From: Kevin Foltinek Subject: Re: infinite series for a periodic function Date: 09 Feb 2001 11:11:34 -0600 Newsgroups: sci.math Summary: Lie theory of differential equations explains trig identities dank@micrologic.com (Dan Kotlow) writes: > [periodicity of cosine from its differential equation] > I think it's instructive to derive the addition formulas as a > consequence of the differential equation -- its translation-invariance > and the uniqueness of solutions. In particular, the constancy of > sin^2 + cos^2 coincides with conservation of energy for solutions. Yes, these are good little exercises. It's actually much deeper (and more "beautiful") than that, though. I think I have mentioned this here before, in a different context, but: If you look at the ODE u''=-u, the natural (jet bundle) formulation is 0 = du - v dx 0 = dv + u dx and then if you compute the Lie algebra of symmetries (the "infinitessimal symmetries"), you find so(2) acting on {(u,v)}; then you can look for other invariants of this Lie algebra action, and observe that u^2+v^2 is an invariant. (The flow of the Lie algebra action happens to correspond to the flow of the solutions of the ODE, so u^2+v^2 is a conserved quantity as well.) The point, of course, is that the ODE naturally gives rise to the standard Euclidean geometry on {(u,v)}, and hence \pi (defined strictly in terms of an ODE, as half the period of a solution) is (even with this ODE definition) inherently linked to Euclidean geometry. Kevin.