Canonical Transformations and the Hamilton-Jacobi Equation
We have seen that the equations of motion for dynamical systems can take a variety of forms depending upon the generalized coordinates used. It will also be recalled that when ignorable coordinates arise the associated equations may be partially integrated. Recognizing the equivalence of different forms and attributing the presence of ignorable coordinates to the particular set of generalized coordinates used, it is natural to enquire whether it is possible to transform the equations of motion, expressed in terms of a given set of coordinates, to another set of coordinates for which all the generalized coordinates are ignorable. In this way the problem of solving the equations of motion reduces to a transformation. The researches of Hamilton and Jacobi revealed that the required transformation may be obtained from a single function which is governed by a differential equation referred to as the Hamilton-Jacobi equation.
KeywordsCanonical Transformation Transformation Equation Momentum Vector Generalize Displacement Total Time Derivative
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