# How do you find conserved quantities for linear second order ODEs?

I have a differential equation of the form

$\frac{d^2 y}{dt^2} + f(t) \frac{dy}{dt} + g(t) y = 0$

where $f$ and $g$ are known functions of time.

Is there a systematic (or otherwise) way of finding the conserved quantities, if there are any?

I've been trying to google this topic, but haven't had much success yet. Perhaps someone with more mathematical knowledge than myself (i.e. almost everyone) can point me in the right direction. Even phrases to search for would help.

## 2 Answers

+1 to Marek's answer. I will provide you with some references.

The canonical reference is without much doubt

I bought this book a few years ago, but haven't found much time to study it thoroughly. But if you know modern differential geometry well (manifolds, vector fields, pull-back, differential forms, ...), then this book is a pleasure to read. For easier/less formal introductions, take a look at

I don't know these two books very well, but I took a quick look at them a few years ago and they seemed quite useful.

• +1 also, I'm very glad for these references. – Marek Jun 20 '11 at 1:12

There is a very general Lie theory that is most useful for partial differential equations but it can also be applied here. The full theory is not easy to explain and I am only familiar with it from one course I attended, so I cannot give a reference from the top of my ahead. I will try to provide it later. Try to look for keywords symmetries and conserved quantities and contact transformation.

The basic point is finding infinitesimal transformations $(t,y) \to (t,y) + \epsilon(t,y)$ that preserve the equation. Under special circumstances (like the possibility of variational formulation of the problem) some of these give rise to conserved quantities via Noether's theorem. But there need not be any associated conserved quantities (indeed, the space of infinitesimal transformations that preserve the equation is infinite-dimensional).