Jet Methods in Time-Dependent Lagrangian Biomechanics

TL;DR

This paper introduces a geometric framework for modeling human biomechanics that accounts for time-dependent energy variations, utilizing jet manifolds and Ricci flow to describe the evolution of biomechanical systems.

Contribution

It develops a novel time-dependent biomechanical model using jet manifolds and Ricci flow, extending traditional autonomous biomechanics to include biochemical and energy non-conservation.

Findings

Framework based on jet manifolds for time-dependent biomechanics

Application of Ricci flow to model biomechanical evolution

Extension of configuration manifold with a real time axis

Abstract

In this paper we propose the time-dependent generalization of an `ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds associated to the extended musculo-skeletal configuration manifold, called the configuration bundle. We start with an ordinary configuration manifold of human body motion, given as a set of its all active degrees of freedom (DOF) for a particular movement. This is a Riemannian manifold with a material metric tensor given by the total mass-inertia matrix of the human body segments. This is the base manifold for standard autonomous biomechanics. To make its time-dependent generalization, we need to extend it with a real time axis. By this extension, using techniques from fibre bundles, we defined the biomechanical configuration bundle. On the biomechanical bundle we define vector-fields, differential forms and affine connections, as well as the associated jet manifolds. Using the formalism of jet manifolds of velocities and accelerations, we develop the time-dependent Lagrangian biomechanics. Its underlying geometric evolution is given by the Ricci flow equation. Keywords: Human time-dependent biomechanics, configuration bundle, jet spaces, Ricci flow