Hamilton's principle

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Hamilton's principle, also called the principle of least action, states that a system evolves in time along a trajectory q_i(t)\, between two configurations q_{i,0} = q_i(t_0)\, and q_{i,1} = q_i(t_1)\,, such that the action functional,

S[ q(t) ] \equiv \int_{t_0}^{t_1}\!dt\, L(q, \dot{q})\,

is stationary. To clarify, the action maps functions q_i(t)\, over the domain [t_0, t_1]\, onto \mathbb{R}\,. The principle is the statement that the functional derivative of S[q_i]\, with respect to q_i(t)\, vanishes if q_i(t)\, describes the physical evolution of the system.

Derivation

Consider all continuous and differentiable trajectories q_i(t)\, between two configurations q_{i,0} = q_i(t_0)\, and q_{i,1} = q_i(t_1)\,. To define the functional derivative at q_i(t)\,, let a neighbouring trajectory be q_i(t) + \epsilon \alpha_i(t)\,, where \alpha_i(t_0) = \alpha_i(t_1) = 0\,. We are therefore looking for solutions q_i(t)\, such that

\left.\frac{d}{d\epsilon} S[q_i(t)+\epsilon \alpha_i(t)] \right|_{\epsilon= 0} = 0\,.

For the sake of brevity, one normally writes the variation as q_i(t) + \delta q_i(t)\,, so that this is written

\frac{\delta S}{\delta q_i} = 0\,.

Thus

\delta S\, = \int_{t_0}^{t_1}\!dt\,\frac{\partial L}{\partial q_i(t)}\delta q_i(t) +\frac{\partial L}{\partial \dot q_i(t)}\delta \dot q_i(t)\,,
= \int_{t_0}^{t_1}\!dt\,\frac{\partial L}{\partial q_i(t)}\delta q_i(t) + \frac{d}{dt}\left( \frac{\partial L}{\partial \dot q_i(t)}\delta q_i(t) \right) - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i(t)} \right)\delta q_i(t)\,,
= \left. \frac{\partial L}{\partial \dot q_i(t)}\delta q_i(t)\right|_{t_0}^{t_1} + \int_{t_0}^{t_1}\!dt\,\left( \frac{\partial L}{\partial q_i(t)} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i(t)} \right) \right]\delta q_i(t)\,,
= \int_{t_0}^{t_1}\!dt\,\left( \frac{\partial L}{\partial q_i(t)} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i(t)} \right) \right]\delta q_i(t)\,, since \delta q_i(t_0) = \delta q_i(t_1) = 0\,.

Since the variation \delta q_i\, is arbitrary, by the fundamental lemma of calculus of variations, the only way \delta S\, can vanish is if the term in brackets vanishes identically:

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i} \right) - \frac{\partial L}{\partial q_i} = 0\,,

which is just the statement that the trajectory q_i(t)\, is a solution to the Euler-Lagrange equations.

back to Lagrangian
on to cyclic variables

References

Refs [1], [2], from the collection Sir William Rowan Hamilton (1805-1865): Mathematical Papers edited by David R. Wilkins, School of Mathematics, Trinity College, Dublin 2, Ireland. (2000); also reviewed as On a General Method in Dynamics.

  1. ↑ W.R. Hamilton (1834). "On a General Method in Dynamics, Part I". Philosophical Transaction of the Royal Society. 
  2. ↑ W.R. Hamilton (1835). "On a General Method in Dynamics, Part II". Philosophical Transaction of the Royal Society. 
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