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Maupertuis's least action principle is written mathematically as the stationary condition = on the abbreviated action [] = , (sometimes written ), where = (,, …,) are the particle momenta or the conjugate momenta of generalized coordinates, defined by the equation = ˙, where (, ˙,) is the Lagrangian.
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
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 ...
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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.
The book is a mathematical introduction to various theoretical physics concepts, such as principle of least action, Lagrangian mechanics, Hamiltonian mechanics, Poisson brackets, and electromagnetism. [3]
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 principle of least action states that in all natural phenomena a quantity called 'action' tends to be minimised. Maupertuis developed such a principle over two decades. For him, action could be expressed mathematically as the product of the mass of the body involved, the distance it had travelled and the velocity at which it was travelling.