Variational Methods Package 内置符号

# 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.
• 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.
 例   (2)
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 y[x] about the x axis has the integrand:
Here f has no explicit dependence on x:
 Out[3]=