Hamilton's Principle and Lagrangian Mechanics

Hamilton's principle reformulates Newtonian mechanics as a variational problem: the true trajectory of a mechanical system extremizes the action functional. This perspective unifies classical mechanics, reveals deep connections to symmetry and conservation, and generalizes seamlessly to field theory and quantum mechanics.

The Action Principle

Definition7.4Hamilton's Principle of Stationary Action

For a mechanical system with generalized coordinates $q = (q_1,\ldots,q_n)$ and Lagrangian $L(q,\dot{q},t) = T - V$ (kinetic minus potential energy), the action is $S[q] = \int_{t_1}^{t_2} L(q(t),\dot{q}(t),t)\,dt$ . Hamilton's principle states that the physical trajectory satisfies $\delta S = 0$ : among all paths $q(t)$ with fixed endpoints $q(t_1) = q_a$ , $q(t_2) = q_b$ , the actual motion makes $S$ stationary. This yields the Euler-Lagrange equations $\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0$ for $i = 1,\ldots,n$ .

ExampleLagrangians for Physical Systems

Simple pendulum: $L = \frac{1}{2}m\ell^2\dot{\theta}^2 + mg\ell\cos\theta$ , giving $\ddot{\theta} + (g/\ell)\sin\theta = 0$ . Central force: $L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2) - V(r)$ in polar coordinates; $\theta$ is cyclic, so $p_\theta = mr^2\dot{\theta} = \text{const}$ (angular momentum conservation). Charged particle in EM field: $L = \frac{1}{2}m\dot{\mathbf{r}}^2 - q\phi + q\dot{\mathbf{r}}\cdot\mathbf{A}$ where $\phi, \mathbf{A}$ are the electromagnetic potentials. The Euler-Lagrange equations give the Lorentz force law.

Constraints and Lagrange Multipliers

Definition7.5Constrained Variational Problems

For a functional $J[y]$ subject to the isoperimetric constraint $K[y] = \int_a^b G(x,y,y')\,dx = \ell$ (constant), introduce the augmented functional $\tilde{J}[y] = J[y] - \lambda K[y] = \int_a^b [L - \lambda G]\,dx$ where $\lambda$ is a Lagrange multiplier. The extremal satisfies the Euler-Lagrange equation for $L - \lambda G$ , and $\lambda$ is determined by the constraint $K = \ell$ . For holonomic constraints $g(q,t) = 0$ in mechanics: the Lagrange multiplier $\lambda$ represents the constraint force.

ExampleClassical Isoperimetric Problem

Among all closed curves of length $\ell$ , find the one enclosing maximum area. Parametrize by arclength: maximize $A = \frac{1}{2}\oint(x\,dy - y\,dx)$ subject to $\oint ds = \ell$ . The augmented Lagrangian gives $\kappa = \lambda = \text{const}$ (constant curvature), so the optimal curve is a circle. The isoperimetric inequality $4\pi A \leq \ell^2$ follows, with equality only for circles.

Hamiltonian Formulation

Definition7.6Legendre Transform and Hamilton's Equations

The conjugate momenta are $p_i = \partial L/\partial\dot{q}_i$ . The Hamiltonian is $H(q,p,t) = \sum_i p_i\dot{q}_i - L(q,\dot{q},t)$ (Legendre transform). Hamilton's principle in phase space yields Hamilton's equations: $\dot{q}_i = \frac{\partial H}{\partial p_i}$ , $\dot{p}_i = -\frac{\partial H}{\partial q_i}$ . These are $2n$ first-order ODEs equivalent to $n$ second-order Euler-Lagrange equations. The Hamiltonian generates time evolution: $\dot{F} = \{F,H\} + \partial F/\partial t$ where $\{F,G\} = \sum_i(\partial F/\partial q_i \partial G/\partial p_i - \partial F/\partial p_i \partial G/\partial q_i)$ is the Poisson bracket.

RemarkFrom Classical to Quantum via the Action

In Feynman's path integral formulation of quantum mechanics, the quantum propagator is $K(q_b,t_b;q_a,t_a) = \int\mathcal{D}[q]\,e^{iS[q]/\hbar}$ where the integral is over all paths from $(q_a,t_a)$ to $(q_b,t_b)$ , not just the classical extremal. In the classical limit $\hbar \to 0$ , the stationary phase approximation selects the classical path $\delta S = 0$ , recovering Hamilton's principle. The Lagrangian and Hamiltonian structures thus bridge classical and quantum physics.