TITLE: Generalizing "variation of parameters"
QUESTION [11 upvotes]: I'm stuck on generalizing an ODE formula and could use your help!
One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here $z(t)\in\mathbb{R}^n$, $A\in{\mathbb R}^{n\times n}$) into formulas for nonlinear problems.  In particular, to solve $y'=Ay+G[y]$ for some  $G:\mathbb{R}^n\rightarrow\mathbb{R}^n$, we can write
$$y(t)=e^{At}y_0+\int_0^t e^{A(t-\tau)}G[y(\tau)]\,d\tau.$$
The integral compensates between the closed-form solution of the linear ODE and the solution of the ODE including $G[\cdot]$.
Suppose instead that we wish to solve $y'=F[y]+G[y]$ for two nonlinear functions $F,G:\mathbb{R}^n\rightarrow\mathbb{R}^n.$  Furthermore, let's say we know how to solve $z'=F[z]$ in closed form via some $\Phi(z,t)$ so that $z(t)=\Phi(z_0,t).$  Is there an analogous formula to variation of parameters in this case?  E.g. something of the form:
$$y(t)=\Phi(y_0,t)+\int_0^t\left[\textrm{something involving $G$}\right]\,d\tau$$
PS:  If it helps, we can assume both $F$ and $G$ come from a Hamiltonian dynamics problem.  So, $n$ is even and contains both velocity and momentum variables, $\Phi_t$ is inverted by $\Phi_{-t}$, $\Phi_t$ is area-preserving, and so on.

REPLY [13 votes]: Yes, this is called the nonlinear variation of constants formula due to Alekseev: “An estimate for the perturbations of the solutions of ordinary differential equations”, in: Vestnik Moskov. Univ. Ser. I Mat. Meh. 2 (1961), pp. 28–36. I don't think that that article is available in English.
It can also be found in the book by V. Lakshmikantham and S. Leela "Differential and integral inequalities: Theory and applications" Vol. I: Ordinary differential equations.Mathematics in Science and Engineering, Vol. 55-I. New York: Academic Press, 1969, pp. ix+390.
The formula is
$$
\tilde{\Phi}(y_0,t) = \Phi(y_0,t) + \int_0^t D\Phi(\tilde{\Phi}(y_0,\tau),t-\tau) G(\tilde{\Phi}(y_0,\tau)) \;d\tau
$$
where $\tilde{\Phi}$ denotes the flow of $F+G$.
See also Appendix E in my book for this, a proof, and a few more details.