Wolfram Function Repository
Instantuse addon functions for the Wolfram Language
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Get the Euler–Lagrange differential equations derived from a given functional
ResourceFunction["EulerEquations"][f,u[x],x] returns the Euler–Lagrange differential equation obeyed by u[x] derived from the functional f, where f depends on the function u[x] and its derivatives, as well as the independent variable x. 

ResourceFunction["EulerEquations"][f,u[x,y,…],{x,y,…}] returns the Euler–Lagrange differential equation obeyed by u[x,y,…]. 

ResourceFunction["EulerEquations"][f,{u[x,y,…],v[x,y,…],…},{x,y,…}] returns a list of Euler–Lagrange differential equations obeyed by u[x,y,…],v[x,y,…],…. 
The Euler equations for the arc length in two dimensions yields a straight line:
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A simple pendulum has the Lagrangian :
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The solution to the pendulum equation can be expressed using the function JacobiAmplitude:
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The Lagrangian of a point particle in two dimensions has two dependent variables and yields Newton's equations:
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The Lagrangian of a point particle in two dimensions with a central potential:
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Second and higherorder derivatives may be included in the integrand. A Lagrangian for motion on a spring using higherorder terms:
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The integrand has several independent variables:
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The Euler equations yield Laplace's equation:
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The Euler equations for the integrand f[y_{xx},y_{x},y,x]:
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The "textbook" answer:
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Check:
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The brachistochrone problem asks for the curve of quickest descent. The time taken for a particle to fall an arc length ds is . If y measures the decrease in height from an initial point of release, then the velocity v satisfies:
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The equation for a curve joining two points, where a particle starting at rest from the higher point takes the least amount of time to reach the lower point:
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It is well known that the solution to the brachistochrone problem is a cycloid:
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The Lagrangian for a vibrating string yields the classical wave equation:
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Wolfram Language 11.3 (March 2018) or above
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