FirstIntegrals

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.

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Wolfram Language code: Needs["VariationalMethods`"]

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

Wolfram Language code: ℒ = (1/2)m(r'[t]^2 + r[t]^2θ'[t]^2) - V[r[t]];

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

Wolfram Language code: FirstIntegrals[ℒ, {r[t], θ[t]}, t]

Wolfram Language code: Needs["VariationalMethods`"]

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

Wolfram Language code: f = 2π y[x]Sqrt[1 + y'[x]^2];

Here f has no explicit dependence on x:

Wolfram Language code: FirstIntegrals[f, y[x], x]

Wolfram Language code: Needs["VariationalMethods`"]

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

Wolfram Language code: FirstIntegrals[(1/2)m(x'[t]^2 + y'[t]^2) - V[x[t], y[t]], {x[t], y[t]}, t]

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