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EulerEquations


returns the Euler-Lagrange differential equation obeyed by derived from the functional f, where f depends on the function and its derivatives as well as the independent variable x.

returns the Euler-Lagrange differential equation obeyed by .

returns a list of Euler-Lagrange differential equations obeyed by .
The Euler equations for the arc length in 2 dimensions yields a straight line:
A simple pendulum has the Lagrangian :
The solution to the pendulum equation can be expressed using the function JacobiAmplitude:
Needs["VariationalMethods`"]
The Euler equations for the arc length in 2 dimensions yields a straight line:
In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
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Needs["VariationalMethods`"]
A simple pendulum has the Lagrangian :
In[2]:=
Click for copyable input
Out[2]=
The solution to the pendulum equation can be expressed using the function JacobiAmplitude:
In[3]:=
Click for copyable input
Out[3]=
The Lagrangian of a point particle in 2 dimensions has 2 dependent variables, and yields Newton's equations:
The Lagrangian of a point particle in 2 dimensions with a central potential:
Second- and higher-order derivatives may be included in the integrand. A Lagrangian for motion on a spring using higher-order terms:
The integrand has several independent variables:
The Euler equations yield Laplace's equation:
The Euler equations for the integrand :
The "textbook" answer:
Check:
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:
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:
It is well known that the solution to the brachistochrone problem is a cycloid:
The Lagrangian for a vibrating string yields the classical wave equation: