The Journal of Geometric Mechanics (JGM)

On a geometric framework for Lagrangian supermechanics

Pages: 411 - 437, Volume 9, Issue 4, December 2017      doi:10.3934/jgm.2017016

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Andrew James Bruce - Mathematics Research Unit, University of Luxembourg, Maison du Nombre 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg (email)
Katarzyna Grabowska - Faculty of Physics, University of Warsaw, Pasteura 5, 02-093, Warszawa, Poland (email)
Giovanni Moreno - Institute of Mathematics, Polish Academy of Sciences, ul. Ĺšniadeckich 8,00-656, Warszawa, Poland (email)

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.

Keywords:  Supermechanics, supermanifolds, Lagrangian systems, Tulczyjew triples, curves.
Mathematics Subject Classification:  Primary: 58A50; Secondary: 70H03.

Received: September 2016;      Revised: March 2017;      Available Online: October 2017.