Adjoint symmetries for graded vector fields
Abstract
Suppose that
is a graded manifold and consider a direct subsheaf
of
and a graded vector field
on
, both satisfying certain conditions.
is used to characterize the local expression of
. Thus we review some of the basic definitions, properties, and geometric structures related to the theory of adjoint symmetries on a graded manifold and discuss some ideas from Lagrangian supermechanics in an informal fashion. In the special case where
is the tangent supermanifold, we are able to find a generalization of the adjoint symmetry method for time-dependent second-order equations to the graded case. Finally, the relationship between adjoint symmetries of
and Lagrangians is studied.
![{\mathcal{M}}=(M,{\mathcal A}_M)](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/eb61d04b184932ad0a1cebe6ff37efdc.png)
![\cd](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/9e911e091747e941030c752831cd0755.png)
![{ Der {\mathcal A}_M}](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/db2c5ac80558b30c37b4f97936e0a35e.png)
![\Gamma](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/07710b5c43702a8bb7b9104eacc6ba71.png)
![{\mathcal{M}}](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/cb719bb834dc741326a9aed6ef8649d2.png)
![\cd](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/9e911e091747e941030c752831cd0755.png)
![\Gamma](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/07710b5c43702a8bb7b9104eacc6ba71.png)
![{\mathcal{M}}](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/cb719bb834dc741326a9aed6ef8649d2.png)
![\Gamma](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/07710b5c43702a8bb7b9104eacc6ba71.png)
DOI Code:
10.1285/i15900932v39n1p33
Keywords:
supermanifold; involutive distribution; second-order differential equation field; Lagrangian systems; adjoint symmetry
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