::Lagrangian mechanics


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Joseph-Louis Lagrange (1736—1813)

Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind,<ref name="Dvorak 2005 page=24">{{#invoke:Footnotes|harvard_citation_no_bracket}}</ref> which treat constraints explicitly as extra equations, often using Lagrange multipliers;<ref>{{#invoke:Footnotes|harvard_citation_no_bracket}}</ref><ref>{{#invoke:Footnotes|harvard_citation_no_bracket}}</ref> or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates.<ref name="Dvorak 2005 page=24"/><ref>{{#invoke:Footnotes|harvard_citation_no_bracket}}</ref> In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.

No new physics is introduced by Lagrangian mechanics; it is actually less general than Newtonian mechanics.<ref>{{#invoke:Footnotes|harvard_citation_no_bracket}}</ref> Newton's laws can include non-conservative forces like friction, however they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces, and some (not all) non-conservative forces, in any coordinate system. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which considerably simplifies describing the dynamics of the system. Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, although only as a special case of Noether's theorem. The theory connects with the principle of stationary action,<ref>{{#invoke:Footnotes|harvard_citation_no_bracket}}</ref> although Lagrangian mechanics is less general because it is restricted to equilibrium problems.<ref></ref> Also, Lagrangian mechanics can only be applied to systems with holonomic constraints, because the formulation does not work for nonholonomic constraints. Three examples<ref>{{#invoke:Footnotes|harvard_citation_no_bracket}}</ref> are when the constraint equations are nonintegrable, when the constraints have inequalities, or with complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may revert to Newtonian mechanics, or use other methods.

The Lagrangian formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics, Hamilton's principle that can be used to derive the Lagrange equation was later recognized to be applicable to much of theoretical physics as well. In quantum mechanics, action and quantum-mechanical phase are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions. If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either special relativity or general relativity. The action principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system. Lagrangian mechanics and Noether's theorem together yield a natural formalism for first quantization by including commutators between certain terms of the Lagrangian equations of motion for a physical system.

Lagrangian mechanics is widely used to solve mechanical problems in physics and engineering when Newton's formulation of classical mechanics is not convenient. Lagrange's equations are also used in optimisation problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind.

Lagrangian mechanics sections
Intro  Introduction  Definition of the Lagrangian  Equations of motion  From Newtonian to Lagrangian mechanics  Properties of the Euler\u2013Lagrange equation  Examples in specific coordinate systems  Non-relativistic examples  Extensions to include non-conservative forces  Applications or extensions of Lagrangian mechanics in other contexts  See also  Footnotes  Notes  References  [[Lagrangian_mechanics?section=Further</a>_reading|Further</a> reading]]  External links  

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