A survey about framing the bases of Impulsive Mechanics of constrained systems into a jet-bundle geometric context

TL;DR

This survey explores how impulsive mechanical systems with constraints can be modeled using jet-bundle geometry, clarifying the geometric and constitutive distinctions and introducing novel constraint types.

Contribution

It provides a comprehensive geometric framework for impulsive constraints, including new types like unilateral and breakable constraints, enhancing the theoretical understanding of impulsive mechanics.

Findings

Geometric description of impulsive constraints in jet-bundle framework

Introduction of unilateral and breakable constraints

Analysis of frame dependence in impulsive reactions

Abstract

We illustrate how the different kinds of constraints acting on an impulsive mechanical system can be clearly described in the geometric setup given by the configuration space--time bundle $\pi_t:\mathcal{M} \to \mathbb{E}$ and its first jet extension $\pi: J_1 \to \mathcal{M}$ in a way that ensures total compliance with axioms and invariance requirements of Classical Mechanics. We specify the differences between geometric and constitutive characterizations of a constraint. We point out the relevance of the role played by the concept of frame of reference, underlining when the frame independence is mandatorily required and when a choice of a frame is an inescapable need. The thorough rationalization allows the introduction of unusual but meaningful kinds of constraints, such as unilateral kinetic constraints or breakable constraints, and of new theoretical aspects, such as the possible dependence of the impulsive reaction by the active forces acting on the system.