EulerEquations

Contributed by: Wolfram Research

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,…],….

Examples

Basic Examples

The Euler equations for the arc length ∫ⅆs in two dimensions yields a straight line:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

A simple pendulum has the Lagrangian 12 m r² θ.²+m g r cos(θ):

In[3]:=

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

Out[3]=

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

In[4]:=

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

Out[4]=

Scope

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

In[5]:=

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

Out[5]=

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

In[6]:=

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

Out[6]=

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

In[7]:=

Out[7]=

The integrand 12∇ϕ.∇ϕ has several independent variables:

In[8]:=

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

Out[8]=

The Euler equations yield Laplace's equation:

In[9]:=

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

Out[9]=

Applications

The Euler equations for the integrand f[y_xx,y_x,y,x]:

In[10]:=

Out[10]=

The "textbook" answer:

In[11]:=

Out[11]=

Check:

In[12]:=

Out[12]=

The brachistochrone problem asks for the curve of quickest descent. The time taken for a particle to fall an arc length ds is dsv. If y measures the decrease in height from an initial point of release, then the velocity v satisfies:

In[13]:=

Out[13]=

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:

In[14]:=

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