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

${\displaystyle \left\{f,g\right\}:=\sum _{k=1}^{s}{\left({\frac {\partial f}{\partial q_{k}}}{\frac {\partial g}{\partial p_{k}}}-{\frac {\partial f}{\partial p_{k}}}{\frac {\partial g}{\partial q_{k}}}\right)}}$

With

• ${\displaystyle f}$and functions of generalized coordinates and canonically conjugated momenta ${\displaystyle g}$ ${\displaystyle q_{k}}$ ${\displaystyle p_{k}}$
• ${\displaystyle s}$Number of degrees of freedom .

In general, the Poisson bracket can also be defined for functions and 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 as indices to the brackets: ${\displaystyle F}$ ${\displaystyle G}$

${\displaystyle \{F,G\}_{ab}:=\sum _{k=1}^{s}\left({\frac {\partial F}{\partial a_{k}}}{\frac {\partial G}{\partial b_{k}}}-{\frac {\partial F}{\partial b_{k}}}{\frac {\partial G}{\partial a_{k}}}\right)}$.

properties

• Bilinearity

${\displaystyle \,\{c_{1}f_{1}+c_{2}f_{2},g\}=c_{1}\{f_{1},g\}+c_{2}\{f_{2},g\}}$

• Antisymmetry

${\displaystyle \{f,g\}=-\{g,f\}\,\Rightarrow \,\{f,f\}=0}$

• Product rule

${\displaystyle \,\{f,gh\}=\{f,g\}h+g\{f,h\}}$

• Jacobi identity

${\displaystyle \,\{f,\{g,h\}\}+\{h,\{f,g\}\}+\{g,\{h,f\}\}=0}$

• Invariance

Physically, 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. Be and two different sets of coordinates by canonical transformation to be transformed, then: ${\displaystyle (\mathbf {q} ,\mathbf {p} )}$ ${\displaystyle (\mathbf {Q} ,\mathbf {P} )}$

${\displaystyle \{f,g\}_{\mathbf {qp} }=\{f,g\}_{\mathbf {QP} }=\{f,g\}}$.

The evidence is elongated so we leave it out here.

Fundamental Poisson brackets

The fundamental Poisson brackets are important for canonical mechanics

${\displaystyle \left\{q_{k},q_{l}\right\}=0}$

${\displaystyle \left\{p_{k},p_{l}\right\}=0}$

${\displaystyle \left\{q_{k},p_{l}\right\}=\delta _{kl}}$( Kronecker Delta ).

They follow from the trivial relationships

${\displaystyle {\begin{alignedat}{2}&{\frac {\partial q_{k}}{\partial q_{l}}}=\delta _{kl}\quad &&{\frac {\partial p_{k}}{\partial q_{l}}}=0\\&{\frac {\partial q_{k}}{\partial p_{l}}}=0\quad &&{\frac {\partial p_{k}}{\partial p_{l}}}=\delta _{kl}\end{alignedat}}}$.

application

Hamilton's equation of motion

With the help of Poisson brackets, the time evolution of any observable in a Hamiltonian system can be expressed. ${\displaystyle f(q_{k},p_{k},t)}$ ${\displaystyle H(q_{k},p_{k})}$

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

${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\sum _{k=1}^{s}\left({\frac {\partial f}{\partial q_{k}}}{\frac {\mathrm {d} q_{k}}{\mathrm {d} t}}+{\frac {\partial f}{\partial p_{k}}}{\frac {\mathrm {d} p_{k}}{\mathrm {d} t}}\right)+{\frac {\partial f}{\partial t}}}$.

Inserting the Hamilton equations

${\displaystyle {\dot {q}}_{k}={\frac {\partial H}{\partial p_{k}}}}$

and

${\displaystyle {\dot {p}}_{k}=-{\frac {\partial H}{\partial q_{k}}}}$

results

${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\sum _{k=1}^{s}\left({\frac {\partial f}{\partial q_{k}}}{\frac {\partial H}{\partial p_{k}}}-{\frac {\partial f}{\partial p_{k}}}{\frac {\partial H}{\partial q_{k}}}\right)+{\frac {\partial f}{\partial t}}}$.

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

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

In particular, this equation can be used to characterize constants of motion ( conserved quantities ). An observable is a conserved quantity if and only if:

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

If it is not explicitly time-dependent , it becomes: ${\displaystyle f}$ ${\displaystyle \left(f(q_{k},p_{k})\neq f(t)\right)}$

${\displaystyle \{f,H\}=0}$

additional

• The Liouville equation , which describes the dynamics of the distribution density in statistical mechanics , is dual to the equation of motion of the observables :

${\displaystyle {\dot {\rho }}=\{H,\rho \}.}$

• In quantum mechanics , in the context of canonical quantization, the Poisson bracket is replaced by the commutator : ${\displaystyle \textstyle \left(-{\frac {\rm {i}}{\hbar }}\right)}$

${\displaystyle \{H,f\}\rightarrow -{\frac {i}{\hbar }}[{\hat {H}},{\hat {f}}]}$

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 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 . ${\displaystyle {\hat {H}}}$

• 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 symplectic form, which is given in local coordinates by , the Poisson bracket of the functions and by: ${\displaystyle \textstyle \omega =\sum _{ij}\omega _{ij}\,\mathrm {d} x^{i}\wedge \mathrm {d} x^{j}}$ ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle \{f,g\}=\sum _{ij}\omega ^{ij}\,\partial _{i}f\,\partial _{j}g\,.}$

• The Poisson bracket can be represented as follows, independent of coordinates: Let the isomorphism described by . Let the vector field for a function be defined as . This then applies ${\displaystyle J:T^{*}M\rightarrow TM}$ ${\displaystyle J^{-1}(v)(w)=\omega (v,w)}$ ${\displaystyle f}$ ${\displaystyle X_{f}}$ ${\displaystyle J(\mathrm {d} f)}$

${\displaystyle {f,g}=\omega (X_{f},X_{g}).}$

Web links

• Eric W. Weisstein : Poisson Bracket . In: MathWorld (English).

Individual evidence

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