In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle .
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In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle .
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The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system.
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Fundamental result in analytical Lagrangian mechanics is D'Alembert's principle, introduced in 1708 by Jacques Bernoulli to understand static equilibrium, and developed by D'Alembert in 1743 to solve dynamical problems.
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An important property of the Lagrangian mechanics is that conserved quantities can easily be read off from it.
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Since is absent from the Lagrangian mechanics, it is a cyclic coordinate.
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Ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations.
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Can be obtained by performing a Legendre transformation on the Lagrangian mechanics, which introduces new variables canonically conjugate to the original variables.
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Lagrangian mechanics can be applied to geometrical optics, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow.
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Lagrangian mechanics can be formulated in special relativity and general relativity.
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Some features of Lagrangian mechanics are retained in the relativistic theories but difficulties quickly appear in other respects.
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In particular, the EL equations take the same form, and the connection between cyclic coordinates and conserved momenta still applies, however the Lagrangian mechanics must be modified and is not simply the kinetic minus the potential energy of a particle.
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The Lagrangian mechanics is then the volume integral of the Lagrangian mechanics density over 3D space.
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Action principle, and the Lagrangian mechanics formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system.
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