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Hamiltonian Mechanics

  • Nicholas M. J. WoodhouseEmail author
Chapter
  • 2.8k Downloads
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

Lagrange’s equations
$$\frac{\mathrm{d}} {\mathrm{d}t}\left ( \frac{\partial L} {\partial {v}_{a}}\right ) - \frac{\partial L} {\partial {q}_{a}} = 0,\qquad \frac{\mathrm{d}{q}_{a}} {\mathrm{d}t} = {v}_{a}$$
(7.1)
determine the trajectories in the extended phase space of a holonomic system subject to conservative forces. We have seen that it is possible to simplify the dynamical analysis by making coordinate transformations of the form
$$\begin{array}{lll} \tilde{{q}}_{a}& =&\tilde{{q}}_{a}(q,t)\\ \\ \\ \tilde{{v}}_{a}& =&\tilde{{v}}_{a}(q,v,t) = \frac{\partial \tilde{{q}}_{a}} {\partial {q}_{b}}{v}_{b} + \frac{\partial \tilde{{q}}_{a}} {\partial t}\\ \\ \\ \tilde{t} & =&t,\end{array}$$
(7.2)
a technique that proved particularly useful for handling constraints.

Keywords

Partial Derivative Analytical Dynamics Poisson Bracket Canonical Transformation Jacobi Equation 
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 London 2009

Authors and Affiliations

  1. 1.Mathematical Institute University of OxfordOxfordUnited Kingdom

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