# Canonical Transformations

• Walter Dittrich
• Martin Reuter

## 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., ℝ2N, ℝ n x (S1) m , n + m = 2N etc.

Torque Librium