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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.
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.
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.
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In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). Maupertuis's principle uses the abbreviated action between two generalized points on a path.
Leonhard Euler corresponded with Maupertuis from 1740 to 1744; [1]: 582 in 1744 Euler proposed a refined formulation of the least action principle in 1744. [10] He writes [ 11 ] "Let the mass of the projectile be M , and let its squared velocity resulting from its height be v {\displaystyle v} while being moved over a distance ds .
The principle of virtual work, which is the form of the principle of least action applied to these systems, states that the path actually followed by the particle is the one for which the difference between the work along this path and other nearby paths is zero (to the first order).
The principle of least action states that the world line between two events in spacetime is that world line that minimizes the action between the two events. In classical mechanics the principle of least action is used to derive Newton's laws of motion and is the basis for Lagrangian dynamics. In relativity it is expressed as