SOLUTIONS

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.
DetailsDetails
 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.
ExamplesExamplesopen allclose all
Basic Examples (2)Basic Examples (2)
In[1]:= 
The Lagrangian of a particle in two dimensions with a central potential:
In[2]:= 
The coordinates with conserved first integrals are the angle and the time , corresponding to conservation of angular momentum and energy:
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In[1]:= 
The area of a surface of revolution obtained by revolving the curve about the axis has the integrand:
In[2]:= 
Here has no explicit dependence on :
In[3]:= 
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