# VariationalIntegralInvariance

 VariationalIntegralInvariance[symmetry, varIntegral, extraVars, jetOrder] gives the condition(s) that must be satisfied for the invariance of the variational integral, defined by the expression varIntegral, under the action of a Lie point or Lie-bäcklund symmetry symmetry.
• symmetry is a symmetry represented by its infinitesimal generator using Partial.
• extraVars is a symbol or a list of symbol designating the name of the functions included in the varIntegral without any derivatives.
• jetOrder is an optional argument for explicitly determining the type and the order of the provided symmetry. 0 is for Lie point symmetries and any positive integer for Lie-bäcklund, hence providing also the order of the symmetry. When omitted, it is set to 0.
Let the invariant integral:
 Out[2]=
and the Lie point symmetry:
 Out[3]=
The invariance condition is:
 Out[4]=
that is this symmetry leaves invariant the variational integral under investigation.

Let the Lie point symmetry
 Out[2]=
and the variational integral
 Out[3]=
The invariance condition is:
 Out[4]=
here we can see that for the symmetry to keep invariant the variational integral defined by [x, t] (u, t-u, xx) the relation between the constants 3+4+5=0 must hold.

Let the Lie-bäcklund symmetry
 Out[2]=
and the variational integral
 Out[3]=
The invariance condition is:
 Out[4]=
the above system must hold for every value of the independent variables, the dependent ones and their derivatives. Thus obtaining a system for the constants.
• SYM and Conservation Laws