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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.

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, cylindrical and spherical coordinates, and where the coordinates do not necessarily refer to position, for which Newton's formulation of classical mechanics is not convenient.

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 holes. 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.

Uses in Engineering

Circa 1963^[when?] Lagrangians were a general part of the engineering curriculum, but a quarter of a century later, even with the ascendency of dynamical systems, they were dropped as requirements for some engineering programs, and are generally considered to be the domain of theoretical dynamics. Circa 2003^[when?] 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.

Circa 2013,^[when?] 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).^[1]

Notes

^ Roger F Gans (2013). Engineering Dynamics: From the Lagrangian to Simulation. New York: Springer. ISBN 978-1-4614-3929-5.

References

David Tong Classical Dynamics (Cambridge lecture notes)

[Engineering_Lagrangians-1] Roger F Gans (2013). Engineering Dynamics: From the Lagrangian to Simulation. New York: Springer. ISBN 978-1-4614-3929-5.

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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.
-==Lagrangian mechanics==
-=== Simple example ===
-The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a (local) minimum.
-The Lagrangian ''L'' can be calculated at several instants of time ''t'', and a graph of ''L'' against ''t'' can be drawn. The area under the curve is the [[action (physics)|action]]. Any different path between the initial and final positions leads to a larger action than that chosen by nature. Nature chooses the smallest action – this is the [[Principle of Least Action]].
-{{quote|If Nature has defined the mechanics problem of the thrown ball in so elegant a fashion, might She<!--sic--> have defined other problems similarly. So it seems now. Indeed, at the present time it appears that we can describe all the fundamental forces in terms of a Lagrangian. The search for Nature's One Equation, which rules all of the universe, has been largely a search for an adequate Lagrangian.|Robert Adair|The Great Design: Particles, Fields, and Creation<ref>The Great Design: Particles, Fields, and Creation (New York: Oxford University Press, 1989), ROBERT K. ADAIR, p.22–24</ref>}}
-Using only the principle of least action and the Lagrangian we can deduce the correct trajectory, by trial and error or the [[calculus of variations]].
 ==Uses in Engineering==

Revision as of 11:28, 5 August 2015

Uses in Engineering

See also

Notes

References