# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

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

Contributed by:
Wolfram Research

ResourceFunction["EulerEquations"][ returns the Euler–Lagrange differential equation obeyed by | |

ResourceFunction["EulerEquations"][ returns the Euler–Lagrange differential equation obeyed by | |

ResourceFunction["EulerEquations"][ returns a list of Euler–Lagrange differential equations obeyed by |

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

In[1]:= |

Out[1]= |

In[2]:= |

Out[2]= |

A simple pendulum has the Lagrangian :

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 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 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 . 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]= |

Wolfram Language 11.3 (March 2018) or above

This work is licensed under a Creative Commons Attribution 4.0 International License