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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.
Building on the early work of Pierre Louis Maupertuis, Leonhard Euler, and Joseph-Louis Lagrange defining versions of principle of least action, [34]: 580 William Rowan Hamilton and in tandem Carl Gustav Jacob Jacobi developed a variational form for classical mechanics known as the Hamilton–Jacobi equation.
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 .
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 [ 1 ] culminating in his 1788 ...
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.
Leonhard Euler is credited of introducing both specifications in two publications written in 1755 [3] and 1759. [4] [5] Joseph-Louis Lagrange studied the equations of motion in connection to the principle of least action in 1760, later in a treaty of fluid mechanics in 1781, [6] and thirdly in his book Mécanique analytique. [5]
Implications of symmetries in a physical situation can be found with the action principle, together with the Euler–Lagrange equations, which are derived from the action principle. An example is Noether's theorem , which states that to every continuous symmetry in a physical situation there corresponds a conservation law (and conversely).
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).