Lagrangian Mechanics

Motivation

Newton’s form $\mathbf{F} = m\ddot{\mathbf{r}}$ is powerful, but for systems with constraints — pendulums, rigid bodies, rotating frames — choosing generalized coordinates $q_i$ collapses the equations into something dramatically cleaner. Lagrangian mechanics gives that framework.

Action and the Lagrangian

Describe the state by generalized coordinates $q = (q_1,\dots,q_n)$. The Lagrangian is the difference between kinetic energy $T$ and potential $V$:

$$L(q,\dot q,t) = T(q,\dot q,t) - V(q,t).$$

The action $S$ is a functional of the path $q(t)$:

$$S[q] = \int_{t_1}^{t_2} L\big(q(t),\dot q(t),t\big)\,dt.$$

The principle of least action

Among paths with fixed endpoints $q(t_1), q(t_2)$, the physical path makes $S$ stationary:

$$\delta S = 0.$$

Carrying the variation through gives

$$\delta S = \int_{t_1}^{t_2} \sum_i \left( \frac{\partial L}{\partial q_i} - \frac{d}{dt}\frac{\partial L}{\partial \dot q_i} \right) \delta q_i \, dt = 0.$$

Since each $\delta q_i$ is independent, the integrand must vanish identically. The Euler–Lagrange equations:

$$\boxed{\;\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = 0.\;}$$

Example: simple pendulum

Length $\ell$, mass $m$, angle $\theta$:

$$T = \tfrac{1}{2} m \ell^2 \dot\theta^2,\quad V = -m g \ell \cos\theta.$$ $$L = \tfrac{1}{2} m \ell^2 \dot\theta^2 + m g \ell \cos\theta.$$

Apply Euler–Lagrange:

$$\frac{d}{dt}(m \ell^2 \dot\theta) + m g \ell \sin\theta = 0 \quad\Longrightarrow\quad \ddot\theta + \frac{g}{\ell}\sin\theta = 0.$$

In the small-angle limit $\sin\theta \approx \theta$, this becomes simple harmonic motion with $\omega = \sqrt{g/\ell}$.

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Symmetries and conservation: Noether’s theorem

Each continuous symmetry of $L$ implies a conserved quantity.

Symmetry	Conserved quantity
Time translation	Energy $E = \sum_i \dot q_i \frac{\partial L}{\partial \dot q_i} - L$
Space translation	Momentum $p_i = \frac{\partial L}{\partial \dot q_i}$
Rotation	Angular momentum

The same logic threads through field theory, quantum mechanics, and the Standard Model.

What to read next

• Maxwell’s Equations — recast as a Lagrangian density on spacetime.
• Schrödinger Equation — Feynman’s path integral quantizes the principle of least action.
• Lorentz Transformation — the relativistic action $S = -mc^2 \int d\tau$ is built from invariants.