Wolfram Function Repository (Under Development)
Instantuse addon functions for the Wolfram Language
Get the EulerLagrange 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 ∫ⅆs in two dimensions yields a straight line:
In[1]:= 

Out[1]= 

In[2]:= 

Out[2]= 

A simple pendulum has the Lagrangian 12 m r^{2} θ.^{2}+m g r cos(θ):
In[3]:= 

Out[3]= 

The solution to the pendulum equation can be expressed using the function JacobiAmplitude:
In[4]:= 

Out[4]= 

The Lagrangian of a point particle in two dimensions has two dependent variables, and yields Newton's equations:
In[5]:= 

Out[5]= 

The Lagrangian of a point particle in two dimensions with a central potential:
In[6]:= 

Out[6]= 

Second and higherorder derivatives may be included in the integrand. A Lagrangian for motion on a spring using higherorder terms:
In[7]:= 

Out[7]= 

The integrand 12∇ϕ.∇ϕ has several independent variables:
In[8]:= 

Out[8]= 

The Euler equations yield Laplace's equation:
In[9]:= 

Out[9]= 

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]:= 

Out[14]= 

It is well known that the solution to the brachistochrone problem is a cycloid:
In[15]:= 

Out[15]= 

The Lagrangian for a vibrating string yields the classical wave equation:
In[16]:= 

Out[16]= 
