Chapter 6: Lagrange, Hamilton & Variational Principles
1788–1843
Lagrange's Mécanique Analytique
Joseph-Louis Lagrange (1736–1813) published his Mécanique Analytique in 1788, declaring proudly that it contained “not a single diagram.” Physics was now pure analysis.
Lagrange reformulated all of mechanics using a single function, the Lagrangian\(L = T - V\) (kinetic minus potential energy), and the principle of least action:
\( \delta S = \delta \int_{t_1}^{t_2} L(q, \dot{q}, t)\,dt = 0 \)
This says: nature chooses the path that makes the action \(S\) stationary. From this single principle, all of Newton's mechanics, all of electromagnetism, all of general relativity, and the entire Standard Model can be derived.
Generalized Coordinates
Lagrange's key insight was generalized coordinates: instead of using\(x, y, z\) for every particle, you choose whatever coordinates \(q_1, q_2, \ldots, q_n\)naturally describe the system. A pendulum needs only one angle \(\theta\). A double pendulum needs two angles. The Lagrangian approach automatically handles constraints.
Example: The Pendulum
For a simple pendulum of length \(\ell\):\(L = \frac{1}{2}m\ell^2\dot{\theta}^2 + mg\ell\cos\theta\). The Euler-Lagrange equation gives \(\ddot{\theta} + \frac{g}{\ell}\sin\theta = 0\) — derived without ever mentioning forces, tensions, or constraints.
Hamilton's Revolution
William Rowan Hamilton (1805–1865) reformulated Lagrangian mechanics by introducing the Hamiltonian \(H = T + V\) (total energy) and replacing second-order Lagrange equations with first-order equations:
\( \dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i} \)
Hamilton's equations treat position \(q\) and momentum \(p\) on equal footing — they live in phase space, a 2n-dimensional space that encodes all possible states of a system.
Bridge to Quantum Mechanics
Hamilton's formulation is the classical skeleton on which quantum mechanics was built. The Hamiltonian becomes the energy operator \(\hat{H}\). Poisson brackets\(\{q,p\} = 1\) become commutators \([\hat{q},\hat{p}] = i\hbar\). Phase space becomes Hilbert space. Dirac called this “the most beautiful and suggestive analogy in all of physics.”
Hamilton's Optico-Mechanical Analogy
Hamilton originally developed his mechanics by analogy with optics. He showed that the equations of ray optics (Fermat's principle: light takes the path of least time) have exactly the same mathematical structure as the equations of mechanics (principle of least action). The Hamilton-Jacobi equation:
\( \frac{\partial S}{\partial t} + H\left(q, \frac{\partial S}{\partial q}, t\right) = 0 \)
This analogy between optics and mechanics was the key that Schrödinger used in 1925 to discover wave mechanics. If classical mechanics is like ray optics, Schrödinger reasoned, then quantum mechanics should be like wave optics. The Hamilton-Jacobi equation became the Schrödinger equation — one of the most extraordinary mathematical bridges in history.
Poisson Brackets & Symplectic Geometry
Siméon Denis Poisson introduced the Poisson bracket:
\( \{f,g\} = \sum_i \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right) \)
This algebraic structure on phase space functions turned out to be the classical limit of quantum commutators. The geometry of phase space — symplectic geometry — became a major branch of modern mathematics, with applications from topology to string theory. What Lagrange and Hamilton built for 18th-century celestial mechanics now underlies 21st-century mathematical physics.