Poisson bracket

The Poisson bracket ( after Siméon Denis Poisson ) is a bilinear differential operator in the canonical ( Hamiltonian ) mechanics. It is defined as

Where and are functions of the generalized coordinates and the canonically conjugate momenta are. The number of degrees of freedom.

Hamiltonian equation of motion

Using the Poisson bracket, the time evolution of an arbitrary observable of a Hamiltonian system can be expressed.

The total time derivative of any observable is

And describes the time evolution of the observables. Inserting the Hamiltonian equations

And

Results

The front part corresponds to the definition of the Poisson bracket:

In general, the Poisson bracket for functions and which do not depend on generalized coordinates and canonical pulses are defined. To clarify which variables the Poisson bracket is meant to refer, it is written as an index to the bracket:

Properties

  • Bilinearity
  • Antisymmetry
  • Product rule
  • Jacobi identity
  • Invariance

Physically, it is natural to assume that the time evolution of a property of a system should not depend on the coordinates used. This means that the Poisson brackets should be independent of the used canonical coordinates. Let and be two different sets of coordinates which are transformed by canonical transformations, the following applies

The proof of the invariance property is elongated, so we omit it here.

Fundamental Poisson brackets

For the canonical mechanics important are the fundamental Poisson brackets

Which simply from the trivial relations

Follow.

It is the Kronecker delta.

Application

  • The Poisson bracket can be used to determine the temporal change of observables by the dynamics of the system. It applies to an observable
  • In particular, one can characterize this equation constants of motion ( conserved quantities ). An observable is in fact a conserved quantity if and only if
  • Dual the equation of motion of the observable is Liouville equation describing the dynamics of the distribution of density in the statistical mechanics:
  • In quantum mechanics, the Poisson bracket is replaced by the commutator times within the framework of the canonical quantization.
  • Both the phase-space functions of canonical mechanics and the operators of quantum mechanics, with their brackets each have a Lie algebra.
Differentialoperator Degrees of freedom (physics and chemistry) Index#Mathematics Bilinearform Kronecker-Delta Equations of motion Liouville's theorem (Hamiltonian) Quantization (physics) Heisenberg picture
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