This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 VARIATIONAL METHODS PACKAGE SYMBOL

# FirstIntegrals

 returns a list of first integrals corresponding to the coordinate and independent variable t of the integrand f. returns a list of first integrals corresponding to the coordinates and independent variable t.
• A first integral is a conserved quantity associated with a coordinate or the independent variable.
• A first integral associated with a coordinate is returned if f is independent of that coordinate, although f may contain derivatives of the coordinate. Such coordinates are typically called cyclic or ignorable coordinates.
• A first integral associated with the independent variable t is returned if f is independent of t and does not contain any second or higher derivatives of the coordinates.
• In mechanics, a first integral corresponding to a coordinate is typically associated with conservation of momentum, and a first integral corresponding to the independent variable is typically associated with conservation of energy.
• returns a list of rules of the form FirstIntegral[u]->c, where u may be either the coordinates or the independent variable t, and c is the conserved quantity.
The Lagrangian of a particle in 2 dimensions with a central potential:
The coordinates with conserved first integrals are the angle and the time t, corresponding to conservation of angular momentum and energy:
The area of a surface of revolution obtained by revolving the curve about the x axis has the integrand:
Here f has no explicit dependence on x:
Needs["VariationalMethods`"]
The Lagrangian of a particle in 2 dimensions with a central potential:
The coordinates with conserved first integrals are the angle and the time t, corresponding to conservation of angular momentum and energy:
 Out[3]=

Needs["VariationalMethods`"]
The area of a surface of revolution obtained by revolving the curve about the x axis has the integrand:
Here f has no explicit dependence on x:
 Out[3]=
For Lagrangians independent of time the first integral associated with time represents energy conservation: