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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.

Keywords

Canonical Transformation Canonical Equation Exchange Transformation Rotate Reference System Eoordinate System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Walter Dittrich
    • 1
  • Martin Reuter
    • 2
  1. 1.Institut für Theoretische PhysikUniversität TübingenTübingenFed. Rep. of Germany
  2. 2.Institut für Theoretische PhysikUniversität HannoverHannover 1Fed. Rep. of Germany

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