13.1: Introduction to Hamilton’s Principle of Least Action

In two papers published in 1834 and 1835, Hamilton announced a dynamical principle upon which it is possible to base all of mechanics, and indeed most of classical physics. Hamilton was seeking a theory of optics when he developed Hamilton’s Principle, plus the field of Hamiltonian mechanics, both of which play a pivotal role in classical mechanics.

Hamilton’s Principle is based on defining the action functional \(S\) of the \(n\) generalized coordinates \(q\) and their corresponding velocities \(\dot{q}\).

\[S = \int _ { t _ { 1 } } ^ { t _ { 2 } } L ( \mathbf { q } , \dot { \mathbf { q } } , t ) d t \label{13.1}\]

Action

The term action functional often is abbreviated to action. It is called Hamilton’s Principal Function in older texts.

The scalar quantity \(S\) is a functional of the Lagrangian \(L( q , \dot { q } , t )\). In principle, higher order time derivatives of the generalized coordinates could be included, but most systems in classical mechanics are described adequately by including only the generalized coordinates, plus their velocities. Note that the definition of the action functional does not limit the specific form of the Lagrangian. That is, it allows for more general Lagrangians than the standard Lagrangian

\[L ( \mathbf { q } , \dot { \mathbf { q } } , t ) = T ( \dot { \mathbf { q } } , t ) - U ( \mathbf { q } , t )\]

that was used throughout chapters 5 − 12. Hamilton stated that the actual trajectory of a mechanical system is given by requiring that the action functional is stationary. The action functional is stationary if the variational principle is written in terms of virtual infinitessimal displacement \(\delta\) to be

\[\delta S = \delta \int _ { t _ { 1 } } ^ { t _ { 2 } } L ( \mathbf { q } , \dot { \mathbf { q } } , t ) d t = 0\]

Typically this stationary point corresponds to a minimum of the action functional. Applying variational calculus to the action functional leads to the Lagrange equations of motion for the system. That is, Hamilton’s Principle, applied to the Lagrangian function \(L ( \mathbf { q } , \mathbf { \dot { q } } , t )\), generates the Lagrangian equations of motion.

\[\frac { d } { d t } \frac { \partial L } { \partial \dot { q } _ { j } } - \frac { \partial L } { \partial q _ { j } } = 0\]

These Lagrange equations agree with those derived using d’Alembert’s Principle, if the

\[\sum _ { k = 1 } ^ { m } \lambda _ { k } \frac { \partial g _ { k } } { \partial q _ { j } } ( \mathbf { q } , t ) + Q^{EX C}_j\]

generalized force terms are ignored.

Hamilton’s Principle can be considered to be the fundamental postulate of classical mechanics. It replaces Newton’s postulated three laws of motion. As illustrated in chapters 6 − 12, Lagrangian mechanics based on the standard Lagrangian \(L = T - U\) provides a remarkably powerful and consistent approach to solving the equations of motion in classical mechanics. This chapter extends the discussion to non-standard Lagrangians.

Chapter 5.12 developed a plausibility argument, based on Newton’s laws of motion, that led to the Lagrange equations of motion using the standard Lagrangian. d’Alembert’s Principle of virtual work was used in chapter 6 to provide a more fundamental derivation of Lagrange’s equations of motion which was based on the standard Lagrangian. An important feature is that Hamilton’s Principle extends Lagrangian mechanics to the use of non-standard Lagrangians.