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Non-commuting Variations in Mathematics and Physics

A Survey

  • Serge Preston

Part of the Interaction of Mechanics and Mathematics book series (IMM)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Non-commuting variations - elementary topics

    1. Front Matter
      Pages 1-1
    2. Serge Preston
      Pages 3-15
  3. Vertical connections and the twisted prolongations

  4. APPENDIX . Fibre bundles,jet bundles and the Noether balance laws

  5. Back Matter
    Pages 231-235

About this book

Introduction

This text presents and studies the method of so –called noncommuting variations in Variational Calculus. This method was pioneered by Vito Volterra  who noticed that the conventional  Euler-Lagrange (EL-)  equations  are not applicable in Non-Holonomic Mechanics and  suggested to modify the basic rule used in Variational Calculus. This book  presents a survey of   Variational Calculus with non-commutative variations and shows  that most  basic properties of  conventional  Euler-Lagrange Equations  are, with some modifications,  preserved for  EL-equations with  K-twisted  (defined by K)-variations.    

Most of the book can be understood by readers without strong mathematical preparation (some knowledge of Differential Geometry is necessary).  In order to make the text more accessible the definitions and several necessary results in Geometry are presented separately in Appendices  I and II Furthermore in Appendix III  a  short presentation of the Noether Theorem describing the relation  between the symmetries of  the differential equations with dissipation   and  corresponding s balance laws is presented.

Keywords

Differential Equations Dissipation Energy-momentum Balance Law Entropy Hamiltonian Systems Mechanics Noether Theorem Non-holonomic Symmetries Variational Calculus Variations

Authors and affiliations

  • Serge Preston
    • 1
  1. 1.PortlandUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-28323-4
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Springer, Cham
  • eBook Packages Engineering
  • Print ISBN 978-3-319-28321-0
  • Online ISBN 978-3-319-28323-4
  • Series Print ISSN 1860-6245
  • Series Online ISSN 1860-6253
  • Buy this book on publisher's site
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