Poisson Bracket - Lie Algebra

Lie Algebra

The Poisson bracket is skewsymmetric/antisymmetric. (Equivalently, viewed as a binary product operation, it is anticommutative.) It also satisfies the Jacobi identity. This makes the space of smooth functions on a symplectic manifold an infinite-dimensional Lie algebra with the Poisson bracket acting as the Lie bracket. The corresponding Lie group is the group of symplectomorphisms of the symplectic manifold (also known as canonical transformations).

Given a smooth vector field X on the tangent bundle, let PX be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket:

This important result is worth a short proof. Write a vector field X at point q in the configuration space as

where the is the local coordinate frame. The conjugate momentum to X has the expression

where the pi are the momentum functions conjugate to the coordinates. One then has, for a point (q,p) in the phase space,

=\sum_{ij} p_i Y^j(q) \frac {\partial X^i}{\partial q^j} - p_j X^i(q) \frac {\partial Y^j}{\partial q^i}

The above holds for all (q,p), giving the desired result.

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