3 ms·
I don’t think the typical “change of variables” definition is bad. You take the derivative of L along the fiber of the tangent bundle. If the derivative is non-
by hansen 7y ago
I don’t think the typical “change of variables” definition is bad.
You take the derivative of L along the fiber of the tangent bundle.
If the derivative is non-singular it defines an isomorphism in each
point of the tangent space with the cotangent space. And that’s the
important thing, going from the tangent bundle to the cotangent
bundle. Now we can use all the beauty of symplectic geometry
- jessriedel 7y agoThe change of variable definition, as actually presented in the textbooks everyone teaches from, is horrible. That's the topic of the blog post. Yes, that definition can be made clear after introducing a bunch of machinery of symplectic geometry, but I'm doubtful this is good pedagogy and I'm confident that, due to time constraints, it could never be taught to most physicists.
- fspeech 7y agoWhat is a good reference from the symplectic geometry angle?
- hansen 7y agoThe standard (but rather heavy and difficult) is Abraham, Marsden: Foundations of Mechanics More gentle is Vladimir Arnold: Mathematical Methods of Classical Mechanics