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

Integrating P- super vectorfields and the super geodesic flow