Editing Poisson bracket (section)

==Constants of motion==
An [[integrable system]] will have [[constants of motion]] in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function <math>f(p, q)</math> is a constant of motion. This implies that if <math>p(t), q(t)</math> is a [[trajectory]] or solution to [[Hamilton's equations of motion]], then along that trajectory:<math display="block">0 = \frac{df}{dt}</math>Where, as above, the intermediate step follows by applying the equations of motion and we assume that <math>f</math> does not explicitly depend on time. This equation is known as the [[Liouville's theorem (Hamiltonian)#Liouville equations|Liouville equation]]. The content of [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]  is that the time evolution of a [[measure (mathematics)|measure]] given by a [[Distribution function (physics)|distribution function]] <math>f</math> is given by the above equation.

If the Poisson bracket of <math>f</math> and <math>g</math> vanishes (<math>\{f,g\} = 0</math>), then <math>f</math> and <math>g</math> are said to be '''in involution'''.  In order for a Hamiltonian system to be [[completely integrable]], <math>n</math> independent constants of motion must be in [[Distribution (differential geometry)#Involutive distributions|mutual involution]], where <math>n</math> is the number of degrees of freedom.

Furthermore, according to '''Poisson's Theorem''', if two quantities <math>A</math> and <math>B</math> are explicitly time independent (<math>A(p, q), B(p, q)</math>) constants of motion, so is their Poisson bracket <math>\{A,\, B\}</math>. This does not always supply a useful result, however, since the number of possible constants of motion is limited (<math>2n - 1</math> for a system with <math>n</math> degrees of freedom), and so the result may be trivial (a constant, or a function of <math>A</math> and <math>B</math>.)