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 x, y, ... and independent variable t.
- To use
, you first need to load the Variational Methods Package using Needs["VariationalMethods`"].
- 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 x, y, ... or the independent variable t, and c is the conserved quantity.
and the time 
, corresponding to conservation of angular momentum and energy:
about the 
axis has the integrand:
has no explicit dependence on 
:
