The important role of invariance principles in physical theory was recognized only relatively recently, after the development of analytical dynamics. In classical mechanics, the equations of motion are completely solved if one knows all the constants of motion, therefore the problem of solving the equations of motion is the same as the problem of finding all the “angle” variables ϕ k , viz those coordinates which the Hamiltonian is independent of, since the canonically conjugate “action” variables J k are constant in time. Since the J k are also the generators of infinitesimal canonical transformations which induce changes of the ϕ k alone, recognition of all the invariance transformations of the Hamiltonian, and identification of the corresponding generators, is equivalent to a complete solution of the equations of motion. In quantum theory, the situation is a little more complex in that there is a limitation of principle in the extent to which one can specify the state of a dynamical system, and therefore a fortiori, in the manner in which one can trace its evolution in time. The corresponding statement is that, if one can find a maximal set of commuting operators, each of which commutes with the Hamiltonian, then the states which are simultaneous eigenstates of all those operators have a particularly simple time dependence.
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