Euler's Solution to the Action Problem

The Origins: Maupertuis and the "Least Action" Principle.

In the 1740s, Pierre-Louis Moreau de Maupertuis, a French polymath and president of the Berlin Academy of Sciences, proposed a bold philosophical idea:

“Nature is economical. It does nothing in vain. It chooses the path that requires the least effort.”

He formulated this as the Principle of Least Action, suggesting that physical systems evolve in such a way that a certain quantity—action—is minimized.

But Maupertuis was vague. He claimed this principle explained everything from optics to planetary motion but provided no real mathematics. Some even accused him of cloaking philosophy in scientific language to gain prestige.

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Enter Euler: Turning Philosophy into Mathematics

Maupertuis’s vague principle reached Leonhard Euler, already one of the greatest mathematicians of his time and also working at the Berlin Academy. Unlike Maupertuis, Euler wanted rigor.

Euler had been working independently on the calculus of variations, inspired by earlier problems like the brachistochrone (the curve of fastest descent) tackled by the Bernoullis.

In 1744, Euler published his "Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes" ("Method for Finding Curved Lines Enjoying Properties of Maxima or Minima")—a masterpiece. Here he formally laid down the Euler–Lagrange equation, providing the mathematical machinery to make Maupertuis’s vague idea precise.

Euler took the philosophical musings about nature’s economy and gave them teeth—hard, analytical mathematics.

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Lagrange: The Next Generation

A few decades later, a young Italian mathematician named Joseph-Louis Lagrange refined and expanded Euler’s ideas. Without the geometric crutches of earlier work, he formalized mechanics using only generalized coordinates and energy functions—leading to the birth of Lagrangian mechanics.

Lagrange credited Euler, saying that his own methods were extensions of Euler’s vision, but the focus had now shifted from curves in space to abstract configuration spaces—an early step toward modern theoretical physics.

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Philosophy, Prestige, and Petty Feuds

Maupertuis and Euler had a somewhat tense relationship at the Berlin Academy. Maupertuis saw Euler as a subordinate mathematician, while Euler increasingly viewed Maupertuis’s claims as unscientific bluster. This tension spilled into the infamous "Vis viva controversy"—a debate over the correct formulation of energy—where Euler’s precise methods eventually won out.

Voltaire even mocked Maupertuis for his vagueness, and Euler, ever the tactful scientist, quietly buried his rival’s reputation by building a framework that actually worked.

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Legacy: From Classical Paths to Quantum Waves

Euler’s resolution of the action problem became:

The backbone of classical mechanics,

The philosophical underpinning of general relativity,

And the literal playground for quantum physics, where the path integral formulation by Feynman treats the action as central.

While Newton told us how things move, Euler (with Maupertuis's spark) told us why they move the way they do—because that’s the path nature “chooses.”

-- Pavitra Kanetkar