Here we introduce the *Poisson bracket*, in terms of which
Hamilton's equations have an elegant and symmetric expression.
Consider a function *F* of time, coordinates, and momenta. The value
of *F* along the path (*t*) = ( *t*, *q*(*t*), *p*(*t*) ) is (*F* o
)(*t*) = *F*(*t*, *q*(*t*), *p*(*t*)). The time derivative of *F* o
is

If the phase-space path is a realizable path for a system with
Hamiltonian *H*, then *D**q* and *D**p* can be reexpressed using
Hamilton's equations:

where the Poisson bracket { *F* , *H* } of *F* and *H* is defined by^{17}

Note that the Poisson bracket of two functions on the phase-state space is also a function on the phase-state space.

The coordinate selector *Q* = *I*_{1} is an example of a function
on phase-state space: *Q*(*t*, *q*, *p*) = *q*. According to
equation (3.79),

but this is the same as Hamilton's equation

Similarly, the momentum selector *P* = *I*_{2} is a function on phase-state
space: *P*(*t*, *q*, *p*) = *p*. We have

which is the same as Hamilton's other equation

So the Poisson bracket provides a uniform way of writing Hamilton's equations:

The Poisson bracket of any function with itself is zero, so we recover the conservation of energy for a system that has no explicit time dependence:

Let *F*, *G*, and *H* be functions of time, position, and momentum,
and let *c* be independent of position and momentum.

The Poisson bracket is antisymmetric:

It is bilinear (linear in each argument):

The Poisson bracket satisfies Jacobi's identity:

All but the last of (3.87-3.92) can immediately be verified from the definition. Jacobi's identity requires a little more effort to verify. We can use the computer to avoid some work. Define some literal phase-space functions of Hamiltonian type:

`(define F
(literal-function 'F
(-> (UP Real (UP Real Real) (DOWN Real Real)) Real)))
(define G
(literal-function 'G
(-> (UP Real (UP Real Real) (DOWN Real Real)) Real)))
(define H
(literal-function 'H
(-> (UP Real (UP Real Real) (DOWN Real Real)) Real)))
`

Then we check the Jacobi identity:

`(print-expression
((+ (Poisson-bracket F (Poisson-bracket G H))
(Poisson-bracket G (Poisson-bracket H F))
(Poisson-bracket H (Poisson-bracket F G)))
(up 't (up 'x 'y) (down 'px 'py))))
`

The residual is zero, so the Jacobi identity is satisfied for any three phase-space state functions with two degrees of freedom.

The Poisson bracket of conserved quantities is conserved.
Let *F* and *G* be time-independent phase-space state functions:
_{0} *F* = _{0} *G* = 0. If *F* and *G* are conserved
by the evolution under *H* then

So the Poisson brackets of *F* and *G* with *H* are zero: { *F*, *H*} =
{ *G*, *H*} = 0. The Jacobi identity then implies

so { *F*, *G* } is a conserved quantity.
The Poisson bracket of two conserved quantities is also a
conserved quantity.

^{17} In traditional notation the Poisson bracket is written