Integrating P- super vectorfields and the super geodesic flow

TL;DR

This paper develops a framework for integral and geodesic flows on P-supermanifolds, introducing parametrization over super algebras, and extends classical mechanics formulations to supermanifolds, demonstrating their equivalence.

Contribution

It introduces the concept of parametrization over super algebras in supermanifold theory and extends classical mechanics to P-Riemannian supermanifolds.

Findings

Parametrization simplifies supermanifold theory.

A version of Palais' theorem for P-supermanifolds is established.

Classical mechanics approaches are formulated and shown equivalent on supermanifolds.

Abstract

Aim of this article is to introduce the notion of integral and geodesic flows on P-supermanifolds as certain partial actions of R . First I introduce the concept of parametrization over a `small' super algebra P, which leads to the notion of P-objects and is superized local deformation theory. It is shown how parametrization makes the theory much easier. A version of Palais' theorem for P-supermanifolds is obtained stating that every infinitesimal P-action of a simply connected P-super Lie group G on a P-supermanifold can be integrated to a whole action of G . Furthermore the faithful linearization of affine P-supermorphisms is proven. Finally I show that Newton's, Lagrange's and Hamilton's approach to mechanics can be formulated also for P- Riemannian supermanifolds and are infact equivalent.