14.1 Classical mechanics and the Poisson bracket

14.1 Classical mechanics and the Poisson bracket

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie bracket

In classical mechanics in the Hamiltonian formalism, the space $M={\mathbf{R}}^{2d}$ that one gets by putting together positions and the corresponding momenta is known as “phase space”. Points in phase space can be thought of as uniquely parametrizing possible initial conditions for classical trajectories, so another interpretation of phase space is that it is the space that uniquely parametrizes solutions of the equations of motion of a given classical mechanical system. The basic axioms of Hamiltonian mechanics can be stated in a way that parallels the ones for quantum mechanics.

Axiom (States)

The state $of$ a classical mechanical system is given by a point in the phase space $M={\mathbf{R}}^{2d}$ , with coordinates $q_j,p_j,forj=1,\dots ,d.$

Axiom (Observables)

The observables of a classical mechanical system are the functions on phase space.

Axiom (Dynamics)

There is a distinguished observable, the Hamiltonian function $h,$ and states evolve according to Hamilton’s equations

$${\dot{q}}_j=\frac{\partial h}{\partial p_j}$$ $${\dot{p}}_j=-\frac{\partial h}{\partial q_j}$$

Specializing to the case $d=1$ , for any observable function $f,$ , Hamilton’s equations imply

$$\frac{df}{dt}=\frac{\partial f}{\partial q}\frac{dq}{dt}+\frac{\partial f}{\partial p}\frac{dp}{dt}=\frac{\partial f}{\partial q}\frac{\partial h}{\partial p}-\frac{\partial f}{\partial p}\frac{\partial h}{\partial q}$$

We can define:

Definition (Poisson bracket)

There is a bilinear operation on functions on the phase space $\dot{M}={\mathbf{R}}^2$ (with coordinates $(q,p),$ called the Poisson bracket, given $by$

$$(f_1,f_2)\to \{f_1,f_2\}=\frac{\partial f_1}{\partial q}\frac{\partial f_2}{\partial p}-\frac{\partial f_1}{\partial p}\frac{\partial f_2}{\partial q}$$

An observable $f$ evolves in time according to

$$\frac{df}{dt}=\{f,h\}$$

This relation is equivalent to Hamilton’s equations since it implies them by taking $f=q$ and $f=p$

$$\dot{q}=\{q,h\}=\frac{\partial h}{\partial p}$$ $$\dot{p}=\{p,h\}=-\frac{\partial h}{\partial q}$$

For a non-relativistic free particle, $\begin{array}{r}h=\frac{p^2}{2m}\end{array}$ and these equations become

$$\dot{q}=\frac{p}{m},\dot{p}=0$$

which says that the momentum is the mass times the velocity, and is conserved. For a particle subject to a potential $V(q)$ one has

$$h=\frac{p^2}{2m}+V(q)$$

and the trajectories are the solutions to

$$\dot{q}=\frac{p}{m},\dot{p}=-\frac{\partial V}{\partial q}$$

This adds Newton’s second law

$$F=-\frac{\partial V}{\partial q}=ma=m\ddot{q}$$

to the relation between momentum and velocity.

One can easily check that the Poisson bracket has the properties

• Antisymmetry

$$\{f_1,f_2\}=-\{f_2,f_1\}$$

• Jacobi identity

$$\{\{f_1,f_2\},f_3\}+\{\{f_3,f_1\},f_2\}+\{\{f_2,f_3\},f_1\}=0$$

These two properties, together with the bilinearity, show that the Poisson bracket fits the definition of a Lie bracket, making the space of functions on phase space into an infinite dimensional Lie algebra. This Lie algebra is responsible for much of the structure of the subject of Hamiltonian mechanics, and it was historically the first sort of Lie algebra to be studied.

From the fundamental dynamical equation

$$\frac{df}{dt}=\{f,h\}$$

we see that

$$\{f,h\}=0⟹\frac{df}{dt}=0$$

and in this case the function $f$ is called a “conserved quantity”, since it does not change under time evolution. Note that if we have two functions $f_1$ and $f_2$ on phase space such that

$$\{f_1,h\}=0,\{f_2,h\}=0$$

then using the Jacobi identity we have

$$\{\{f_1,f_2\},h\}=-\{\{h,f_1\},f_2\}-\{\{f_2,h\},f_1\}=0$$

This shows that if $f_1$ and $f_2$ are conserved quantities, so is $\{f_1,f_2\}$ . As a result, functions $f$ such that $\{f,h\}=0$ make up a Lie subalgebra. It is this Lie subalgebra that corresponds to “symmetries” of the physics, commuting with the time translation determined by the dynamical law given by $h$.

Chapter contents · Next section: 14.2 The Poisson bracket and the Heisenberg Lie algebra · 中文版本

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