On a geometric framework for Lagrangian supermechanics

Andrew James Bruce; Katarzyna Grabowska; Giovanni Moreno

doi:10.3934/jgm.2017016

Article Contents

2017, Volume 9, Issue 4: 411-437. Doi: 10.3934/jgm.2017016

This issue Previous Article Next Article Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems

On a geometric framework for Lagrangian supermechanics

1.
Mathematics Research Unit, University of Luxembourg, Maison du Nombre 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg
2.
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093, Warszawa, Poland
3.
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656, Warszawa, Poland

* Corresponding author: Andrew James Bruce
* Corresponding author: Andrew James Bruce

Received: September 2016

Revised: March 2017

Published: November 2017

KG was supported by the Polish National Science Centre grant DEC-2012/06/A/ST1/00256. GM supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska–Curie grant agreement No 654721 'GEOGRAL'.

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Abstract

We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, we use arbitrary supermanifolds by following the categorical approach to supermanifolds.

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Mathematics Subject Classification: Primary: 58A50; Secondary: 70H03.

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