Principle of Least Action

Drag the path — watch action S = ∫(T−V)dt change. Nature always picks the minimum.

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Action S (your path)

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S_min (true path)

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S / S_min

Projectile

Scenario

Mass: 2 kg Gravity: 9.8 m/s²

Hamilton's Principle: nature evolves along paths that make the action S = ∫L dt stationary. The Euler-Lagrange equations ∂L/∂q − d/dt(∂L/∂q̇) = 0 are satisfied exactly on the true (green) path. All other paths have larger action.