VariationalMethods`
VariationalMethods`

# FirstIntegrals

FirstIntegrals[f,x[t],t]

returns a list of first integrals corresponding to the coordinate x[t] and independent variable t of the integrand f.

FirstIntegrals[f,{x[t],y[t],},t]

returns a list of first integrals corresponding to the coordinates x, y, ... and independent variable t.

# Details and Options

• To use FirstIntegrals, 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 x[t], y[t], ... 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.
• FirstIntegrals 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.

# Examples

open allclose all

## Basic Examples(2)

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

The coordinates with conserved first integrals are the angle θ and the time t, corresponding to conservation of angular momentum and energy:

The area of a surface of revolution obtained by revolving the curve y[x] about the axis has the integrand:

Here f has no explicit dependence on x:

## Properties & Relations(1)

For Lagrangians independent of time, the first integral associated with time represents energy conservation: