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| (1) |
Consider a PDE problem with two independent variables and one dependent variable,
, and consider the list of infinitesimals of a symmetry group
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| (2) |
In the input above you can also enter the symmetry
without infinitesimals' labels, as in
. The corresponding infinitesimal generator is
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| (3) |
A
is invariant under the symmetry transformation generated by
in that
, where, in this formula,
represents the prolongation necessary to act on
(see InfinitesimalGenerator).
The actual form of this finite, one-parameter, symmetry transformation relating the original variables
to new variables,
, that leaves invariant any PDE system admitting the symmetry represented by
above is obtained via
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| (4) |
where
is a (Lie group) transformation parameter. To express this transformation using jetnotation use
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| (5) |
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| (6) |
That this transformation leaves invariant any PDE system invariant under
above is visible in the fact that it also leaves invariant the infinitesimals
; to verify this you can use ChangeSymmetry
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| (7) |
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| (8) |
which is the same as
(but written in terms of
instead of
). So to this list of infinitesimals corresponds, written in terms of
, this infinitesimal generator
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| (9) |
which is also equal to
, only written in terms of
.
If the new variables,
, are not indicated, variables prefixed by the underscore _ to represent the new variables are introduced
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| (10) |
An example where the Lie group parameter
appears only through the subexpression
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| (11) |
A symmetry transformation with the parameter redefined
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| (12) |