What is the origin of the Poisson brackets

Poisson bracket

The Poisson bracket, named after Siméon Denis Poisson, is a bilinear differential operator in canonical (Hamiltonian) mechanics. It is an example of a Lie bracket, i.e. a multiplication in a Lie algebra.

definition

The Poisson bracket is defined as

$ \ left \ {f, g \ right \}: = \ sum_ {k = 1} ^ {s} {\ left (\ frac {\ partial f} {\ partial q_k} \ frac {\ partial g} {\ partial p_k} - \ frac {\ partial f} {\ partial p_k} \ frac {\ partial g} {\ partial q_k} \ right)} $

With

In general, the Poisson bracket can also be defined for functions $ F $ and $ G $ that do not depend on generalized coordinates and canonical impulses. To make it clear which variables the Poisson brackets should refer to, these are written to the brackets as indices:

$ \ {F, G \} _ {ab}: = \ sum ^ s_ {k = 1} \ left (\ frac {\ partial F} {\ partial a_k} \ frac {\ partial G} {\ partial b_k} - \ frac {\ partial F} {\ partial b_k} \ frac {\ partial G} {\ partial a_k} \ right) $.

properties

$ \, \ {c_1 f_1 + c_2 f_2, g \} = c_1 \ {f_1, g \} + c_2 \ {f_2, g \} $
$ \ {f, g \} = - \ {g, f \} \, \ Rightarrow \, \ {f, f \} = 0 $
$ \, \ {f, gh \} = \ {f, g \} h + g \ {f, h \} $
$ \, \ {f, \ {g, h \} \} + \ {g, \ {h, f \} \} + \ {h, \ {f, g \} \} = 0 $
From a physical point of view, it seems reasonable to assume that the time evolution of a property of a system should not depend on the coordinates used; thus the Poisson brackets should also be independent of the canonical coordinates used. Let $ (\ mathbf {q}, \ mathbf {p}) $ and $ (\ mathbf {Q}, \ mathbf {P}) $ be two different sets of coordinates that are transformed by canonical transformations, then:
$ \ {f, g \} _ {\ mathbf {qp}} = \ {f, g \} _ {\ mathbf {QP}} = \ {f, g \} $.
The evidence is elongated so we leave it out here.

Fundamental Poisson brackets

The fundamental Poisson brackets are important for canonical mechanics

$ \ left \ {q_k, q_l \ right \} = 0 $
$ \ left \ {p_k, p_l \ right \} = 0 $
$ \ left \ {q_k, p_l \ right \} = \ delta_ {kl} $ (Kronecker delta).

They follow from the trivial relationships

$ \ begin {alignat} {2} & \ frac {\ partial q_k} {\ partial q_l} = \ delta_ {kl} \ quad && \ frac {\ partial p_k} {\ partial q_l} = 0 \ & \ frac {\ partial q_k} {\ partial p_l} = 0 \ quad && \ frac {\ partial p_k} {\ partial p_l} = \ delta_ {kl} \ end {alignat} $.

application

Hamilton's equation of motion

The time evolution of any observable $ f (q_k, p_k, t) $ of a Hamiltonian system $ H (q_k, p_k) $ can be expressed with the help of Poisson brackets.

This time evolution of any observable is described by the total derivative with respect to time:

$ \ frac {\ mathrm {d} f} {\ mathrm {d} t} = \ sum_ {k = 1} ^ s \ left (\ frac {\ partial f} {\ partial q_k} \ frac {\ mathrm { d} q_k} {\ mathrm {d} t} + \ frac {\ partial f} {\ partial p_k} \ frac {\ mathrm {d} p_k} {\ mathrm {d} t} \ right) + \ frac { \ partial f} {\ partial t} $.

Inserting the Hamilton equations

$ \ dot {q} _k = \ frac {\ partial H} {\ partial p_k} $

and

$ \ dot {p} _k = - \ frac {\ partial H} {\ partial q_k} $

results

$ \ frac {\ mathrm {d} f} {\ mathrm {d} t} = \ sum ^ s_ {k = 1} \ left (\ frac {\ partial f} {\ partial q_k} \ frac {\ partial H } {\ partial p_k} - \ frac {\ partial f} {\ partial p_k} \ frac {\ partial H} {\ partial q_k} \ right) + \ frac {\ partial f} {\ partial t} $.

The front part corresponds to the definition of Poisson brackets:

$ \ frac {\ mathrm {d} f} {\ mathrm {d} t} = \ {f, H \} + \ frac {\ partial f} {\ partial t} $.

In particular, with this equation Constants of motion (Conservation quantities) characterize. An observable is a conserved quantity if and only if:

$ \ {f, H \} + \ frac {\ partial f} {\ partial t} = 0 $

If $ f $ is not explicitly time-dependent $ \ left (f (q_k, p_k) \ neq f (t) \ right) $, then this becomes:

$ \ {f, H \} = 0 $

additional

$ \ dot {\ rho} = \ {H, \ rho \}. $
$ \ {H, f \} \ rightarrow- \ frac {i} {\ hbar} [\ hat {H}, \ hat {f}] $
In addition, observables are represented by operators. The above equation of the time evolution of an observable leads to the time evolution of operators of a quantum mechanical system with the Hamilton operator $ \ hat {H} $ in the Heisenberg picture. This equation of motion is called Heisenberg's equation of motion. The Liouville equation finds its equivalent in Von-Neumann's equation of motion.
  • Both the phase space functions of canonical mechanics and the operators of quantum mechanics each form a Lie algebra with their brackets.
  • In general, one defines on a symplectic manifold with a symplectic form, which is given in local coordinates by $ \ textstyle \ omega = \ sum_ {ij} \ omega_ {ij} \, \ mathrm dx ^ i \ wedge \ mathrm dx ^ j $, the Poisson brackets of the functions $ f $ and $ g $ by:
$ \ {f, g \} = \ sum_ {ij} \ omega ^ {ij} \, \ partial_i f \, \ partial_j g \ ,. $

Web links

Individual evidence

  1. ↑ Hong-Tao Zhang: A Simple Method of Calculating Commutators in Hamilton System with Mathematica Software, arxiv: quant-ph / 0204081