Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 72 / 4 / a

A constant of motion is a quantity preserved along the evolution, possibly with explicit time dependence. For $C(\mathbf{x},\mathbf{v},t)$ in a gravitational potential $\Phi$, the preservation condition isFor free motion, $\mathbf{x}-\mathbf{v}t$ is a useful explicitly time-dependent example. Such initial-position labels should not be counted as stationary isolating integrals of a galactic orbit.

In the distinction relevant here, an integral of motion is a single-valued function $I(\mathbf{x},\mathbf{v})$ of instantaneous phase space, with no explicit time dependence, constant along the orbit. In an autonomous Hamiltonian system this is $\{I,H\}=0$, where the bracket is the Poisson bracket. An integral must be defined consistently on the phase space region under discussion; an arbitrary multivalued formula obtained by solving one trajectory does not supply a global integral. Some conventions call explicitly time-dependent conserved functions integrals too; the stationary interpretation is needed for the following counts.

An isolating integral restricts the accessible phase space to a lower-dimensional regular level set. For functionally independent integrals of motion $I_1,\dots ,I_N$, the differentials $d{I}_j$ must be independent, and $I_j=c_j$ impose $N$ independent restrictions. In six-dimensional phase space the common regular level set has dimension $6-N$. Nonisolating orbit labels or multivalued constants do not further reduce the closure explored by the motion. This is the isolating integrals and orbital closure dimension distinction.

Zero stationary isolating integrals are possible when time dependence removes conservation of energy and no other symmetry or invariant survives. For instance, a generic time-dependent nonsymmetric gravitational potential need not have a stationary integral. This is not possible in an ordinary time-independent galactic gravitational potential: $E=\frac{1}{2}{v}^2+\Phi$ is already an isolating integral on a regular specific orbital energy surface. An explicitly time-dependent constant that encodes an initial condition does not contradict the stationary count.

One isolating integral occurs when specific orbital energy is the only independent stationary invariant. An autonomous chaotic orbit that densely explores a connected five-dimensional specific orbital energy surface is the standard case. Every continuous integral of motion must then be constant on that surface, so it is locally a function of specific orbital energy rather than an additional independent integral. Irregular motion need not mean zero integrals; specific orbital energy still isolates the orbit from other specific orbital energy surfaces.

More than three occur when symmetries or frequency degeneracies reduce the orbit's invariant closure below a generic three-dimensional invariant torus. For a nonradial orbit in a spherical gravitational potential, specific orbital energy and the three components of specific angular momentum are generically four independent isolating integrals. Their values fix the orbital plane, angular momentum magnitude and radial specific orbital energy, but a nonclosed rosette still fills a two-dimensional invariant torus. Thus more than three does not by itself imply closure. Orbital resonances can further reduce the number of independent orbital phases. The count concerns independent integrals, not arbitrarily many functions of the same invariants.

Five independent isolating orbit labels describe a regular closed, nonstationary orbit in three-dimensional space: its phase space closure is one-dimensional. Five is the maximum at a nonzero flow vector, since every conserved differential annihilates that vector in the six-dimensional tangent space. A global superintegrable Hamiltonian system example is the Kepler orbit, where specific orbital energy, angular momentum and the Laplace-Runge-Lenz vector have two vector/scalar relations leaving five independent invariants. A circular or radial orbit requires a separate treatment of any rank-degenerate formula for these invariants. Five integrals do not force an escaping trajectory to be periodic: unbound Kepler orbits also have the conserved apsidal vector. The closure criterion concerns recurrent bound motion.

Closure of one resonant orbit need not imply five smooth independent integrals on an open neighborhood of all nearby orbits. The orbit-wise count and global superintegrability must be distinguished; the distinction matters in part (b).