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14.S: Hamilton's Principle of Least Action (Summary)

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  • This chapter introduced Hamilton’s use of least action to derive Hamilton’s Principle, and its application to Lagrangian and Hamiltonian mechanics. Gauge invariance of the Lagrangian was discussed. The concept of alternative standard, and non-standard, Lagrangians was introduced and their applicability was illustrated. The following summarizes the conclusions.

    Hamilton’s Principle. Hamilton’s Principle is based on use of variational calculus to determine the equa- tions of motion for which the action functional \(S\) has a stationary solution, where

    \[ S = \int _ { t _ { 1 } } ^ { t _ { 2 } } L ( \mathbf { q } , \dot { \mathbf { q } } , t ) d t\]

    That is

    \[\delta S = \delta \int _ { t _ { 1 } } ^ { t _ { 2 } } L d t = 0 \label{13.2}\]

    Hamilton’s Principle of least action leads directly to the Lagrange-Euler equations without assuming that the Lagrangian is of the standard form. That is, Hamilton’s Principle allows for a wide range of allowable functional forms for the Lagrangian.

    Hamilton’s Principle leads to a direct relation between the generalized momentum and the action.

    \[p _ { j } = \frac { \partial S } { \partial q _ { j } }\]

    It was shown that Hamilton’s Principle of least action predicts Hamilton’s equations of motion
    \[\dot { p } _ { j } + \frac { \partial H } { \partial q _ { j } } = 0\]


    \[- \dot { q } _ { j } + \frac { \partial H } { \partial p _ { j } } = 0\]

    In addition, it predicts the Hamiltonian-Jacobi equation.

    \[\frac { \partial S } { \partial t } + H ( \mathbf { q } , \mathbf { p } , t ) = 0\]

    Gauge invariance of the standard Lagrangian: It was shown that there is a continuum of equivalent standard Lagrangians that lead to the same set of equations of motion for a system. This feature is related to gauge invariance in mechanics. The following transformations change the standard Lagrangian, but leave the equations of motion unchanged.

    1. The Lagrangian is indefinite with respect to addition of a constant to the scalar potential which cancels out when the derivatives in the Euler-Lagrange differential equations are applied.
    2. Similarly the Lagrangian is indefinite with respect to addition of a constant kinetic energy.
    3. The Lagrangian is indefinite with respect to addition of a total time derivative of the form \(L + \frac { d } { d t } \left[ \Lambda \left( q _ { i } , t \right) \right]\) for any differentiable function \(\Lambda \left( q _ { i } t \right)\) of the generalized coordinates, plus time, that has continuous second derivatives.

    Non-standard Lagrangians: The flexibility and power of Lagrangian mechanics can be extended to a broader range of dynamical systems by employing an extended definition of the Lagrangian that is allowed by Hamilton’s variational action principle, equation \ref{13.2}. It was illustrated that the inverse variational calculus formalism can be used to identify non-standard Lagrangians that generate the required equations of motion. These non-standard Lagrangians can be very different from the standard Lagrangian and do not separate into kinetic and potential energy components. These alternative Lagrangians can be used to handle dissipative systems which are beyond the range of validity when using standard Lagrangians. That is, it was shown that several very different Lagrangians and Hamiltonians can be equivalent for generating useful equations of motion of a system. Currently the use of non-standard Lagrangians is a narrow, but active, frontier of classical mechanics.