Recap of Historical Context

Last time we saw how Maupertuis first guessed that nature “chooses” the simplest path, and how Euler began tinkering with small variations of a curve to turn that poetic idea into real equations. Now we’ll dig into the nuts and bolts of those variational methods and see how Lagrange polished them into the form we use today.

Euler’s Variational Methods

Euler’s key insight was: if you wiggle a path $q(t)$ ever so slightly, the true path should make the first-order change in your action vanish. Concretely, suppose you define $$ S[q] = \int_{t_1}^{t_2} F\bigl(q(t),\dot q(t)\bigr)\mathrm{d}t, $$ where $F$ is some function of the position $q$ and velocity $\dot q$. Replace $q(t)$ by $q(t) + \varepsilon\eta(t)$, with $\eta(t_1)=\eta(t_2)=0$, and expand:

$$ \delta S = \varepsilon \int_{t_1}^{t_2} \Bigl(\frac{\partial F}{\partial q}\eta + \frac{\partial F}{\partial \dot q}\dot\eta\Bigr)\mathrm{d}t + \mathcal{O}(\varepsilon^2). $$

Integrating the $\dot\eta$ term by parts and using $\eta\bigl(t_1,t_2\bigr)=0$ turns this into

$$ \delta S = \varepsilon \int_{t_1}^{t_2} \Bigl(\frac{\partial F}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\bigl(\tfrac{\partial F}{\partial \dot q}\bigr)\Bigr)\eta\mathrm{d}t. $$

Since $\eta(t)$ is arbitrary, the bracket must vanish:

$$ \frac{\partial F}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\frac{\partial F}{\partial \dot q}\Bigr) = 0. $$

That relation, usually called the Euler–Lagrange equation, is the backbone of variational mechanics.

Lagrange’s Synthesis

Lagrange took Euler’s general method and chose a very specific $F$, namely the Lagrangian: $$ L(q,\dot q,t) = T(\dot q) - V(q), $$ where $T$ is kinetic energy and $V$ is potential energy. Plugging $F=L$ into the Euler–Lagrange equation gives $$ \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\frac{\partial L}{\partial \dot q}\Bigr) = 0, $$ or more explicitly, $$ \frac{\partial V}{\partial q} + \frac{\mathrm{d}}{\mathrm{d}t}(m\dot q) = 0 \quad\Longrightarrow\quad m\ddot q + V’(q) = 0, $$ which is just Newton’s second law $m\ddot q = -V’(q)$. The magic is that one equation covers everything: from springs to planets.

Applications in Classical Mechanics

Simple Pendulum

For a pendulum of length $\ell$ and mass $m$, angle $\theta$: $$ T = \tfrac12m(\ell\dot\theta)^2,\quad V = m g \ell (1 - \cos\theta). $$ Thus $$ L = \tfrac12 m\ell^2\dot\theta^2 - mg\ell\bigl(1-\cos\theta\bigr), $$ and the Euler–Lagrange equation yields $$ m\ell^2\ddot\theta + mg\ell\sin\theta = 0, $$ the familiar pendulum equation.

Planetary Orbits

In polar coordinates $(r,\phi)$ with central potential $V(r)$: $$ L = \tfrac12 m\bigl(\dot r^2 + r^2\dot\phi^2\bigr) - V(r). $$ Two Euler–Lagrange equations (for $r$ and $\phi$) give conservation of angular momentum and the radial equation that leads to Kepler’s laws. No need to juggle vector forces, everything pops out of one tidy package.

References & Further Reading

  • Euler, L. (1748). Methodus inveniendi lineas curvas…
  • Lagrange, J.-L. (1788). Mécanique Analytique.
  • Goldstein, H. (1980). Classical Mechanics. Addison–Wesley.
  • Lanczos, C. (1970). The Variational Principles of Mechanics. University of Toronto Press.