For a configuration manifold M we expect there to be a bracket on C∞(T∗M) endowing the space with a Lie algebra structure. Here we assume that T∗M is also equipped with a non-degenerate symplectic 2-form ωH and that the bracket can be written ωH(Xf,Xg) for two Hamiltonian vector fields Xf,Xg∈χ(T∗M). We note that there exists a 1-form θH on T∗M such that ωH=−dθ and that θH can be pulled back to TM to give θL. In charts we may write,
θL=∂L∂˙qidqi
Hence we expect the exterior derivative to give an induced 2-form on TM.
ωL=∂2∂q∂vdq∧dq+∂2L∂v∂vdq∧dv
At this point I am interested in evaluating ωL(Xf,Xg) in order to see what expression is generated. My attempt is to assume that Xf and Xg are now Lagrangian vector fields on TM and hence can be written,
Xf=˙q∂f∂q+˙v∂f∂v
This approach however does not successfully lead anywhere (could be my shoddy maths skills to blame however, so I would be extremely interested to see if it did in fact lead somewhere?). If we cannot develop a bracket on TM does this mean that f,g∈C∞(TM) do not have a Lie algebra structure, and at what point would they inherit such a structure during their transition to the cotangent bundle?
Thank you for your time and I hope this question is appropriate for this forum (?) apologies if not!