# Generalizing "variation of parameters"

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.

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 $F+G$.
• Incidentally, to make sure I understand, $D\Phi$ here is the Jacobian of $\Phi$ with $t$ fixed, right? [That is, only differentiation in $y$?] Sep 15, 2014 at 18:10
• @Justin: indeed, or in geometric terms the tangent map of the diffeomorphism $\Phi(\cdot,t)$. Sep 15, 2014 at 21:35