# Variational Equation in Perturbation Theory Chapter of Arnold *Geometric Methods in ODE*

In the preface to Chapter 4 Perturbation Theory" in Arnold's book Geometric Methods in the Theory of Ordinary Differential Equations available for preview here

https://www.springer.com/gp/book/9780387966496

he writes:

If the size of the perturbation is characterized by a small parameter $\varepsilon$, then the effect of perturbations over time of order 1 leads to a change of order $\varepsilon$ of the solution. This change can be calculated approximately by solving a variational equation along the unperturbed solution."

He then goes on to discuss the asymptotic methods necessary to discuss the validity of approximation at long times, but I am left wondering:

Question: What is the variational equation along the unperturbed solution'' referred to here?

I would guess that we are finding for finite time the next-order correction term in $\varepsilon$ to the leading order behavior (given by the solution of the unforced equation) but I am not sure about this and also not sure how this leads to a variational problem. Any help would be greatly appreciated!!

• This is a name given to the (systems of) linear ODEs satisfied by the derivative of the solution operator with respect to initial conditions, parameters, etc. For example, if $\varphi(\cdot)$ is a solution of a scalar ODE $x'=f(t,x)$ then the variational equation is $y'=\frac{\partial f}{\partial x}(t,\varphi(t))y$. A small piece of advice: One should have a good command of the basic theory of ODEs before endeavoring to read V. I. Arnol'd's Geometric Methods .... – user539887 May 28 '18 at 7:54
• thank you - I was confused because I expected the name "variational equation" to mean that the equation will come from the variational principle as Euler-Lagrange equations for some functional. Does this interpretation hold? and can you recommend a reference besides encyclopediaofmath.org/index.php/Variational_equations – Swallow Tail May 28 '18 at 19:03
• Regarding the first question: No, it has nothing to do with the calculus of variations (and nothing to do with the "variation of constants" either). Perhaps there were some historical links, but I am not aware of them. – user539887 May 28 '18 at 19:57
• Now to the second: I found it in J. K. Hale's Ordinary Differential Equations, pp. 21-24 (but it is not mentioned in the index). – user539887 May 28 '18 at 20:06
• Ok great thank you for settling my first question, that was my main confusion – Swallow Tail May 28 '18 at 20:19