Application of the Action Principles

  • Walter Dittrich
  • Martin Reuter

Abstract

We begin this chapter by deriving a few laws of nonconservation in mechanics. To this end we first consider the change of the action under rigid space translation δx i = δε i and δt(t1,2) = 0. Then the noninvariant part of the action,
$$ S = \;\int_{t1}^{t2} {dt} \left[ {{p_i}\frac{{d{x_i}}}{{dt}} - \frac{{p_i^2}}{{2m}} - V({x_i},t)} \right] $$
(2.1)
is given by
$$ \delta V({x_i},\;t) = \frac{{\partial V}}{{\partial {x_i}}}\delta {x_i} $$
and thus it immediately follows for the variation of S that
$$ \delta S = \int_{t1}^{t2} {dt} \left[ { - \frac{{\partial V({x_i},\;t)}}{{\partial {x_i}}}\delta {x_i}} \right] = \;{G_2} - {G_1} = \int_{t1}^{t2} {dt} \frac{d}{{dt}}({p_i}\delta {x_i})\;, $$
or
$$ \int_{t1}^{t2} {dt} \left[ {\frac{d}{{dt}}{p_i} + \frac{{\partial V}}{{\partial {x_i}}}} \right]\;\delta {\varepsilon _i} = 0\;. $$

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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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