[hansen]

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]

The 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]

What is a good reference from the symplectic geometry angle?

[hansen]

The standard (but rather heavy and difficult) is

Abraham, Marsden: Foundations of Mechanics

More gentle is

Vladimir Arnold: Mathematical Methods of Classical Mechanics