# American Institute of Mathematical Sciences

December  2017, 9(4): 411-437. doi: 10.3934/jgm.2017016

## 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

Received  September 2016 Revised  March 2017 Published  October 2017

Fund Project: 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'.

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.

Citation: Andrew James Bruce, Katarzyna Grabowska, Giovanni Moreno. On a geometric framework for Lagrangian supermechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 411-437. doi: 10.3934/jgm.2017016
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