Integrals of Motion

When a particle moves under the influence of a potential, there may be physical quantities assiciated with the particle which are constant in time. These quantities are known as integrals of motion. A simple example should make the idea clear. In a time-independent potential, the total energy of a particle in the potential is conserved and is an integral of motion. For potentials which are spherically symmetric, there are additional analytical integrals of motion. One is the magnitude of the angular momentum vector. Another is the z-component of the angular momentum vector.

As the geometrical symmetries of the potential vanish, the number of analytically known integrals decrease. If the potential is axisymmetric, then the z-component of the angular momentum, as well as the energy, are still integrals. The integral that disappears is the magnitude of the angular momentum vector. The interesting thing is that some axisymmetric potentials, like Richstone's scale-free logarithmic potential (Richstone 1982), still posess a third integral. However, the precise nature (dynamical or geometrical) and analytical form of this integral is unknown.

There are many different ways to attack the problem of the form of the third integral. One way is to try to find a coordinate system in which the Lagrangian has cyclic coordinates. For example, from the Lagrangian of a particle moving in a spherical potential written in cartesian coordinates, it is not at all obvious that the z-component of angular momentum is conserved. However, when the same Lagrangian is written in spherical coordinates, the azimuthal angle is cyclic and so it's conjugate momentum is conserved. This is the idea followed by Joel Tohline in his on-line book, "The Structure, Stability, and Dynamics of Self-Gravitating Systems".

Another method to crack the third integral is to utilize the fact that the Poisson bracket of the Hamiltonian and an interal of motion equals zero. In a general curvilinear coordinate system, the Hamiltonian will contain scale factors. If a particular form of integral is used (as done when using Noether's theorem), then some information about the scale factors may be gained. This information may lead to a nontrivial coordinate system that accomplishes the goal of the previous paragraph.