Introduction

When we talk about action in physics, we mean a single quantity––denoted $S$––that somehow “summarizes” the dynamics of a system over time. By finding the path that makes $S$ extremal (usually minimal), you automatically recover all of the familiar equations of motion. This idea, called the Principle of Least Action, sits at the heart of everything from classical mechanics to quantum field theory.

Before diving into modern uses of action, it’s worth taking a stroll through the 18th and 19th centuries to see how this elegant principle took shape.

Early Ideas (Mid-18th Century)

  • Pierre-Louis Maupertuis (1744)
    In 1744, Maupertuis proposed that nature operates in the “simplest” way possible, which he phrased as the quantity
    $$ S = \int_{t_1}^{t_2} p,\mathrm{d}q $$
    being minimal along the true path of a particle. Though his definition of $S$ wasn’t quite the modern one, the philosophical seed was planted.
    Reference: Maupertuis, P.-L. de (1744). Vénus physique.

  • Leonhard Euler (1748)
    Euler recast Maupertuis’ idea using what we’d now call a variational approach: he considered small “wiggles” of a path $q(t)$ and required $$ \delta S = 0 $$ to derive differential equations for $q(t)$.
    Reference: Euler, L. (1748). Methodus inveniendi lineas curvas…

Lagrange and the Mécanique Analytique (1788)

Joseph-Louis Lagrange refined these ideas in his landmark 1788 work. He defined the action functional $$ S[q] = \int_{t_1}^{t_2} L\bigl(q(t), \dot q(t), t\bigr),\mathrm{d}t $$ where $L$ is the Lagrangian (typically $L = T - V$, kinetic minus potential energy). Requiring $\delta S=0$ leads to the celebrated Euler–Lagrange equation: $$ \frac{\partial L}{\partial q} - \frac{\mathrm{d}}{\mathrm{d}t}!\Bigl(\frac{\partial L}{\partial \dot q}\Bigr) ;=; 0. $$ This single formula unifies Newton’s laws, Kepler’s orbits, small oscillations, and much more.
Reference: Lagrange, J.-L. (1788). Mécanique Analytique.

19th-Century Developments

  • William Rowan Hamilton (1834–1835)
    Hamilton introduced an alternative formulation via what we now call the Hamiltonian, $H(q,p,t)$, obtained by a Legendre transform of $L$. His principle of stationary action is mathematically equivalent but often more convenient for later developments.

  • Hamilton–Jacobi Theory
    By the mid-19th century, HJ theory recast mechanics as a first-order partial differential equation for a function $S(q,t)$. This paved the way toward both wave mechanics and, eventually, quantum theory.

Why It Matters Today

The 18th- and 19th-century pioneers showed that a single variational principle could encode all of classical mechanics. This deep unification is what makes “action” so powerful:

  • Generalization to fields: the same extremal-action idea applies to electromagnetic fields, fluids, and more.
  • Quantum foundations: Feynman’s path integral is nothing but a sum over all possible actions, weighted by $e^{iS/\hbar}$.
  • Modern geometry: symplectic and variational geometry trace their roots back to these classical formulations.

In upcoming posts, we’ll see how action evolves into Hamiltonian mechanics, modern geometric formulations, and ultimately quantum and field-theoretic frameworks.

References

  • Maupertuis, P.-L. de. 1744. Vénus physique.
  • Euler, L. 1748. Methodus inveniendi lineas curvas.
  • Lagrange, J.-L. 1788. Mécanique Analytique.
  • Hamilton, W.R. 1834–1835. “On a General Method in Dynamics.” Philosophical Transactions of the Royal Society.