principle of least actionprinciple
Even in antiquity, Heron of Alexandria noticed that light reflects at equal angles, as if seeking the shortest path. In 1744, Pierre Louis Maupertuis formulated a general principle: nature always acts in the most economical way. Leonhard Euler and Joseph-Louis Lagrange developed the mathematics: for mechanical systems there is a Lagrange function L (difference between kinetic and potential energy), and its integral over time — the action S — must be minimal on the real trajectory. William Hamilton reformulated mechanics in terms of action, and Emmy Noether showed that from the invariance (unchangingness) of the action under transformations follow conservation laws — of energy, momentum, charge. Today, the principle of least action is the foundation of all theoretical physics.
How it works
This principle explains why a thrown ball flies in a parabola: the parabola is the trajectory with the least action. It works everywhere: from planetary motion to scattering of elementary particles.
💡 Using this principle, Maxwell's equations for electromagnetism and the Schrödinger equation in quantum mechanics are derived.
S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt
S — action (functional on trajectories), L — Lagrangian (function of coordinates q, velocities \dot{q} and time t, usually the difference between kinetic and potential energy), t_1, t_2 — initial and final moments of time.
\delta S = 0
δ — variation, δS = 0 means that the action takes an extreme (usually minimum) value on the true trajectory.
\frac{d}{dt} \frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0
q — generalized coordinate, \dot{q} — its time derivative, L — Lagrangian. This equation describes the evolution of the system in time.