SYM Paclet Symbol

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.

MORE INFORMATION

To use VariationalIntegralInvariance, you first need to load the SYM Package using Needs["SYM`"].

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:

In[2]:=

Out[2]=

and the Lie point symmetry:

In[3]:=

Out[3]=

The invariance condition is:

In[4]:=

Out[4]=

that is this symmetry leaves invariant the variational integral under investigation.

In[1]:=

Let the Lie point symmetry

In[2]:=

Out[2]=

and the variational integral

In[3]:=

Out[3]=

The invariance condition is:

In[4]:=

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.

In[1]:=

Let the Lie-bäcklund symmetry

In[2]:=

Out[2]=

and the variational integral

In[3]:=

Out[3]=

The invariance condition is:

In[4]:=

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.