- #1
jillnorth@hotmail.com
Symplectic structure vs. metric structure
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.
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.