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Fermat's solution was a landmark in that it unified the then-known laws of geometrical optics under a variational principle or action principle, setting the precedent for the principle of least action in classical mechanics and the corresponding principles in other fields (see History of variational principles in physics). [42]
In quantum mechanics, action and quantum-mechanical phase are related via the Planck constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions. In 1948, Feynman discovered the path integral formulation extending the principle of least action to quantum mechanics for electrons and ...
In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis, 1698 – 1759) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). [1] It is a special case of the more generally stated principle of least action.
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The path for which action is least is the path taken by the system. From this principle, all equations of motion in classical mechanics can be derived. This approach can be extended to fields rather than a system of particles (see below), and underlies the path integral formulation of quantum mechanics , [ 11 ] [ 12 ] and is used for ...
The names of action principles have evolved over time and differ in details of the endpoints of the paths and the nature of the variation. Quantum action principles generalize and justify the older classical principles. Action principles are the basis for Feynman's version of quantum mechanics, general relativity and quantum field theory.
The action is a functional rather than a function, since it depends on the Lagrangian, and the Lagrangian depends on the path q(t), so the action depends on the entire "shape" of the path for all times (in the time interval from t 1 to t 2). Between two instants of time, there are infinitely many paths, but one for which the action is ...
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