Canonical Transformations

Abstract

Let q₁, q₂,... , q _N , p₁.p₂,... p _N be 2N independent canonical variables, which satisfy Hamilton’s equations:

$$ {\dot q_i} = \frac{{\partial H}}{{\partial {p_i}}},\;\;\;{\dot p_i} = - \frac{{\partial H}}{{\partial {p_i}}},\;\;\;i = 1,\;2,\; \ldots ,N\;{\rm{.}} $$

(4.1)

We now transform to a new set of 2N coordinates Q₁,... Q _N , P₁,... P _N , which can be expressed as functions of the old coordinates:

$$ {Q_i} = {Q_i}({q_i},{p_i};\;t)\;,\;\;\;{P_i} = {P_i}({q_i},{p_i};\;t)\;{\rm{.}} $$

(4.2)

These transformations should be invertible. The new coordinates Q _i , P _i are then exactly canonical if a new Hamiltonian K(Q, P, t) exists with

$$ {\dot Q_i} = \frac{{\partial K}}{{\partial {P_i}}}\;,\;\;\;\;{\dot P_i} = - \frac{{\partial K}}{{\partial {Q_i}}}\;{\rm{.}} $$

(4.3)

Our goal in using the transformations (4.2) is to solve a given physical problem in the new coordinates more easily. Canonical transformations are problem-independent; i.e., (Q _i , P _i ) is a set of canonical coordinates for all dynamical systems with the same number of degrees of freedom, e.g., for the two-dimensional oscillator and the two-dimensional Kepler problem. Strictly speaking, for fixed N, the topology of the phase space can still be different, e.g., ℝ^2N, ℝⁿ x (S¹)^m, n + m = 2N etc.