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one last transfer of content to Lagrangian mechanics, for the time being write a note saying where the term is used, keeping the link to the current Lagrangian DAB page, before this page gets redirected soon |
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{{About|the Lagrangian function in [[Lagrangian mechanics]]}} |
{{About|the Lagrangian function in [[Lagrangian mechanics]]}} |
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The Lagrangian is a function used in [[Lagrangian mechanics]] and [[Lagrangian field theory]]. |
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The '''Lagrangian''', ''L'', of a [[dynamical system]] is a [[mathematical function]] that summarizes the dynamics of the system. For a simple mechanical system, it is the value given by the kinetic energy of the particle minus the potential energy of the particle but it may be generalized to more complex systems. It is used primarily as a key component in the [[Euler-Lagrange equations]] to find the path of a particle according to the [[principle of least action]]. |
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<!---For the time being this article is just a note saying where the term is used, keeping the link to the current Lagrangian DAB page, before this page gets redirected soon---> |
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The Lagrangian is named after Italian mathematician and astronomer [[Joseph Louis Lagrange]]. The concept of a Lagrangian was introduced in a reformulation of [[classical mechanics]] introduced by Lagrange known as [[Lagrangian mechanics]] in 1788. This reformulation was needed in order to explore mechanics in alternative systems to [[cartesian coordinates]] such as [[polar coordinate system|polar]], [[Cylindrical coordinate system|cylindrical]] and [[Spherical coordinate system|spherical]] coordinates, and where the coordinates do not necessarily refer to position, for which [[Newtonian mechanics|Newton's formulation]] of classical mechanics is not convenient. |
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The Lagrangian has since been used in a method to find the acceleration of a particle in a Newtonian gravitational field and to derive the [[Einstein field equations]]. This led to its use in applying [[electromagnetism]] to curved [[spacetime]] and in describing charged [[black hole]]s. It also has additional uses in [[Mathematical formalism]] to find the [[functional derivative]] of an action, and in [[engineering]] for the analysis and optimisation of dynamic systems. |
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==Uses in Engineering== |
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Circa 1963{{When|date=January 2015}} Lagrangians were a general part of the engineering curriculum, but a quarter of a century later, even with the ascendency of [[dynamical system]]s, they were dropped as requirements for some engineering programs, and are generally considered to be the domain of theoretical dynamics. Circa 2003{{When|date=January 2015}} this changed dramatically, and Lagrangians are not only a required part of many ME and EE graduate-level curricula, but also find applications in finance, economics, and biology, mainly as the basis of the formulation of various path integral schemes to facilitate the solution of parabolic partial differential equations via random walks. |
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Circa 2013,{{When|date=January 2015}} Lagrangians find their way into hundreds of direct engineering solutions, including [[robotics]], turbulent flow analysis ([[Lagrangian and Eulerian specification of the flow field]]), [[signal processing]], microscopic component contact and [[nanotechnology]] (superlinear convergent augmented Lagrangians), gyroscopic forcing and dissipation, [[semi-infinite]] supercomputing (which also involve [[Lagrange multipliers]] in the subfield of [[semi-infinite programming]]), [[chemical engineering]] (specific heat linear Lagrangian interpolation in reaction planning), civil engineering (dynamic analysis of traffic flows), optics engineering and design (Lagrangian and [[Hamiltonian optics]]) aerospace (Lagrangian interpolation), force stepping integrators, and even [[airbag]] deployment (coupled Eulerian-Lagrangians as well as SELM—the [[stochastic Eulerian Lagrangian method]]).<ref name="Engineering Lagrangians">{{cite book |isbn= 978-1-4614-3929-5 |url= http://www.amazon.com/gp/product/1461439299/ref=olp_product_details?ie=UTF8&me=&seller=|title= Engineering Dynamics: From the Lagrangian to Simulation |author=Roger F Gans |location=New York|publisher=Springer|year=2013}}</ref> |
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==See also== |
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{{multicol}} |
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*[[Calculus of variations]] |
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*[[Covariant classical field theory]] |
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*[[Einstein–Maxwell–Dirac equations]] |
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*[[Euler–Lagrange equation]] |
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*[[Functional derivative]] |
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*[[Functional integral]] |
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*[[Generalized coordinates]] |
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*[[Hamiltonian mechanics]] |
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{{multicol-break}} |
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*[[Lagrangian and Eulerian coordinates]] |
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*[[Lagrangian mechanics]] |
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*[[Lagrangian point]] |
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*[[Lagrangian system]] |
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*[[Noether's theorem]] |
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*[[Onsager–Machlup function]] |
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*[[Principle of least action]] |
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*[[Scalar field theory]] |
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{{multicol-end}} |
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==Notes== |
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<references /> |
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==References== |
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* David Tong [http://www.damtp.cam.ac.uk/user/tong/dynamics.html Classical Dynamics] (Cambridge lecture notes) |
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[[Category:Concepts in physics]] |
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[[Category:Lagrangian mechanics| ]] |
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[[Category:Dynamical systems]] |
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[[Category:Mathematical and quantitative methods (economics)]] |
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Revision as of 11:41, 5 August 2015
The Lagrangian is a function used in Lagrangian mechanics and Lagrangian field theory.