Poisson Brackets and Conservation Laws

A very important object of the Hamiltonian formalism for both the investigation of conservation theorems and the transition towards quantum theory are the Poisson brackets. Consider two arbitrary p/zflse space functions u = u q, p, t) and V = v q, p, t) depending only on the variables of phase space q and p and the time t. The Poisson bracket between u and v is defined as 

It is always defined with respect to the given set of canonical variables which is sometimes symbolized by suitable indices. The result, i.e., the Poisson bracket itself is, in general, another phase space function. In anticipation of the next section 2.3.3, all Poisson brackets are invariant under canonical transformations, i.e., their value does not depend on the choice of canonical variables. The indices q and p attached to the Poisson brackets are therefore often suppressed. 

Since also the canonical variables q and p are phase space functions, we may calculate their Poisson brackets. They are given by 

The last Eq. (2.84) is called Jacobi s identity it is simple but tedious to prove. We note in passing that the Poisson brackets of classical mechanics presented here will be promoted to the commutators of quantum mechanics (scaled by —i/h) by the transition to quantum theory in chapter 4. 

Poisson brackets can be employed to cast the time dependence of an arbitrary phase space function u = u q,p,t) in a more compact form, which may be useful for the exhibition of conservation laws. Since the total time derivative of u is given as 

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