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andresB
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- TL;DR 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\}?##
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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