Poincaré and the shortcomings of the Hamilton-Jacobi method for classical or quantized mechanics

Abstract

This chapter discusses the limits of accuracy in classical mechanics and explains the foundations of statistical mechanics. The canonical equations of celestial mechanics do not admit, except for some exceptional cases, any analytical and uniform integral besides the energy integral. The Poincare theorem proves that the (n − 1) integrals have very different properties. The total energy is the only expression that is represented by a well-behaved mathematical function, and all other so-called integrals are not analytical; they can have a very strange behavior, with discontinuities, and cannot be compared with the energy. The integrals representing momentum or moment of momentum are comprised in the other exceptional cases. When variables cannot be separated, the problem becomes much more involved. The Hamilton–Jacobi equations yield a possibility of finding angular variables w k and the corresponding action variables J K . These variables are often called Delaunay variables. The Hamilton–Jacobi procedure has never been a way to solve a practical problem. It is very useful for general theorems and elegant discussions but not for actual solution.