Integral of motion

An integral of motion is a function on phase space that remains constant along every trajectory. If it has no explicit time dependence, $I(q,p)$ is an integral exactly when its Poisson bracket with the Hamiltonian vanishes: $\{I,H\}=0$.

For $\Phi (R,\varphi -\Omega t,z)$ with constant pattern angular velocity $\Omega$, ordinary specific energy satisfies $\dot{E}={\partial }_t\Phi =-\Omega {\partial }_{\varphi }\Phi$ and specific axial angular momentum satisfies ${\dot{L}}_z=-{\partial }_{\varphi }\Phi$. Their combination $E_J$ is conserved. This is the rotating-frame integral of motion; in the circular restricted three-body problem, the usual Jacobi constant is an affine rescaling of it. Thus time dependence in an inertial frame does not alone imply the absence of all integrals.

An isolating integral is a regular single-valued integral of motion whose fixed value restricts accessible phase space to a lower-dimensional level set. Functional independence of several such integrals means their differentials are linearly independent on the region considered. A central potential has specific orbital energy and the three components of specific angular momentum as four generically independent integrals. A Kepler orbit has a fifth independent integral supplied by its fixed apsidal direction, encoded by the Laplace-Runge-Lenz vector; its vector components obey two constraints with energy and angular momentum. Circular and radial orbits are degenerate cases of this count.

In three-dimensional autonomous mechanics, a regular common level set of $N$ independent isolating integrals has dimension $6-N$ in phase space. If an orbit is dense in a $d$-dimensional invariant torus and the integrals characterize its closure, $N=6-d$. A generic three-frequency regular orbit has three integrals; a spherical nonclosed rosette has energy and three components of specific angular momentum, with two-dimensional closure and four integrals. A periodic nonstationary orbit has one-dimensional closure and can be isolated by five independent orbit labels on the invariant family. The maximum is five wherever the Hamiltonian vector field is nonzero, since all integral differentials annihilate that field. A single resonant orbit need not yield an additional smooth single-valued integral on an open neighborhood; global superintegrability is stronger than orbital closure.

A conserved quantity is a quantity that remains constant along the time evolution of a system. In Hamiltonian mechanics, a possibly time-dependent phase-space function $F$ is conserved exactly when ${\partial }_{t}F+\{F,H\}=0$.