Conservation Poisson brackets

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

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... 

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... 

---