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Euler-Lagrange equation

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Variational calculus

variational calculus

Differential geometric version

Derived differential geometric version

Physics

physics, mathematical physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

Equality and Equivalence

equivalence

principle of equivalence

equation

Contents

Idea

The Euler-Lagrange equations of a functional characterize in variational calculus the extrema of that functional, hence its critical locus.

This originates from and is mainly used in physics, where the functional in question is the action functional of a physical system, and where its critical points encode the physically observable particle trajectories and field configurations by the principle of extremal action.

Details

Given a Lagrangian L=L(q,q⋅,t) of a classical mechanical system the corresponding action principle can be expressed as Euler-Lagrange equations: for all i,

ddt(∂L∂q⋅ i)−∂L∂q i=0\frac{d}{dt} \left( \frac{\partial L}{\partial \stackrel{\cdot}{q}_i} \right) - \frac{\partial L}{\partial {q}_i} = 0

If there is only one degree of freedom then one talks in singular: Euler-Lagrange equation.

Unlike in the usual classical mechanical systems, in some other problems of calculus of variations one has Lagrangians involving higher time derivatives; the Euler-Lagrange equations have then correspondingly more terms. Similarly, one can have dependence on more parameters than a single time parameter, what can also be easily incorporated.

Examples

References

  • Wikipedia, Euler-Lagrange equation.
  • Robert Bryant, Phillip Griffiths, Daniel Grossman, Exterior differential systems and Euler-Lagrange partial differential equations, Chicago Lectures in Mathematics 2003, 205+xiv pp. math.DG/0207039 bookpage

Revised on January 12, 2013 23:06:57 by Urs Schreiber (89.204.138.6)
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