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jillnorth@hotmail.com

A question about the relationship between the phase space of the

Hamiltonian formulation of classical mechanics and of the Lagrangian

formulation; that is, between the cotangent bundle of configuration

space, T*Q, which has a natural symplectic structure given by the

canonical symplectic form, and the tangent bundle, TQ, which has a

natural metric structure given by the Riemannian line element, viz. a

quadratic differential form of the q dots.

It seems there's a sense in which the former has *less structure*

than the latter. Hamiltonian phase space has symplectic structure

(this determines a volume element), not metric structure. And in

general, metric structure determines, or presupposes, a volume

structure, but not the other way around. A metric would then add

another level of structure to what's needed for the Hamiltonian

equations of motion. (``Level of structure'' in the sense that,

starting with a set of points, we can define mathematical objects on

it, some of which presuppose others; in this sense a topological

space has more structure than a set of points, a metric space has

more structure than a topological space--a metric induces a topology--

and so on.)

It also seems the Lagrangian formulation needs this metric structure,

for the quadratic differential form is the invariant quantity of the

Lagrangian transformations, and a symplectic manifold is ``floppy'',

having no local notion of curvature that would distinguish one

symplectic manifold from another locally (from Darboux's theorem).

This is because of the two sets of generalized coordinates used by

the Lagrangian as opposed to the Hamiltonian formulation: for the

Hamiltonian, the canonically conjugate q's and p's, treated as

independent variables on the phase space (so that the energy function

is linear in each); for the Lagrangian, the generalized coordinates

and their first time derivatives, the generalized velocities (giving

rise to the quadratic differential form of the q dots). (Would a more

coordinate-free version of Lagrangian mechanics be able to get by

without the Riemannian metric structure on the base manifold? I

would've thought not, given the above, but I'm not sure.)

It seems this should mean that not every symplectic manifold

(similarly, not every cotangent bundle) is isomorphic to a Riemannian

manifold (tangent bundle). However, there is a natural isomorphism

between T*Q and TQ, given by the Legendre transformation. On the

other hand, the isomorphism is non-canonical (there is no basis-

independent isomorphism). More generally, then, how can we compare

the structures of these two spaces, given that they are, after all,

*different* spaces? Is there a structure-preserving map between the

two? How should one go about trying to find such a thing?

Any and all feedback or references would be extremely helpful! I am

in philosophy of physics, struggling with a paper, and having trouble

figuring out an answer on the basis of the books and online

references I've seen so far. Profuse apologies if I have simply

gotten myself rather mixed up.