Application of the Action Principles

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(t_1,2) = 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

$$ \int_{t1}^{t2} {dt} \left[ {\frac{d}{{dt}}{p_i} + \frac{{\partial V}}{{\partial {x_i}}}} \right]\;\delta {\varepsilon _i} = 0\;. $$