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Hamilton's Principle

Source lecture(s): PHY622 Lec2

Intuition

Physics chooses the path that makes the action stationary, not the path of greatest or least anything.

Formal Definition

The action \(S=\int_{t_0}^{t_1} L(q,\dot{q},t)\,dt\) is stationary for the true trajectory.

Mathematical Formulation

\[\delta S = \delta\int_{t_0}^{t_1} L(q,\dot{q},t)\,dt = 0\]

Derivation

Apply Euler-Lagrange to the Lagrangian \(L=T-V\). The resulting equations are Newton's second law in generalized coordinates.

Worked Example

For a simple harmonic oscillator, \(L=\frac{1}{2}m\dot{x}^2-\frac{1}{2}kx^2\), giving \(m\ddot{x}+kx=0\).

Common Mistakes

  • Thinking 'stationary' means 'minimum'—saddles exist.
  • Applying to dissipative systems without modification.

Quiz

Q1. Is Hamilton's principle valid for dissipative forces?

Answer

No, without adding Rayleigh's dissipation function.