Parallel transport on a symplectic space

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Summary:

How to use the Poisson Bracket to write the curvature 2-form of a connection that parallels transport local coordinates in a symplectic space?
Sorry if the question is not rigorously stated.


Statement: Let ##(q,p)## be a set of local coordinates in 2-dimensional symplectic space. Let ##\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})## be a set of local coordinates of certain open set of a differentiable manifold ##\mathcal{M}.## For some ##\lambda_{0}## we define two differentiable functions ##Q_{0}=Q_{0}(q,p,\lambda_{0})## and ##P_{0}=P_{0}(q,p,\lambda_{0})## such that ##dq\wedge dp=dQ_{0}\wedge dP_{0}##.

For the other values of ##\lambda## we also want functions ##(Q_{\lambda},P_{\lambda})## such that ##dq\wedge dp=dQ_{\lambda}\wedge dP_{\lambda}.## However, we don't want ##(Q_{\lambda},P_{\lambda})## to depend only on a given value of ##\lambda## but they should depend also on the path taken from ##\lambda_{0}## to ##\lambda.## This is, we want a notion of parallel transport.

Assume we have an operator ##G=\left\{ \cdot,g\right\} =\partial_{q}g\partial_{p}-\partial_{p}g\partial_{q},## where ##g=g_{i}d\lambda_{i}## and ##g_{i}=g_{i}(q,p,\lambda_{i})##and the bracket is the Poisson bracket. For ##\lambda## very close to ##\lambda_{0}##we define parallel transport as $$Q_{\lambda} =Q_{0}+G[Q_{0}]=Q_{0}+\left\{ Q_{0},g\right\},$$
$$P_{\lambda} =P_{0}+G[P_{0}]=P_{0}+\left\{ P_{0},g\right\}.$$

I guess ##G##(or maybe ##g##?) is some form of a connection 1-form. I'm interested in the curvature 2-form of ##G.## In particular, can the components of the of the curvature can be expressed using the Poisson bracket as ##F_{ij}=\left\{ \cdot,f_{ij}\right\}?##
 
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Answers and Replies

  • #2
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I guess you should use the Lie bracket instead of the Poisson bracket, considering that there is an antihomorphism from the Poisson Algebra of functions to the Lie algebra of vector fields.
Then, if G is the diffeomorphism group of a manifold M, the exponential map of G is given by parallel transport of the natural connection and the curvature of the natural connection is the Lie bracket of vector fields on M.
 

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