- #1
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I've been reading about the abstract formulation of dynamics in terms of symplectic manifolds, and it's amazing how naturally everything falls out of it. But one thing I can't see is why the generalized momenta should be cotangent vectors. I can see why generalized velocities are tangent vectors, making the Lagrangian a function on the tangent bundle of configuration space. But then the book I'm reading claims that since:
[tex] p_i = \frac{\partial L}{\partial \dot q_i} [/tex]
it follows that pi is clearly a cotangent vector.
To me it seems like what pi is is a function assigning numbers to vectors in the "tangent space of the tangent space of a point in the manifold". Note that by what's in the quotes, I don't mean "the tangent space of the tangent bundle" (which would put the momentum in [itex]T^*(TM)[/itex] ), because it doesn't depend on the change in configuration.
But it also shouldn't be in [itex]T^*M[/itex] unless the Lagrangian is a linear function of velocity, so that its partial derivative with respect to velocity doesn't depend on your location in the tangent space, and so can be taken as a uniform linear functional on the tangent space. What am I missing here?
[tex] p_i = \frac{\partial L}{\partial \dot q_i} [/tex]
it follows that pi is clearly a cotangent vector.
To me it seems like what pi is is a function assigning numbers to vectors in the "tangent space of the tangent space of a point in the manifold". Note that by what's in the quotes, I don't mean "the tangent space of the tangent bundle" (which would put the momentum in [itex]T^*(TM)[/itex] ), because it doesn't depend on the change in configuration.
But it also shouldn't be in [itex]T^*M[/itex] unless the Lagrangian is a linear function of velocity, so that its partial derivative with respect to velocity doesn't depend on your location in the tangent space, and so can be taken as a uniform linear functional on the tangent space. What am I missing here?