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.



Generalizing "variation of parameters"
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](http://www.jaapeldering.nl/includes/get.php?file=NHIM-noncompact-book.pdf) for this, a proof, and a few more details.


