Euler-Lagrange equation

This may be derived by using the Euler-Lagrange equations.

Using Euler-Lagrange equation, the equations of motion for can be derived.

This functional can be minimized by solving the associated Euler-Lagrange equations.

Using the Euler-Lagrange equations, this can be shown in polar coordinates as follows.

Straight lines can formally be obtained by solving the Euler-Lagrange equations for the above functional.

In this case Euler-Lagrange equations are not independent.

Substituting this into the Euler-Lagrange equation of motion for a field:

It has as a special case the Euler-Lagrange equation of the calculus of variations.

These equations are called the Euler-Lagrange equations for the variational problem.

He also invented the calculus of variations including its best-known result, the Euler-Lagrange equation.

The Euler-Lagrange equation plays a prominent role in classical mechanics and differential geometry.

The Euler-Lagrange equation is used to determine the function that minimizes the functional .

This leads to solving the associated Euler-Lagrange equation.

It could also be obtained from the Euler-Lagrange equation of motion, noting that the action depends on but not its derivative.

Using the Euler-Lagrange equation, we arrive at the equation of motion:

From this we get the force equation (equivalent to the Euler-Lagrange equation)

The Euler-Lagrange equations follow directly from Hamilton's principle, and are mathematically equivalent.

An example of a necessary condition that is used for finding weak extrema is the Euler-Lagrange equation.

The Euler-Lagrange equation for this problem is nonlinear:

Writing the Euler-Lagrange equation for the metric tensor one obtains that:

The condition that the first variation vanish at an extremal may be regarded as a weak form of the Euler-Lagrange equation.

The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler-Lagrange equation.

This is the Euler-Lagrange equation.

This algebra is quotiented over by the ideal generated by the Euler-Lagrange equations.

Noether's theorem states that (as one may explicitly check by substituting the Euler-Lagrange equations into the left hand side).