How are Poisson brackets and the variational principle related?

Question

In the lecture Space and spaces, Segal argues that the origin of non-commutativity in classical mechanics “which is encoded in the Poisson Bracket” is the fact that the evolution of classical states is governed by the variational principle. Can anyone shed more light on this? What is the connection between the variational principle and non-commutativity?

Answer 1

The direct connection between Poisson bracket and non-commutativity is pretty clear, at least if you agree that the (later introduced) Lie bracket $[X,Y]$ of vector fields measures the non-commutativity of their flows. The original setting is symplectic manifolds, where each function (“hamiltonian”) $a$ defines a vector field $X_a$ (its “symplectic gradient”) defined by $\omega(X_a,\cdot)=-da$ where $\omega$ is the symplectic form. Indeed one has $$ [X_a,X_b]=X_{\{a,b\}} $$ where $\{a,b\}$ is the Poisson bracket $\omega(X_a,X_b)$. (In Darboux variables where $\omega=\sum dp_i\wedge dq_i$, the symplectic gradient $X_a$ has the expression $\bigl(-\frac{\partial a}{\partial q_i},\frac{\partial a}{\partial p_i}\bigr)$ familiar in Hamilton’s equations.)

Segal’s detour through variational principles is justified by results to the effect that spaces of extremals of variational principles are symplectic manifolds; but it is IMHO unwarranted in the sense that not every symplectic manifold arises in this way. (Those that do have $\omega$ not only closed but exact.)