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2017 | OriginalPaper | Chapter

2. Mathematical Preliminaries

Authors : Tijana Ivancevic, Leon Lukman, Zoran Gojkovic, Ronald Greenberg, Helen Greenberg, Bojan Jovanovic, Aleksandar Lukman

Published in: The Evolved Athlete: A Guide for Elite Sport Enhancement

Publisher: Springer International Publishing

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Abstract

There is a single mathematical, physical and biomechanical concept that underpins most of the formal derivations presented in this book.

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previous chapter Introduction
next chapter Multi-physical Structure of Human Fascia: Harmony of Biomechanics and Neurodynamics
Footnotes
1
The second variation of the Lagrangian:
$$\begin{aligned} \delta ^{2}L=\frac{\partial L}{\partial q^{i}}\,\delta q^{i}+\frac{\partial L }{\partial \dot{q}^{i}}\,\delta \dot{q}^{i}+\frac{\partial L}{\partial \ddot{ q}^{i}}\,\delta \ddot{q}^{i} \end{aligned}$$
is used only in more sophisticated optimal control algorithms.
 
2
Recall that Newton’s fundamental equation of force: \(\varvec{f}= \varvec{\dot{p}}=m\varvec{a}=m\varvec{\dot{v}}=m \varvec{\ddot{x}}\), states that the application of the force vector \( \varvec{f}\) to a particle \(\bullet \) of mass m, causes \(\bullet \) to move with the momentum \(\varvec{p}=m\varvec{v}\), acceleration \( \varvec{a}=\varvec{\ddot{x}}\) and velocity \(\varvec{v}= \varvec{\dot{x}}\) in the direction \(\varvec{x}\).
 
3
Note that in a more general, nonlinear Riemannian elasticity, the displacement vector is defined as the deformation covector (i.e., one-form): \(\varvec{u}=u_{i}dx^{i}\).
 
4
In Riemannian elasticity, there are actually two strain tensors: the Cauchy-Green strain tensor, an infinitesimal tensor field generated during deformation, given by: \(e_{ik}^{\mathrm {CG}}=g_{ik}\,dx^{i}dx^{k},\) and the relative, or Green-Lagrange strain tensor, measuring the metric-change between the undeformed and deformed states, given by: \(e_{ik}^{ \mathrm {GL}}=\frac{1}{2}(g_{ik}-\delta _{ik})\,dx^{i}dx^{k}.\)
 
5
In case of large (or, finite) deformations, the Cauchy stress tensor generalizes to the (first and second)Piola-Kirchhoff stress tensors.
 
6
Physical properties of isotropic media are independent of directions in the 3D Euclidean space.
 
7
We actually present a generic simulator with three more nameless attractor systems, to demonstrate how easy it is to extend this simulator for other applications.
 
Literature
1.
2.
Fung, F.C.: A First Course in Continuum Mechanics, 3rd edn. Prentice-Hall, Englewood Cliffs (1994)
3.
Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Butterworth Heinemann, Oxford (1986)
4.
Weisstein, E.W.: Lorenz Attractor, Wolfram MathWorld (2015)
5.
6.
7.
Metadata
Title
Mathematical Preliminaries
Authors
Tijana Ivancevic
Leon Lukman
Zoran Gojkovic
Ronald Greenberg
Helen Greenberg
Bojan Jovanovic
Aleksandar Lukman
Copyright Year
2017
Book
The Evolved Athlete: A Guide for Elite Sport Enhancement

Print ISBN: 978-3-319-57927-6

Electronic ISBN: 978-3-319-57928-3

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-57928-3_2

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