Cosmos can be considered genuinely philosophical by our standards. (Alex beta 19:32, 24 Sep 2004 (UTC))

Deleting comments on the discussion page
Is it OK to delete these comments on the discussion page? Some of them are obsolete. Does anyone know the policies in reference to this? (Alex beta 15:16, 23 December 2006 (UTC))

First Year Logic
Hello. I may be able to help put some kind of first-year logic course together, but I'm not going to do it all myself. --publunch 23:26, 18 Oct 2004 (UTC) i could help

For the spanish-speaking
Estoy en correspondencia con Gustavo Bueno Sánchez, Coordinador del Proyecto Filosofía en Español (pfe arroba filosofia punto org). Se muestra dispuesto a colaborar con nosotros. Si alguien de aquí quiere coordinarse con ellos, seguro que puede salir algo productivo. -- 4lex 09:44, 16 Nov 2004 (UTC)

Objectivism
I Noticed that the outline of the philosophy main page is not very organized. I would like to recommend that we reorganize it in several groups. introduction to Philosophy, Schools of philosophy, types of philosophy (Metaphysics, epistemology,ethics,aesthetics,politics,etc.), philosophy through history, and finally the philosophers themselves. Also under the schools of philosophy, I would be willing to create a section entirely devoted to objectivism. I know a great deal about Objectivism and its creator Ayn Rand and would love to teach it. Judging from what I see on the main page, my concern now is where I would put such a thing. Any comments?

School of Philosophy on Wikiversity

 * Department of Philosophy I've started a Department of Philosophy on Wikiversity. Koavf 17:36, 20 August 2006 (UTC)

Classical Philosophy
Please make sure that Greek and Latin Philosophy are coordinated with the School of Classics, or at least inform them of the classes so that they can benefit from them, as well. The Jade Knight 02:07, 13 October 2006 (UTC)

MIT OpenCouseWare 24.00 Problems of Philosophy, Fall 2005
Well, since things have ostensibly stoped moving forward here. I'm going to start going through MITs OpenCouseWare [24.00 Problems of Philosophy, Fall 2005]. Not to take anything away from this project but I'm not getting anywhere with the sparse framwork that is in place at the moment. Hopefully whilst going through the coursework over there I can contribute something back here. It would be nice to interact with some other people going the same route though. If anyone is interested in working along please throw up a message on my talk page. There's an board that MIT links to for discussions on the projects, but it seems to be dead as well. RichMac 02:07, 21 October 2006 (UTC)

Ayn Rand
Am I allowed to post here? Sorry, if not.

I suggest you, for credibility's sake, not teach Ayn Rand or list "Philosophy: Who Needs It?" on the reading list. There is a reason she is not taken seriously by most academicians. She was a clever ranter, but a very poor philosopher (or pseudo-philosopher). <IP_ADDRESS> 02:55, 16 November 2007 (UTC)
 * I think it would be great to have a learning project that explores Randian "philosophy"....you could participate and present your negative views. Maybe there could be a more general project that explores differences between the views of academic philosophers and the rest of us. --JWSchmidt 04:53, 16 November 2007 (UTC)

confused about how the course system works
Hello everyone i’m very sorry if i’m not suppose to post here i’m not sure where too i'm not very use to wiki editing but I think this is a good place to ask the department a question and I’d love it if I got some responses. i'm confused about how the course system works, there are reading lists, and things like that could you please tell my the curriculum, or what your suppose to do as a student? Really anything that would help me better understand all this. I am very interested in learning I just don't know how or where to start.

Thanks a ton Wikilife, 05:14, 9 March 2008
 * Hi Wikilife, there is a section called Where to start, which might help. I like especially this comment: "The best place to start is by asking yourself why you came here..." In general it is recommendable to do what you prefer, we are a wiki. So just be bold. A curriculum might be there or not and even if so, it is just a suggestion.
 * Perhaps it is also so, that people from this school are also in vacation or alike ? You could e.g. find out, if you go to the peoples userpage (see list) and on their user page on the left navigation bar called "toolbox" there is the link to: User contributions - with this you can see when the users last edit was done.
 * If you have any more questions (and surely there are always), please feel free to ask. Erkan Yilmaz Wikiversity:Chat  09:55, 9 March 2008 (UTC)

Thanks A ton Erkan you have been very helpful Wikilife

a school to work with Philosophy and Nonkilling studies
I have created an esoteric Pythagorean-Platonic section of this school: Ahimsa Academy. Some of you will like the idea; a few of of you may think it is pointless; a few of you might not like it or (hopefully not) think it should be deleted. It is modelled after the Philosophy school page, because the Academy picture is appropriate, and certainly because I like most of the courses posted here so far. However, we should look at some of the science schools and Wikimedia science project and portal pages for ways to improve the layout and efficiency of the Philosophy schools.--Dchmelik 09:05, 6 April 2009 (UTC)

 * Indented line Why do the topics of Ahimsa Academy need a new section?(Matousekd 21:56, 23 October 2010 (UTC))

Hey
I'm new to Wikiversity. How active is the School of Philosophy?

(Matousekd 21:48, 23 October 2010 (UTC))

 * Hi! I'm pretty new here too. As far as I can tell, a lot of the schools are not very active. But we can help change that. :) So if there's a topic within philosophy that you know a lot about, please help build up the courses! Spock of Vulcan 15:57, 15 November 2010 (UTC)

Course List
The Department of philosophy's has a short course list. Any suggestions? Matousekd 20:12, 29 October 2010 (UTC)

Ethical theory
I know a fair bit about ethics, so I should be able to develop some of the ethics courses. :) I'm just learning about the whole structure of this site a bit first. I'm an editor on Wikipedia, but this site is more different than I expected. Hopefully I can get on that soon! I could probably work on applied ethics as well. Spock of Vulcan 16:04, 15 November 2010 (UTC)

Organization
So, we've got some big organizational issues here. I finally figured out how everything is supposed to be organized, at least. My biggest question is this: What's up with Topic:Philosophy? That means that there is a department of philosophy within the school of philosophy, which isn't in itself a problem if it's where there will be lessons talking about philosophy in general, or something like that. But it isn't even linked to from this page right now, and all the links on it are to Wikipedia pages, not lessons. Also, this school's main page is really confusing. So I'm gonna work on that a bit. If anyone else who works on this school has any issues with stuff I change, or if I step on any toes, please feel free to educate me. :) I'm still pretty new to Wikiversity. Hopefully we can all work together and get this school looking as good as a school of philosophy should. Happy learning! Spock of Vulcan 23:45, 15 November 2010 (UTC)

Wow
Just dropping by to say that this is such a cool school. Keep up the good work! Heartwarming (discuss • contribs) 02:49, 27 December 2015 (UTC)

But, what is philosophy in the end? What we dope out when we are bored?
Please see the theme.
 * That's part of it, sure. It's also a love of wisdom. —Justin ( koavf ) ❤T☮C☺M☯ 04:31, 20 November 2016 (UTC)
 * Or, something to think about after a good meal and a pint of dark ale! The kind you can't see through! --Marshallsumter (discuss • contribs) 00:15, 21 November 2016 (UTC)

Suggesting Oxford’s Free Introduction to Philosophy: Stream 41 Lectures as a resource
The freely available courses at Oxford’s Free Introduction to Philosophy can be a useful resource for students and course developers. --Lbeaumont (discuss • contribs) 14:58, 30 May 2020 (UTC)

title: School talk:Philosophy


text:
Botany/Contributors

Contributors
While Wikibooks offers somewhat clearer opportunities for "authorship" than Wikipedia, there remains the fact that anything put here is really just a contribution, and everyone who furthers the effort is a contributor. In this respect there are no "authors". If you are interested in developing a particular subject within the field of botany as part of this textbook, please start the module and link it here. We will work out the chapter arrangement after you get started. Of course, you can also just expand on an existing chapter.

 * Eric Guinther is a biologist and environmental consultant living in Kane'ohe, Hawai'i. He envisioned this particular textbook "project" and developed the approach of creating a textbook around articles found in Wikipedia. By then working specifically on those articles referred to, the quality and quantity of the Study Guide to the Science of Botany was gradually enhanced. These efforts were done with the steady help of numerous talented contributors at Wikipedia who, collectively, would have no interest in working on a textbook project per se.  Ultimately, the user of the guide benefits from the textbook's approach: 1) by having more control over the depth into any subtopic pursued and 2) being limited in that pursuit only by that considerable information available at Wikipedia and other linked websites.

 * Ransom Akinlade is a final year student of the department of Botany, Obafemi Awolowo University, Ile Ife. He developed a keen interest in this project and he regularly contributes to the articles foi=und therein.

 * Other anonymous Wikibook contributors (see individual page histories).

 * Numerous other Wikipedia contributors.

<<Return to Contents Page

title: Botany/Contributors


text:
Jovellanos

Jovellanos is a municipality and town in the Matanzas Province of Cuba.

Overview
The municipality is divided into the barrios of Asunción, Isabel, Realengo and San José.

It was founded in 1842 as Corral de la Bemba on the location of an old ranch called Bemba. It took its current name in 1870, in honor of the Spanish writer Gaspar Melchor de Jovellanos. The same year it achieved the status of villa (town).

Jovellanos is also called "Bemba" by its inhabitants and locals. "Bemba" means "big lip" in Cuban Spanish; this term refers to the black population of the city. To outsiders, it may have negative racial overtones, but in the city this is not the case. Inside Jovellanos, the black and white population for the most part live harmoniously, side by side. Therefore, "Bemba" is an affectionate name for the town used by locals.

Jovellanos is an industrial town, with large and small factories, while sugar cane is cultivated in the surrounding areas.

Demographics
In 2022, the municipality of Jovellanos had a population of 57,216. With a total area of 505 km2, it has a population density of 110 /km2.

Notable people

 * Esteban Lazo Hernández (born 1944), Cuban politician and current President of the National Assembly
 * Manuel Navarro Luna (1894–1966), Cuban poet and journalist

title: Jovellanos


text:
Advanced Classical Mechanics/Rigid Bodies

Definition

 * In a rigid body, no part of the body moves relative to another part of the body regardless of the force applied.
 * A rigid body is equivalent to a system of particles restricted to have fixed distances apart.

There are two methods for solving for the motion of rigid bodies.

 * 1) Write out the kinetic energy $$T$$ and use the Lagrangian to derive the equations of motion
 * 2) Use Euler's equations to get the motion of the center of mass and the rotation.

The Motion of a Rigid Body
The restrictions of rigid body motion require that distance between any two particles within the body remain constant. For example

$$ l_{ij}^2 = \left ( \vec r_i - \vec r_j \right )^2 = \left (x_i - x_j \right)^2 + \left (y_i - y_j \right)^2 + \left (z_i - z_j \right)^2 $$

will be constant during the motion. Let's take the time derivative of this quantity,

$$ \frac{d l_{ij}^2}{d t} = 2 \left ( \vec r_i - \vec r_j \right ) \cdot \left ( \vec v_i - \vec v_j \right ) $$

For this to vanish either $$\vec v_i - \vec v_j$$ must vanish or the dot product must vanish. The first corresponds to a translation of the object. For the dot product to vanish, the velocity difference must be perpendicular to the position difference. Specifically,

$$ \vec v_i - \vec v_j = \vec \omega_{ij} \times \left ( \vec r_i - \vec r_j \right ). $$

At first it might appear that you can have a different $$\vec \omega_{ij}$$ for each pair of particles. It turns out that this is not the case. Let's calculate

$$ \vec v_i - \vec v_k = \left ( \vec v_i - \vec v_j \right ) - \left ( \vec v_k - \vec v_j \right ) = \vec \omega_{ij} \times \left ( \vec r_i - \vec r_j \right ) - \vec \omega_{kj} \times \left ( \vec r_k - \vec r_j \right ). $$

On the other hand we could have calculated it as follows,

$$ \vec v_i - \vec v_k = \vec \omega_{ik} \times \left ( \vec r_i - \vec r_k \right ) = \vec \omega_{ik} \times \left ( \vec r_i - \vec r_j \right ) - \vec \omega_{ik} \times \left ( \vec r_k - \vec r_j \right ). $$

For these expressions to be equal for arbitrary values of $$\vec r_i$$, $$\vec r_j$$ and $$\vec r_k$$, the three values of $$\vec \omega$$ must be equal to each other so we will drop the subscripts and call this quantity the angular velocity of the rotation.

To summarize, a rigid body has two types of motion:

 * 1) Translation of the entire body, and
 * 2) Uniform rotation about a particular axis (the direction of $$\vec \omega$$ and an angular speed given by the magnitude of $$\vec \omega$$.

Angular Velocity
We can use some point within or outside the body as the origin and denote its position at any time by $$\vec r_0$$. The velocity of a particular part of the body is

$$ \vec v = \vec v_0 + \vec \omega \times \left (\vec r - \vec r_0 \right ) $$

where $$\vec r$$ is the location of the part of the body and $$\vec v_0$$ is the velocity of the origin.

Let's move the origin to $$\vec r_1$$. The velocity of the particle does not change so we have

$$ \vec v = \vec v_0 + \vec \omega \times \left (\vec r - \vec r_0 \right ) = \vec v_1 + \vec \omega \times \left (\vec r - \vec r_1 \right ) $$

so

$$ \vec v_1 = \vec v_0 + \vec \omega \times \left ( \vec r_1 - \vec r_0 \right ). $$

The vector $$\vec \omega$$ is constant with respect to changes in the origin. A useful example of this is a wheel that rolls without slipping.

Moment of Inertia
Rigid bodies are such an important part of classical mechanics that we have developed special techniques to calculate their kinetic energies and angular momenta. Let's calculate the kinetic energy of a mass in rigid body motion. Its velocity is

$$ \vec v_i = \vec v_0 + \vec \omega \times \vec r_i $$

where we have written the coordinates relative to the origin ($$\vec r_0$$).

Kinetic Energy
Its kinetic energy is

$$ T_i = \frac{1}{2} m_i \left ( \vec v_i \cdot \vec v_i \right ) = \frac{1}{2} m_i \left ( \vec v_0 + \vec \omega \times \vec r_i  \right ) \cdot \left ( \vec v_0 + \vec \omega \times \vec r_i  \right ). $$

and multiplying out the terms we get

$$ T_i = \frac{1}{2} m_i \left [ v_0^2 + 2 \vec v_0 \cdot \left ( \vec \omega \times \vec r_i \right )  + \left ( \vec \omega \times \vec r_i\right )^2 \right ]. $$

We can simplify the result using the following fact

$$ \left( \vec A \times \vec B \right )^2 = A^2 B^2 \sin^2 \theta = A^2 B^2 \left ( 1 - \cos^2 \theta \right ) = A^2 B^2 \left[ 1 - \frac{\left (\vec A \cdot \vec B\right )^2}{A^2 B^2} \right ] = A^2 B^2 - \left (\vec A \cdot \vec B\right )^2 $$

to get

$$ T_i = \frac{1}{2} m_i \left [ v_0^2 + 2 \vec v_0 \cdot \left ( \vec \omega \times \vec r_i \right )  + \vec \omega^2 \vec r_i^2 - \left ( \vec \omega \cdot \vec r_i \right )^2 \right ]. $$

Angular Momentum
Let's look at the angular momentum of the particle, we have

$$ \vec L_i = m_i \vec r_i \times \vec v_i = m_i \vec r_i \times \vec v_0 + m_i \vec r_i \times \left ( \vec \omega \times \vec r_i \right ). $$

Here we can use the vector triple product to simplify the result

$$ \vec A \times \left ( \vec B \times \vec C \right ) = \vec B \left ( \vec A \cdot \vec C \right ) - \vec C \left ( \vec A \cdot \vec B \right ) $$

which for us yields

$$ \vec L_i = m_i \vec r_i \times \vec v_i = m_i \vec r_i \times \vec v_0 + m_i \left [ \vec \omega r^2 - \vec r \left ( \vec \omega \cdot \vec r \right ) \right ]. $$

If $$\vec v_0=0$$ then

$$ T_i = \frac{1}{2} \vec \omega \cdot \vec L_i. $$

In this case, let's write out $$\vec L_i$$ component by component, we have

$$ \vec L_i = m_i \left [ \begin{matrix} \omega_x r_i^2 - x_i \left ( \omega_x x_i + \omega_y y_i + \omega_z z_i \right ) \\ \omega_y r_i^2 - y_i \left ( \omega_x x_i + \omega_y y_i + \omega_z z_i \right ) \\ \omega_z r_i^2 - z_i \left ( \omega_x x_i + \omega_y y_i + \omega_z z_i \right ) \end{matrix} \right ] = m_i \left [ \begin{matrix} y_i^2 + z_i^2 & -x_i y_i & -x_i z_i \\ -x_i y_i & x_i^2 + z_i^2 & -y_i z_i \\ -x_i z_i & -y_i z_i & x_i^2 + y_i^2 \end{matrix} \right ] \left [ \begin{matrix} \omega_x \\ \omega_y \\ \omega_z \end{matrix} \right ] \equiv \vec \vec I_i \cdot \vec \omega $$

defining the moment of inertia matrix for that particular particle.

Putting It All Together
Because the quantities $$\vec v_0$$ and $$\vec \omega$$ are the same for every particle, we can sum up the kinetic energy and angular momentum for all of the particles to get

$$ T = \sum_i T_i = \frac{1}{2} M v_0^2 + \frac{1}{2} \vec \omega \cdot \vec I \cdot \vec \omega + M \vec v_0 \cdot \left ( \vec \omega \times \vec r_{CM} \right ) $$

where $$M$$ is the total mass, $$\vec r_{CM}$$ is the location of the centre of mass and $$\vec \vec I = \sum_i \vec \vec I_i$$ is the moment of inertia of the body about the origin. The total angular momentum is

$$ \vec L = M \vec r_{CM} \times \vec v_0 + \vec \vec I \cdot \vec \omega $$

Using the result for the scalar triple product we can show that

$$ \vec L = \frac{\partial T}{\partial \omega} $$

so the angular momentum is the generalized momentum conjugate to the angular velocity; if the potential does not depend on angle, the angular momentum is a first integral of the motion.

Principal Axes
The moment of inertia is a positive definite matrix. We know this because when we multiply it by any non-zero vector $$\vec \omega$$, we get the kinetic energy of the body that is necessarily positive. Such a matrix can always be diagonalized. From a physical point of view, that means that the eigenvector of the matrix are known as the principal axes of the body. If the angular velocity points along one of the principal axes, then the angular momentum is parallel to the angular velocity. Furthermore, if one chooses the principal axes of the body to be one's coordinate axes, the diagonal element of the moment of inertia matrix vanish, so for example,

$$ \vec L = \left [ \begin{matrix} I_{xx} \omega_x \\ I_{yy} \omega_y \\ I_{zz} \omega_z \end{matrix} \right ] $$

and the kinetic energy is

$$ T = \frac{1}{2} M v_0^2 + M \vec v_0 \cdot \left ( \vec \omega \times \vec r_{CM} \right )    +  \frac{1}{2}I_{xx} \omega_x^2 +  \frac{1}{2}I_{yy} \omega_y^2 +  \frac{1}{2} I_{zz} \omega_z^2 $$

Change of Origin
Sometimes it is easiest to calculate the moment of inertia about a particular point but you are interested in the moment of inertia about another point. There is a straightforward prescription to achieve this. First, let's use the center of mass of the system as the origin, so we have

$$ T = \frac{1}{2} M v_{CM}^2 + \frac{1}{2} \vec \omega \cdot \vec \vec I_{CM} \cdot \vec \omega $$

Let's write this same equation with respect to a new origin $$\vec r_0$$ so we have

$$ T = \frac{1}{2} M \left ( \vec v_0 + \vec \omega \times \vec r_{CM}\right )^2 + \frac{1}{2} \vec \omega \cdot \vec \vec I_{CM} \cdot \vec \omega $$

$$ = \frac{1}{2} M v_0^2 + M \vec v_0 \cdot \left ( \vec \omega \times \vec r_{CM}\right ) + \frac{1}{2} M \left( \vec \omega \times \vec r_{CM} \right )^2 + \frac{1}{2} \vec \omega \cdot \vec \vec I_{CM} \cdot \vec \omega. $$

If we compare this result to the earlier result we find

$$ \vec \omega \cdot \vec \vec I \vec \omega = \vec \omega \cdot \vec \vec I_{CM} \cdot \vec \omega + M \left( \vec \omega \times \vec r_{CM} \right )^2. $$

On a component by component basis we have

$$ \vec \vec I = \vec \vec I_{CM} + M \left [ \begin{matrix} y_{CM}^2 + z_{CM}^2 &  x_{CM} y_{CM} &  x_{CM} z_{CM} \\ x_{CM} y_{CM} & x_{CM}^2 + z_{CM}^2 & M y_{CM} z_{CM} \\ x_{CM} z_{CM} & y_{CM} z_{CM} & x_{CM}^2 + y_{CM}^2 \end{matrix} \right ] $$

To change from one origin to another you first use the equation above to get the moment of inertia relative to the centre of mass and then use it a second time to get the moment of inertia about the new origin.

Calculating a Moment of Inertia
Let's calculate the moment of inertia of an ellipsoid. Let's assume that the axes of the ellipsoid are line up along the coordinate axes. Along the $$x-$$axis the ellipsoid spans from $$-a$$ to $$a$$. Along the $$y$$ and $$z-$$axes, the bounds are $$b$$ and $$c$$ respectively.

Because the coordinate axes are the principal axes of the ellipsoid, the off-diagonal components of the matrix will vanish. We will calculate the sum of $$z^2$$ over the ellipsoid first. We have

$$ \sum_i m_i z_i^2 = \int_{\rm Ellipsoid} \rho \, z^2 \, dV = \int_{\rm Ellipsoid} \rho \, z^2 \, dx dy dz. $$

The region of integration is rather complicate in the Cartesian coordinates but this gives us a place to start. We would like to simplify things a bit by defining

$$ x = a u, y = b v, z = c w. $$

In these new coordinates the ellipsoid becomes a sphere of unit radius. We also have to include the Jacobian of the coordinate transformation

$$ \frac{\partial \left ( x, y, z \right )}{\partial \left ( u, v, w \right )} = a b c. $$

Now our integral looks like

$$ \sum_i m_i z_i^2 = \int_{\rm Unit Sphere} \rho \, a b c(c w)^2 \, du dv dw. $$

Let's calculate the density of the ellipsoid in terms of its $$M$$ and volume. The volume of the ellipsoid is

$$ V = \int_{\rm Ellipsoid} \, dx dy dz = \int_{\rm Unit Sphere} a b c \, du dv dw = \frac{4}{3} \pi a b c $$

so we can rewrite the density in the moment of inertia integral

$$ \sum_i m_i z_i^2 = \int_{\rm Unit Sphere} \frac{3 M}{4 \pi a b c} a b c(c w)^2 \, du dv dw = \frac{3 M c^2}{4\pi} \int_{\rm Unit Sphere} w^2 \, du dv dw. $$

The easiest way to integrate over a unit sphere is to use spherical coordinates. Let's do it:

$$ \sum_i m_i z_i^2 = \frac{3 M c^2}{4\pi} \int_0^1 dr \int_0^\pi d\theta \int_0^{2\pi} d\phi \, r^2 \sin\theta \, r^2 \cos^2\theta = \frac{3 M c^2}{4\pi} \frac{2}{5} \pi \int_{-1}^1 d\left (\cos \theta\right ) \cos^2 \theta = \frac{3 M c^2}{4\pi} \frac{4}{15} \pi = \frac{1}{5} M c^2. $$

By symmetry we can calculate the results for the other two coordinates. We have

$$ \sum_i m_i x_i^2 = \frac{1}{5} M a^2, \sum_i m_i y_i^2 = \frac{1}{5} M b^2 $$

and the moment of inertia matrix about the centre of mass have the following non-zero components:

$$ I_{xx} = \frac{M}{5} \left ( b^2 + c^2 \right ), I_{yy} = \frac{M}{5} \left ( a^2 + c^2 \right ), I_{zz} = \frac{M}{5} \left ( a^2 + b^2 \right ). $$

This is an example of reducing a general shape to a symmetric one through a transformation of variables. One important result of this technique is |Routh's rule.

Ellipsoid of Inertia
We saw earlier that if the velocity of the origin vanished, then we could write $$T = \frac{1}{2} \vec L \cdot \vec \omega$$. Now if there are no forces on the body, the kinetic energy of the body is conserved. If there are no forces on the body, there are no torques, so the angular momentum is conserved as well. Does this mean that the angular velocity is constant?

The answer is of course no because the equation only says that the component of the angular velocity along the direction of the angular momentum is constant. The other components can change. For example, if the angular velocity lies along one of the principal axes (eigenvectors of the moment of inertia matrix), then the angular momentum points in the same direction and the angular velocity must stay constant in direction and magnitude.

A usual picture to understand the force-free motion of a rigid body is the ellipsoid of inertia. Let's use the principal axes of the body as a coordinate frame to write the kinetic energy

$$ T = \frac{1}{2} \left (I_{xx} \omega_x^2 + I_{yy} \omega_y^2 + I_{zz} \omega_z^2 \right ). $$

The ellipsoid of inertia is the locus of values of $$\vec \omega$$ that give a kinetic energy of the body.

Ellipsoid of Inertia of an Ellipsoid
Let's calculate the shape of the ellipsoid of inertia of an ellipsoid. We have

$$ T = \frac{1}{2} \frac{M}{5} \left [ \left (b^2 + c^2\right) \omega_x^2 + \left (a^2 + c^2\right) \omega_y^2 + \left (a^2 + b^2\right) \omega_z^2  \right ], $$

so along the $$x$$, $$y$$ and $$z$$ directions, the axes of the ellipsoid of inertia are

$$ \sqrt { \frac{10 T}{M (b^2 + c^2)}}, \sqrt { \frac{10 T}{M (a^2 + c^2)}}, \sqrt { \frac{10 T}{M (a^2 + b^2 )}}. $$

The size of the ellipsoid of inertia increases as $$T$$ and decreases with the mass and size of the actual ellipsoidal body. The shape of the ellipsoid of inertia reflects the shape of the physical ellipsoid. The largest axis of the ellipsoid is also the largest axis of the corresponding ellipsoid of inertia. If two axes of the ellipsoid are the same size, the corresponding axes of the ellipsoid of inertia will be equal as well.

Motion of the Ellipsoid of Inertia
To conserve energy the angular velocity must remain on the ellipsoid of inertia and the orientation of ellipsoid of inertia determines the orientation of the body. We can use the conservation of angular momentum to orient the ellipsoid. If we remember that $$\vec L = \partial T/\partial \vec \omega$$ we see that the normal to the ellipsoid of inertia is the angular momentum; therefore, as the ellipsoid of inertia rotates about the angular velocity, it must remain tangent to a plane perpendicular to the angular momentum, the invariable plane. Furthermore, the angular velocity is restricted to move along the ellipsoid of inertia along curves of constant $$\left | \vec L \right|$$. This curve is called the polhode, and it traces the path of the angular velocity through the body. The angular velocity also traces a path in the invariable plane called the herpolhode. The herpolhode is the path of the angular velocity through space. We can also write that $$2 T=\vec \omega \cdot \vec L$$ so the centre of the ellipsoid of inertia must remain a constant distance above the invariable plane. The height of the centre of the ellipsoid is $$2T/\left |\vec L\right|$$, so when the ellipsoid is down low the angular momentum is high relative to the kinetic energy.

Keeping the ellipsoid of inertia tangent to the invariable plane preserves the direction of the angular momentum. The magnitude of the angular momentum is also conserved. This defines a second ellipsoid through the equation

$$ \left | \vec L \right |^2 = I_{xx}^2 \omega_x^2 + I_{yy}^2 \omega_y^2 + I_{zz}^2 \omega_z^2 $$

so the polhodes are the intersections of the ellipsoid of inertia with the angular momentum ellipsoid. The angular momentum ellipsoid deviates more from a sphere than the ellipsoid of inertia; this gives an idea of where the polhodes lie for small and large angular momenta.

If the body has two principal axes with equal moments of inertia, the polhodes are circles centered on the axis with the unique moment of inertia, and the herpolhodes are circles in the invariable plane. If the unique moment of inertia is larger than the others, the body is called oblate. Otherwise it is prolate -- in analogy to the spheroids with the similar properties. In this case, the free rotation of the body consists of a constant precession of the angular velocity in a circle about the angular momentum in the space frame (the space cone) and about the unique principal axis in the body frame (the body cone). In the oblate case the body cone rolls within the space cone, and in the prolate case, the body cone rolls on the outside of the space cone.

If no two moments of inertia are equal to each other, the polhodes are much more complicated. The polhodes near the axes with the largest and smallest moment of inertia are closed, but those near the intermediate axis are not; consequently, rotation about the intermediate axis is not stable as we shall see when we analyze the free rotation using Euler's equations.

Euler's Equations
So far we have tried to find a graphical description of the motion of an object without any torques. Although "the polhode rolls on the herpolhode without slipping" might paint a nice Victorian picture, it doesn't exploit the mathematical language of physics to which we are now accustomed. We know that the change in the angular momentum is equal to the torques on the body

$$ \frac{d \vec L}{d t} = \vec N. $$

Let's write the angular momentum in terms of the principal axes of the body. Here we will be looking at how the principal axes move so we will use the unit vectors $$\hat 1, \hat 2$$ and $$\hat 3$$ to denote the current direction of the principal axes of the body. We can write

$$ \frac{d \vec L}{dt} = \frac{\partial L_1}{\partial t} \hat 1 + \frac{\partial L_2}{\partial t} \hat 2 +  \frac{\partial L_3}{\partial t} \hat 3 + L_1 \frac{\partial \hat 1}{\partial t} + L_2 \frac{\partial \hat 2}{\partial t} + L_3 \frac{\partial \hat 3}{\partial t} $$

$$ = \frac{\partial L_1}{\partial t} \hat 1 + \frac{\partial L_2}{\partial t} \hat 2 +  \frac{\partial L_3}{\partial t} \hat 3 + L_1 \vec \omega \times \hat 1  + L_2 \vec \omega \times \hat 2  + L_3 \vec \omega \times \hat 3 $$

Let's define the angular momentum relative to the body axes as

$$ \vec L_b = \left [ \begin{matrix} L_1 \\ L_2 \\ L_3 \end{matrix} \right ] = \left [ \begin{matrix} \omega_1 I_1 \\ \omega_2 I_2 \\ \omega_3 I_3 \end{matrix} \right ] $$

to get

$$ \frac{d \vec L}{dt} = \frac{d \vec L_b}{d t} + \vec \omega \times \vec L_b = \vec N $$

where the torque and angular velocity are written relative to the body axes. One can imagine using Euler's equations would be quite cumbersome if one had to include the torques because one would always have to transform the torques from the inertia frame to the body frame. Let's write out these equations component by component

$$ I_1 \dot \omega_1 - \omega_2 \omega_3 \left (I_2 - I_3 \right) = N_1 $$

$$ I_2 \dot \omega_2- \omega_3 \omega_1 \left (I_3 - I_1 \right) = N_2 $$

$$ I_3 \dot \omega_3 - \omega_1 \omega_2 \left (I_1 - I_2 \right) = N_3 $$

These equations are most powerful when there are no torques, so the right-hand sides are zero. In this case we can see immediately that if any two moments of inertia differ than the angular velocity must lie along one of the principal axes to remain constant.

If two of the moments of inertia are equal to each other (let's take $$I_1=I_2$$), we can solve the torque-free Euler's equations exactly. We have

$$ I_1 \dot \omega_1 = \left ( I_1 - I_3 \right ) \omega_3 \omega_2 $$

$$ I_2 \dot \omega_2 = -\left ( I_1 - I_3 \right ) \omega_3 \omega_1 $$

$$ I_3 \dot \omega_3 = 0 $$

We can solve this with

$$ \dot \omega_1 = - \Omega \omega_2, \dot \omega_2 = \Omega \omega_1 $$

with

$$ \Omega = \frac{I_3 - I_1}{I_1} \omega_3, $$

yielding the solution

$$\omega_1 = A \cos \Omega t,\, \omega_2 = A \sin \Omega t.$$

This result coincides with the graphical picture that the polhodes of a symmetric body are circles centered on the axis with the unique moment of inertia.

Chandler Wobble
We can calculate the expected rate for the Earth to precess due to its oblatness. Let's estimate the ratio of the moments of inertia in terms of the radii of the Earth at the equator and at the poles, we have

$$ \frac{I_3 - I_1}{I_1} = \frac{2 R_{\rm eq}^2 - \left ( R_{\rm eq}^2 + R_{\rm pole}^2 \right )}{R_{\rm eq}^2 + R_{\rm pole}^2} $$

$$ = \frac{\left ( R_{\rm eq} - R_{\rm pole} \right )\left ( R_{\rm eq} + R_{\rm pole} \right )}{R_{\rm eq}^2 + R_{\rm pole}^2} \approx \frac{R_{\rm eq} - R_{\rm pole}}{R_{\rm eq}}. $$

where in the last step we have assume that the difference between the polar and equatorial radius is small. We have $$R_{\rm eq} \approx 6378 {\rm km}, R_{\rm pole} \approx 6356 {\rm km}$$, so the ratio is about $$1/290$$. What is $$\omega_3$$ for the Earth? It is $$2\pi/1 {\rm day}$$, so the period of the free precession of the Earth should be about 290 days. The period is somewhat longer about 433 days Chandler_wobble. You can check out a plot of the wobble at the Paris Observatory. Why the Earth wobbles is a bit of a puzzle since because the Earth is not a solid body, without a driving force it would have damped long ago. Since the period of the wobble is similar to a year, seasonal changes are thought to be to blame.

I first learned about the wobble while spending a summer at the Pulkovo Astronomical Observatory near St. Petersburg. I saw a plot of the location of the Earth's pole on the surface of the Earth -- it moves in a irregular circular shape with a radius of 3 to 15 meters. The wobble can be measured by accurate measurements of the locations of stars relative to landmarks on the Earth. I asked my hosts why did they need to keep track the wobble so accurately. "To aim our missiles at your cities -- of course" came the reply.

Triaxial Body
Unfortunately, we can't find such a straightforward to the solution of the motion of a triaxial body. In the symmetric case, one of the components of the angular velocity was constant with time. This is not the case for a triaxial body. We have the following Euler's equations

$$ I_1 \dot \omega_1 = \left (I_2 -I_3\right ) \omega_3 \omega_2 $$

$$ I_2 \dot \omega_2 = \left (I_3 -I_1\right ) \omega_3 \omega_1 $$

$$ I_3 \dot \omega_3 = \left (I_1 -I_2\right ) \omega_1 \omega_2. $$

Let's take $$I_3 > I_2 > I_1$$ and define the following three positive ratios of moments of inertia

$$ A = \frac{I_3-I_2}{I_1}, B = \frac{I_3-I_1}{I_2}, C = \frac{I_2-I_1}{I_3} $$

giving the following equations

$$ \dot \omega_1 = - A \omega_3 \omega_2 $$

$$ \dot \omega_2 = + B \omega_3 \omega_1 $$

$$ \dot \omega_3 = - C \omega_1 \omega_2 $$

Major Axis
Let's assume that the rotation is nearly about the major axis, so $$\omega_1 \gg\ \omega_2, \omega_3$$. We will only include the latter components of the angular velocity to first order, so we have

$$ \dot \omega_1 \approx 0, \dot \omega_2 = B \omega_3 \omega_1, \dot \omega_3 = - C \omega_2 \omega_1. $$

If we define $$\Omega = \sqrt{BC} \omega_1$$ we have the following solution for $$\omega_2$$ and $$\omega_3$$,

$$ \omega_3 = \sqrt{\frac{B}{C}} K \cos \Omega t, \omega_2 = K \sin \Omega t, $$

so the angular velocity travels in an ellipse centered on the major axis.

Minor Axis
Let's assume that the rotation is nearly about the minor axis, so $$\omega_3 \gg\ \omega_1, \omega_2$$. We will only include the latter components of the angular velocity to first order, so we have

$$ \dot \omega_3 \approx 0, \dot \omega_1 = -A \omega_3 \omega_2, \dot \omega_2 = B \omega_3 \omega_1. $$

If we define $$\Omega = \sqrt{AB} \omega_3$$ we have the following solution for $$\omega_1$$ and $$\omega_2$$,

$$ \omega_1 = \sqrt{\frac{A}{B}} K \cos \Omega t, \omega_2 = K \sin \Omega t, $$

so the angular velocity travels in an ellipse centered on the minor axis.

Intermediate Axis
Let's assume that the rotation is nearly about the intermediate axis, so $$\omega_2 \gg\ \omega_1, \omega_3$$. We will only include the latter components of the angular velocity to first order, so we have

$$ \dot \omega_2 \approx 0, \dot \omega_1 = -A \omega_3 \omega_2, \dot \omega_3 = - C \omega_1 \omega_2. $$

If we define $$\alpha = \sqrt{AC} \omega_2$$ we have the following solution for $$\omega_1$$ and $$\omega_3$$,

$$ \omega_1 = K_1 e^{\alpha t}, \omega_3 = K_3 e^{\alpha t},\, $$

so the angular velocity diverges exponetially away from the intermediate axis.

Lagrangian Treatment of Rigid-Body Motion
Although it is possible to include the effects of torques on the motion of a rigid body within the framework of Euler's equations, it is rather cumbersome. Here we will develop the Lagrangian treatment of a top -- a symmetric rigid body with a torque from the Earth's gravity.

Euler Angles
In a Lagrangian treatment it is necessary to find a set of independent coordinates to describe the position of the system at any time. For rigid bodies a convenient set of such coordinates are called the Euler angles. There are several different conventions for the Euler angles. The convention used here makes the analysis of the top simpler.

To get from one set of coordinates to the other, we have to do a series of three rotations. We imagine that the primed coordinates are the body coordinates and the unprimed coordinates are the inertia coordinates; therefore, it is straightforward to write the kinetic energy in the primed coordinates and the potential energy in the unprimed coordinates. Here are the three steps in words and in terms of a rotation matrix

$$D=\left [ \begin{matrix} \cos\phi & \sin\phi & 0 \\ -\sin\phi & \cos\phi & 0 \\ 0        & 0        & 1 \end{matrix} \right ] $$ $$C=\left [ \begin{matrix} 1 & 0 & 0 \\ 0 & \cos\theta & \sin\theta \\ 0 & -\sin\theta & \cos\theta \end{matrix} \right ] $$ $$B=\left [ \begin{matrix} \cos\psi & \sin\psi & 0 \\ -\sin\psi & \cos\psi & 0 \\ 0        & 0        & 1 \end{matrix} \right ] $$
 * 1) Rotate by an angle $$\phi$$ about the initial $$z-$$axis.  The final position of the $$x-$$axis is called the line of nodes.
 * 1) Rotate about the line of nodes by an angle $$\theta$$.   This changes the direction of the $$z-$$axis and pitches the $$y-$$axis out of the initial $$x-y-$$plane.
 * 1) Finally rotate about the new $$z-$$axis by an angle $$\psi$$.  This takes the $$x-$$axis out of the initial $$x-y-$$plane.

Euler proved that any rotation can be decomposed into these three rotations. One can see this mathematically by constructing the rotation matrix for these three successive transformations ($$A=BCD$$) and verifying that it is the most general rotation matrix.

The Kinetic Energy
To calculate the kinetic energy of the body, we have to find the angular velocity in terms of the coordinates, $$\phi, \theta$$ and $$\psi$$ and their time derivatives. In any coordinate system, the angular velocity is directed along the axis of rotation. We can construct the total angular velocity vector using the following information:

 * 1) Changes in $$\phi$$ produce rotations about the $$z-$$axis (inertial frame).
 * 2) Changes in $$\psi$$ produce rotations about the $$z'-$$axis (body frame).
 * 3) Changes in $$\theta$$ produce rotations about the line of nodes.

We can write the angular velocity relative to the inertial frame to get

$$ \vec \omega = \left [ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right ] \dot \phi + \left [ \begin{matrix} \cos \phi \\ \sin \phi \\ 0 \end{matrix} \right ] \dot \theta + \left [ \begin{matrix} \sin\theta \sin \phi \\ -\sin\theta \cos\phi \\ \cos\theta \end{matrix} \right ] \dot \psi $$

or relative to the body frame

$$ \vec \omega = \left [ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right ] \dot \psi + \left [ \begin{matrix} \cos \psi \\ -\sin \psi \\ 0 \end{matrix} \right ] \dot \theta + \left [ \begin{matrix} \sin\theta \sin \psi \\ \sin\theta \cos\psi \\ \cos\theta \end{matrix} \right ] \dot \phi. $$

The first form is useful to visualize what the body is doing, while the second form is good for calculating the kinetic energy because we have used the primed coordinate system to characterize the location of the principal axes of the body.

For simplicity let's specialize to symmetric bodies with $$I_{x'} = I_{y'}$$, so we have

$$ T = \frac{1}{2} I_{x'} \left (\omega_{x'}^2 + \omega_{y'}^2 \right ) + \frac{1}{2} I_{z'} \omega_{z'}^2. $$

Because the primed frame is fixed to the body, the moments of inertia in the kinetic energy are constant with time. Let's first calculate

$$ \omega_{x'}^2 + \omega_{y'}^2 = \left ( \cos\psi \dot \theta + \sin \psi \sin \theta \dot \phi \right )^2 + \left ( -\sin\psi \dot \theta + \cos\psi \sin\theta \dot \phi \right )^2 $$

$$ = \cos^2\psi \dot \theta^2 + \sin^2 \psi \sin^2 \theta \dot \phi^2 + 2 \sin\psi \cos\psi \sin\theta \dot \theta \dot \phi + \sin^2 \psi \dot\theta^2 + \cos^2\psi \sin^2\theta \dot \phi^2 - 2 \sin\psi \cos\psi \sin\theta \dot\theta \dot\phi $$

$$ = \dot\theta^2 + \sin^2\theta \dot \phi^2. $$

This simplication only obtains for a symmetric body. The $$z'-$$component is

$$ \omega_{z'}^2 = \left ( \dot \psi + \dot \phi \cos\theta \right )^2. $$

Combining these results yields the kinetic energy

$$ T = \frac{I_{x'}}{2} \left ( \dot \theta^2 + \dot \phi^2 \sin^2\theta \right ) + \frac{I_{z'}}{2} \left ( \dot \psi + \dot \phi \cos\theta \right )^2 $$

The Potential Energy
The potential energy is pretty simple using the Euler angles. If $$l$$ is the distance between the pivot point and the center of mass of the top, we have

$$ V = M g l \cos\theta $$

The Lagrangian
The Lagrangian is the difference between the kinetic and the potential energy, so we have

$$ L = T - V = \frac{I_{x'}}{2} \left ( \dot \theta^2 + \dot \phi^2 \sin^2\theta \right ) + \frac{I_{z'}}{2} \left ( \dot \psi + \dot \phi \cos\theta \right )^2 - M g l \cos\theta. $$

We notice immediately that $$\psi$$ and $$\phi$$ don't appear in the Lagrangian. The system looks the same as in turns around the $$z-$$ and $$z'-$$axes. This means that we have two conserved quantities, reducing the three dimensional problem to a single dimension (we will choose this to be $$\theta$$). Because the Lagrangian does not depend on time, the Hamiltonian is conserved as well so we can use the techniques outlined in Linear Motion to understand the top.

Integrals of the Motion
To calculate the conserved momenta we have to calculate the partial derivatives of the Lagrangian with respect to the generalized velocities. We have

$$ p_\psi = \frac{\partial L}{\partial \dot \psi} = I_{z'} \left ( \dot \psi + \dot \phi \cos\theta \right ) = I_{z'} \omega_{z'} \equiv I_{x'} a $$

where the last step defines the conserved quantity $$a$$ that has units of angular velocity. Similarly for the coordinate $$\phi$$, we have

$$ p_\phi = \frac{\partial L}{\partial \dot \phi} = \left ( I_{x'} \sin^2 \theta + I_{z'} \cos^2\theta \right ) \dot \phi + I_{z'} \cos\theta \dot \psi \equiv I_{x'} b. $$

Finally we have the Hamiltonian that is also conserved,

$$ H = T + V = \frac{I_{x'}}{2} \left ( \dot \theta^2 + \dot \phi^2 \sin^2\theta \right ) + \frac{I_{z'}}{2} \left ( \dot \psi + \dot \phi \cos\theta \right )^2 + M g l \cos\theta. $$

As we mentioned earlier, these three integrals are sufficient to solve for the motion. We can use the two conserved momenta to solve for the generalized velocities, $$\dot \psi$$ and $$\dot \phi$$, in terms of the conserved quantities $$a$$ and $$b$$ and the value of the third coordinate $$\theta$$. We have

$$ \dot \phi = \frac{b - a\cos\theta}{\sin^2\theta}, $$

$$ \dot \psi = \frac{I_{x'} a}{I_{z'}} - \cos\theta \frac{b - a\cos\theta}{\sin^2\theta}. $$

Conservation of Energy
Since both the Hamiltonian and the angular velocity along the $$z'-$$axis are conserved, let's define the conserved quantity

$$ E' = E - \frac{I_{z'} \omega_{z'}^2}{2} = \frac{I_{x'} \dot \theta^2}{2} + \frac{I_{x'}}{2} \frac{\left( b -a \cos\theta \right )^2}{\sin^2\theta} + M g l \cos\theta $$

which we can rewrite as

$$ E' = \frac{I_{x'} \dot \theta^2}{2} + V'\left (\theta\right ) $$

where

$$ V'(\theta) = \frac{I_{x'}}{2} \frac{\left( b -a \cos\theta \right )^2}{\sin^2\theta} + M g l \cos\theta. $$

We can make further progress by looking at $$u=\cos\theta$$ which will simplify the potential. Let's write

$$ \dot u^2 = \sin^2 \theta \dot \theta^2 = \left ( 1 - u^2 \right ) \dot \theta^2 $$

so we can write $$E'$$ in terms of $$u$$ and its time derivative

$$ E' = \frac{I_{x'}}{2} \frac{\dot u^2}{1-u^2} + \frac{I_{x'}}{2} \frac{\left( b -a u \right )^2}{1-u^2} + M g l u. $$

What remains is to solve for $$\dot u^2$$ to get the boundaries of the motion

$$ \dot u^2 = \left (E' - M g l u \right ) \frac{2}{I_{x'}} \left ( 1 - u^2 \right ) - \left (b - a u\right )^2 $$

The Motion of a Symmetric Top
The equation for $$\dot u^2$$ is cubic so it will generally have three roots (designated at $$u_{1,2,3}$$ -- it must have at least one root because for large positive values of $$u$$ $$\dot u^2$$ is positive and for large negative values of $$u$$ $$\dot u^2$$ is negative.  As the top moves from straight up to straight down, $$u$$ ranges from 1 to -1, so only the roots within this range are important.  Furthermore, the equation for $$\dot \phi$$ has a root at $$u_0=b/a$$.

The figure shows a typical curve for $$\dot u^2$$ as a function of $$\theta$$. The motion is bounded between $$u=0.5$$ and $$u=0.8$$. Depending on the value of $$b/a$$, the motion in the $$\phi$$-direction can be monotonic or not. If $$b/a=0.8$$, the motion will have a cusp at the upper end of the range.

The figures to the right show the motion of the $$z'-$$axis as a function of time for several values of $$b/a$$ with the curve for $$\dot u^2$$ above. The bounds of the motion in all three cases is from $$u=0.5$$ to $$u=0.8$$. How can we set up a top to execute these three classes of motion? If we release the axis of the top from a stationary position, the axis will first drop (nutation) because of gravity. As it drops the angular momentum about the vertical axis must be conserved so the tip of the top starts moving sideways as well (precession) -- yielding the motion in the middle figure. On the other hand if we release the tip of the top with some motion in the direction of the precession, we will get the lower figure. Finally to get the rather freaky looking motion in the top figure, we release the top with some motion opposing the direction of the precession.

Specifically let's examine the case where we release the top from a stationary position, so $$u_0=b/a=u_2$$ where $$u_2$$ is the middle root of the $$\dot u^2$$ curve. Initially we have

$$ E' = M g l \cos \theta = M g l \frac{b}{a} $$

so we can write

$$ \dot u^2 = \left (\frac{b}{a} - u \right ) M g l \frac{2}{I_{x'}} \left ( 1 - u^2 \right ) - \left ( b - a u \right)^2 $$

$$ = \left (u_0- u \right ) a^2 \left [ \frac{2 M g l }{I_{x'} a^2} \left ( 1 - u^2 \right ) -\left ( u_0 - u \right ) \right ] $$

The Fast Top
Unfortunately, the motion of the top in general cannot be solved in closed form, but we can make some progress in understanding the motion of the so-called fast top. If we look at the expression above, we have the ratio $$ 2M g l/(I_{x'} a^2). $$ The numerator is the potential energy of the top while the denominator is the related to the kinetic energy of the top about its spin axis. The condition for a fast top is

$$ I_{x'} a^2 = I_{z'} \omega_{z'}^2 \frac{I_{z'}}{I_{x'}} \gg\ 2 M g l. $$

In a fast top, the kinetic energy is much larger than the potential energy; consequently, the extent of the nutation ($$u_0-u$$) is small, so

$$ 1 - u^2 \approx 1 - u_0^2 = \sin^2 \theta_0 $$

so we can write

$$ \dot u^2 \approx \left (u_0- u \right ) a^2 \left [ \frac{2 M g l }{I_{x'} a^2} \sin^2 \theta_0 -\left ( u_0 - u \right ) \right ]. $$

The extent of the nutation is

$$ u_0 - u_1 \approx \frac{2 M g l }{I_{x'} a^2} \sin^2 \theta_0 = \frac{I_{x'}}{I_{z'}} \frac{2 M g l }{I_{z'} \omega_{z'}^2} \sin^2 \theta_0 $$

and we can rewrite the equation for $$\dot u^2$$ as

$$ \dot u^2 = \left (u_0 - u \right ) \left (u - u_1\right ) a^2. $$

Let's take the time derivative of both sides to yield

$$ 2 \dot u \ddot u = a^2 \left [ \left ( u_0 - u\right ) \dot u - \left (u - u_1\right ) \dot u \right ] $$

$$ \ddot u = a^2 \left ( \frac{u_0+u_1}{2} - u \right ). $$

Let's define $$y=u-(u_0+u_1)/2$$ so we have

$$ \ddot y = -a^2 y $$

with the solution

$$ y = \frac{ u_1 - u_0 }{2} \cos a t $$ and $$ u = \frac{u_1+u_0}{2} + \frac{ u_1 - u_0 }{2} \cos a t. $$

The frequency of the nutation is

$$ a = \frac{I_{z'}}{I_{x'}} \omega_{z'} $$

and

$$ \dot \phi = \frac{a \left (u_0 - u\right )}{\sin^2 \theta} = \frac{M g l}{I_{z'} \omega_{z'}} \left ( 1 - \cos a t \right ) $$

and

$$ \phi = \phi_0 + \frac{M g l}{I_{z'} \omega {z'}} \left ( t - \frac{1}{a}\sin a t \right ). $$

Sleeping Top
A sleeping top is a top that is set in motion with its spin axis vertical and the axis remains vertical. When the top is vertical it is impossible to distinguish changes in $$\phi$$ from changes in $$\psi$$ so the relationship between $$\dot \phi$$ and $$a$$ and $$b$$ isn't very useful. Let's look at the equation for $$\dot u^2$$. We have

$$ \dot u^2 = \left ( E' - M g l u \right ) \frac{2}{I_{x'}} \left (1 - u^2 \right ) - \left ( b - a u \right )^2. $$

When the top points upwards, $$u=1$$. We let go of the top with $$\dot u=0$$ so we have

$$ 0 = \left ( E' - M g l \right ) \frac{2}{I_{x'}} \left (1 - 1^2 \right ) - \left ( b - a  \right )^2. $$

so

$$b=a$$ and

$$ \dot u^2 = \left ( 1 - u \right ) a^2 \left [ \left ( E' - M g l u \right ) \frac{2}{I_{x'} a^2} \left (1 + u \right ) - \left ( 1 - u \right ) \right ]. $$

Using the definition of $$E'$$ we find

$$ E' = E - \frac{I_{z'} \omega_{z'}^2}{2} = \frac{I_{x'} \dot \theta^2}{2} + \frac{I_{x'}}{2} \frac{\left( b -a \cos\theta \right )^2}{\sin^2\theta} + M g l \cos\theta = M g l $$

and

$$ \dot u^2 = \left ( 1 - u \right )^2 a^2 \left [ \frac{2 M g l}{I_{x'} a^2} \left (1 + u \right ) - 1 \right ] =  \left ( 1 - u \right )^2 a^2 \frac{2 M g l}{I_{x'} a^2 } \left [ u - \left (\frac{I_{x'} a^2}{2 M g l} - 1 \right ) \right ], $$

so there is a double root at $$u=1$$ and another root at

$$ u = \frac{I_{x'} a^2}{2 M g l} - 1. $$

If this root is at $$u>1$$, then the top will sleep, so

$$ I_{x'} a^2 > 4 M g l $$

is the condition for a sleeping top.


text:
User:Jim MacDougald/sandbox

Ghia 450/SS

The Ghia 450/SS was built by Carrozia Ghia of Turin, Italy in 1966 and 1967. It was based on the concept car designed by Ghia's design chief, Sergio Sartorelli, and was first introduced at the Automobile Show in Turin in 1963. It was originally intended as a possible replacement for the Ghia-designed Fiat 2300 S then being produced under contract by Ghia's sister-company, Officini Stampaggi Industriali (OSI) for Fiat. Fiat did not choose to put the car in production and Ghia subsequently presented an improved version of the 230S at the 1964 Turin show, this time seeking a manufacturer to provide engines, drivetrains, and distribution for the Ghia/OSI production of the 230S. A photo of the 230S concept car was featured on the cover of Road & Track magazine in March 1965. Burt Sugarman, a wealthy automobile aficionado in Beverly Hills, saw the cover photo and flew to Turin to order a 230S for himself. As the result of discovering during his his visit that Ghia did not have a manufacturing partner to provide engines and drivetrains, Sugarman contacted Bob Anderson of the Plymouth Division of Chrysler Corporation and convinced him to provide engines, drivetrains, and factory warranty on a new derivation of the 230S, to be known as the 450/SS, the "450" derived from the approximate the 4500cc displacement of the 235 hp Plymouth "Commando S" small-block V-8 to be utilized in the automobile.

title: User:Jim MacDougald/sandbox


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New data reported. --COIBot (talk) 13:26, 19 September 2017 (UTC)


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George Corrin

George Corrin (1922–2015) was an American scenic and graphic designer, known for creating the sets for presidential debates between Richard Nixon and John F. Kennedy.

Early life and education
Corrin was born in Long Branch, New Jersey and first started designing sets in the 1940s. His first set was for the Studio Players of Essex County, New Jersey in 1942.

Corrin received a BFA from Carnegie Mellon University, where he was the first African-American student at the School of Drama. Corrin was nearly denied admission by the dean, who extended the offer after an outpouring of recommendation letters.

Corrin served for three years during World War II. He spent some time in the Special Services, where he worked on productions for troops in the South Pacific.

Corrin went on to receive an MFA from the Yale University School of Drama.

Corrin is the uncle of actor Michelle Shay.

Career
Corrin designed sets for theater and television. In the 1950s he moved to New York City to work for ABC Television. At ABC-TV, he designed the Election Night Studio in 1964 and 1966. He also designed sets for shows including Voice of Firestone, Paul Whiteman Show, The Nurses, and Peter Jennings With The News.

After ABC, he worked for Reeves Teletape, where he managed the construction of the set for Sesame Street.

In the 1970s he worked independently for clients including AT&T, the Shubert Organization, the Insurance Information Institute, and Steelcase. He continued to design sets for Off-Broadway shows. He was art director for the 1979 film, The Hitter. Corrin also managed restoration of the Shubert Theater in 1989.

title: George Corrin


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Isabelle Demongeot

Isabelle Demongeot (born 18 September 1966) is a former professional tennis player from France, who turned professional on 1 May 1983. She lived in Saint-Tropez in the French Riviera in the early stages of her career and later settled nearby in Gassin.

Demongeot won her only WTA Tour singles title in Purchase in New York state in 1991. She achieved her best Grand Slam singles performance by reaching the fourth round of Wimbledon in 1986. She won a total of nine WTA Tour doubles titles. Her best Grand Slam doubles performance was reaching the French Open quarter-finals with Nathalie Tauziat of France in 1987 and 1992. She also represented her country from 1985 to 1993 in the Federation/Fed Cup and in the 1988 and 1992 Olympic Games. She retired from the WTA Tour in 1996.

In her 2007 book, Service volé, she accused her coach, Régis de Camaret, of rape and sexual abuse. She would later be joined by several former pupils who made similar allegations. Although her own case was time-barred, she acted as a witness in the 2012 trial of de Camaret for offences against two young players. The 70-year-old coach was sentenced to eight years in prison, increased to ten years on appeal in 2014.

title: Isabelle Demongeot


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Mergers and Acquisitions/Introduction

Welcome to Introduction to Mergers and Acquisitions, M&A 1000. This course is being designed to introduce the basic skills needed in the field of mergers and acquisitions.

Course Outline

 * Course Outline Course Outline & Syllabus

Sessions

 * Session 1: Basic Principles and Concepts
 * Session 1 will focus on.

title: Mergers and Acquisitions/Introduction


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Louis De Jaegher

Louis de Jaegher (30 July 1908 – 22 February 1973) was a Belgian colonial administrator. He was governor of Kasaï Province in the Belgian Congo from 1958 to 1960, just before the country became independent as the Republic of the Congo (Léopoldville).

Early years (1908–1930)
Louis Auguste Jules Frans De Jaegher was born in Nieuwpoort, Belgium, on 30 July 1908. His parents were François Joseph Jules Marie de Jaegher and Juliette Louise Marie Paul. He married Cécile Dutrieu. He studied the humanities at the Royal Atheneum in Antwerp. In October 1926 he was admitted to the Colonial University of Belgium, graduating in July 1930.

Territorial administrator (1930–1946)
On 1 September 1930 De Jaegher was appointed territorial administrator 2nd class. He reached Boma on 26 September 1930. He was assigned for his internship to Baseka Bongwalanga Territory in Lulonga District, Équateur Province. In August 1931 he was appointed head of the Ekota Bosaka Territory in Équateur District. In June 1932 he transferred to the Bakutu Bosaka Territory in Tshuapa District. He left Boma on leave on 6 October 1933, and returned on 23 March 1934. His second term was spent in the territories of Ekota Bakutu and Boende in Tshuapa District.

De Jaegher left on leave from 23 April to 8 October 1937, then returned to Boende Territory in Tshuapa District. In addition to his functions as territorial administrator he was deputy judge of the district court and auxiliary judge at the Court of First Instance in Coquilhatville. During World War II (1939–1945) Belgium was invaded on 10 May 1940. Between then and 20 April 1947 De Jaegher took only three short leaves totalling nine months, two in South Africa and one in Kivu. On 1 July 1942 De Jaegher was promoted to senior territorial administrator. He was assigned to the Congo-Ubangi District on 12 June 1944. He was made administrator of Bumba Territory.

District commissioner (1946–1955)
On 12 April 1946 De Jaegher turned to Tshuapa as intern for the grade of district commissioner, and on 20 March 1947 his internship ended and he became district commissioner of Tshuapa. On 20 April 1947 he flew to Belgium for leave. He married on 12 July 1947, and left for the Congo with his wife on 17 January 1948. He was temporarily appointed assistant to the commissioner of the Congo-Ubangi District, then on 15 April moved to Equateur District to assume his duties in Coquilhatville, where his daughter Suzanne was born. De Jaegher left with his family for Antwerp on 23 October 1951, returning on 8 November 1951. This term he was assigned to the Tanganika and Lualaba districts of Katanga Province. On 1 July 1951 he was confirmed as District Commissioner, and on 15 March 1952 was assigned to the Sankuru District in Kasai Province. He went on leave on 13 February 1955.

Provincial commissioner / governor (1955–1960)
On 1 July 1955 De Jaegher was appointed provincial commissioner, and reached Luluabourg by air plane on 26 September 1955 with his family. On 14 January 1958 he left for Belgium on leave, returning to Luluabourg on 4 June 1958. The governor, Antoine Lamborelle, retired and De Jaegher was appointed governor of the Province of Kasai on 1 July 1958. On 20 June 1959 the journal KASAI published a letter from chief Kalamba Mangole Sylvestre and Barthélemy Mukenge, president of the Lulua–Frères association. It called on the Belgian government to unite the Lulua in a single kingdom under one chief, and to recognise the property rights of land belonging to the Lulua. It also stated that the Baluba should recognize Lulua rights and submit to Lulua customary authority, or else return to their original lands.

At De Jaegher's request the assistant district commissioner M.A. Dequenne, the first burgomaster M. Hentgen and the Provincial Director of Indigenous Affairs met on 29 June to discuss the Lulua-Baluba conflict and the Lulua claims. A 9-page draft confidential report was completed on 8 July. It was stolen that evening, communicated to activists of the Muluba Solidarity Movement, then put back where it had been found. The report caused outrage among the Baluba. On 9 July a three page letter from Evariste Kalonji and Justin Kasanda violently protesting "the scandalous attitude of the Belgian Colonial Administration" was sent to the presidents of the Belgian Chamber and Senate, with copies to King Baudouin, the Minister of Congo, the Governor General in Kalina and the Governor of the Province of Kasai. On 7 August De Jaegher received a delegation of leaders from Léopoldville, consisting of Patrice Lumumba (Mouvement National Congolais), Bertin-Onésime Tumba (Interfédérale) and Barthélemy Mujanayi (Mouvement solidaire Muluba).

Last years
De Jaegher held office as governor until going on leave on 17 February 1960. He was replaced by L. Henrotaux as acting governor. He died in Ixelles on 22 February 1973. He had been made a member of the Order of Leopold II on 18 April 1949, and an officer of the order on 9 May 1950.

title: Louis De Jaegher