1.6 The variational view: Lagrangian mechanics

Newton’s $\mathbf F = m\mathbf a$ has a deep reformulation as a principle about whole trajectories rather than instantaneous forces. The two are equivalent for a point particle, but the variational form generalises cleanly to constrained systems and to fields, and it reveals the conservation laws as consequences of symmetry.

The principle of stationary action

Define the Lagrangian as the difference of kinetic and potential energy, $L = T - U$ , and the action as its time-integral along a candidate trajectory,

S[\mathbf r(t)] \;=\; \int_{t_1}^{t_2} L\, dt.

Of all the trajectories that connect the fixed endpoints $\mathbf r(t_1)$ and $\mathbf r(t_2)$ , the one a particle actually follows is the one that makes the action stationary — a maximum, minimum, or saddle under small variations of the path. Setting the first variation $\delta S = 0$ yields the Euler–Lagrange equation, one for each coordinate $q$ :

\frac{d}{dt}\frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q} \;=\; 0.

▶ Euler–Lagrange recovers F = ma Derivation

For a single particle in a potential, $L = \tfrac12 m\dot q^2 - U(q)$ . The two pieces of the Euler–Lagrange equation are

\frac{\partial L}{\partial \dot q} = m\dot q \;\Rightarrow\; \frac{d}{dt}\frac{\partial L}{\partial \dot q} = m\ddot q, \qquad \frac{\partial L}{\partial q} = -\frac{dU}{dq} = F.

Substituting into $\dfrac{d}{dt}\dfrac{\partial L}{\partial\dot q} - \dfrac{\partial L}{\partial q} = 0$ gives

m\ddot q - F = 0 \quad\Longleftrightarrow\quad F = m\ddot q.

Newton’s second law is exactly the condition for the action to be stationary. ✓

Why bother

For a free particle in Cartesian coordinates the variational route is more work than Newton. Its advantages appear elsewhere:

Constraints. A bead on a wire or a pendulum on a rod is awkward in Newtonian terms, because the constraint forces are unknown. Choosing generalised coordinates that already respect the constraint (the angle of the pendulum, the position along the wire) makes the constraint forces drop out — they do no work, so they never enter $L$ .
Symmetry and conservation. Each symmetry of the Lagrangian gives a conserved quantity (Noether’s theorem): invariance under time-translation gives energy conservation, under spatial translation gives momentum, under rotation gives angular momentum. The conservation laws of the earlier lessons are corollaries.
Fields. Replacing the discrete coordinates by a continuous field and $L$ by an integral of a Lagrangian density extends the whole machinery to continuum systems, giving a variational route to the wave equation and the other field equations.

The Hamiltonian formulation is a further step, trading the coordinates and velocities $(q,\dot q)$ for coordinates and momenta $(q,p)$ and replacing the second-order Euler–Lagrange equation with Hamilton’s two first-order equations. It is the natural language for phase space, for the connection to statistical mechanics, and for the transition to quantum mechanics.

⏳ The history — Newton, Euler, Lagrange, and the refinement of mechanics

Isaac Newton’s Philosophiæ Naturalis Principia Mathematica (1687) does not state his three laws in the form taught today. It states them in Latin prose — Lex I, Lex II, Lex III — and then uses them through geometric demonstrations in the style of Euclid, with no calculus notation: every theorem is proved by limits of inscribed and circumscribed figures.

The modern algebra of mechanics post-dates Newton. Leonhard Euler (1736, Mechanica) was the first to write mechanics systematically as differential equations. Jean d’Alembert (1743) recast dynamics as a principle of virtual work; Joseph-Louis Lagrange (Mécanique analytique, 1788) reduced all of mechanics to a single variational principle and the equation that bears his name, famously boasting that his treatise contained not a single diagram; William Rowan Hamilton (1834) gave the phase-space formulation. Each reformulation is mathematically equivalent to Newton’s three laws, but each makes a different structure manifest — constraints for Lagrange, phase space and conservation for Hamilton — and together they are the apparatus on which the rest of theoretical physics is built.

Read the original: Philosophiæ Naturalis Principia Mathematica (Isaac Newton (tr. Andrew Motte, 1729), 1687)