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Principle of least action
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Actual paths make the action stationary.
S = ∫L dtThe actual path of a system makes the action S = ∫L dt stationary compared with neighbouring paths. This unifies mechanics, optics (Fermat) and field theories.
A thrown ball follows the parabola that extremises S.
It is "stationary", not always a minimum.
In more detail
The principle of least action says that the actual path of a system is the one that makes the action, the integral of the Lagrangian over time, stationary. It unifies mechanics, electromagnetism and quantum theory.
Real-world uses
- Physicists use the action principle to derive the laws of gravity and electromagnetism.
- Quantum path integrals sum over paths weighted by their action.
- The principle of least action explains why light takes the quickest path between two points, which is the basis of optics.
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Frequently asked questions
Does a particle really test all paths?
In quantum mechanics, many paths contribute. In classical mechanics, the actual path makes the action stationary.
Is least action always minimal?
Not always. The action is stationary, which may be a minimum, maximum or saddle point.
Does a particle really try all paths?
No. The principle says the actual path makes the action stationary, a more compact way to describe the motion.
Related topics in Work, Energy & Power
Hamiltonian as total energy · Lagrangian mechanics intro · Euler–Lagrange equations · Noether's theorem