Function Repository Resource:

EulerEquations

Get the Euler–Lagrange differential equations derived from a given functional

Contributed by: Wolfram Research

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,…],….

Examples

Basic Examples

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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$ResourceFunction["EulerEquations"][ 1/2 m r^2 \[Theta]'[t]^2 + m g r Cos[\[Theta][t]], \[Theta][t], t]$

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The solution to the pendulum equation can be expressed using the function JacobiAmplitude:

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$DSolve[%, \[Theta][t], t]$

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Scope

The Lagrangian of a point particle in two dimensions has two dependent variables and yields Newton's equations:

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$ResourceFunction["EulerEquations"][ 1/2 m (x'[t]^2 + y'[t]^2) - V[x[t], y[t]], {x[t], y[t]}, t]$

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The Lagrangian of a point particle in two dimensions with a central potential:

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$ResourceFunction["EulerEquations"][ 1/2 m (r'[t]^2 + r[t]^2 \[Theta]'[t]^2) - V[r[t]], {r[t], \[Theta][t]}, t]$

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Second- and higher-order derivatives may be included in the integrand. A Lagrangian for motion on a spring using higher-order terms:

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The integrand has several independent variables:

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$1/2 \!$ \*SubscriptBox[\(\[PartialD]$, ${{x, y, z}}$]$\[Phi][x, y, z]$\).\!$ \*SubscriptBox[\(\[PartialD]$, ${{x, y, z}}$]$\[Phi][x, y, z]$\)$

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The Euler equations yield Laplace's equation:

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$ResourceFunction["EulerEquations"][%, \[Phi][x, y, z], {x, y, z}]$

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Applications

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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$ResourceFunction["EulerEquations"][Sqrt[( x'[\[Theta]]^2 + y'[\[Theta]]^2)/( 2 g y[\[Theta]])], {x[\[Theta]], y[\[Theta]]}, \[Theta]]$