Basic Examples

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

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

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

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Scope

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

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

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

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

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The Euler equations yield Laplace's equation:

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Applications

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

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The "textbook" answer:

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Check:

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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:

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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:

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It is well known that the solution to the brachistochrone problem is a cycloid:

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The Lagrangian for a vibrating string yields the classical wave equation:

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