# 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} \$.

### 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 \$