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Interaction of Mechanics and Mathematics

Non-commuting Variations in Mathematics and Physics

A Survey

Authors: Preston, Serge

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  • A survey of non-commuting Variations in Mathematics and Physics
  • Presents and develops methods of analysis, potential classification and of study of dissipative patterns of behavior using classical methods of differential geometry and variational calculus
  • Presents a large number of examples of geometrical description of different dynamical behavior in the evolutional systems of partial and ordinary differential equations and characteristics of their irreversible behavior
  • Demonstrates that a large variety of irreversible dynamical behavior in physical, mechanical, etc. systems is covered by the Lagrangian formalism with non-commutative variations
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About this book

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.

Table of contents (9 chapters)

Table of contents (9 chapters)
  • Basics of the Lagrangian Field Theory

    Pages 3-15

    Preston, Serge

  • Lagrangian Field Theory with non-commuting variations

    Pages 17-49

    Preston, Serge

  • Vertical connections, K-twisted prolongations and the NC-variations

    Pages 53-78

    Preston, Serge

  • Twisted prolongations and the NC-variations

    Pages 79-109

    Preston, Serge

  • Applications: Holonomic and non-Holonomic Mechanics, H.Kleinert Action principle, Uniform Materials, Non commutative variations and the Dissipative potentials

    Pages 111-138

    Preston, Serge

Buy this book

eBook $99.00
price for USA in USD
  • ISBN 978-3-319-28323-4
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $129.99
price for USA in USD
  • ISBN 978-3-319-28321-0
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Non-commuting Variations in Mathematics and Physics
Book Subtitle
A Survey
Authors
Series Title
Interaction of Mechanics and Mathematics
Copyright
2016
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-28323-4
DOI
10.1007/978-3-319-28323-4
Softcover ISBN
978-3-319-28321-0
Series ISSN
1860-6245
Edition Number
1
Number of Pages
XIV, 235
Number of Illustrations
11 b/w illustrations
Topics

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