Non-commuting Variations in Mathematics and Physics, 1st ed. 2016 A Survey Interaction of Mechanics and Mathematics Series

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.

Basics of the Lagrangian Field Theory.- Lagrangian Field Theory with the Non-commuting (NC) Variations.- Vertical Connections in the Congurational Bundle and the NCvariations.- K-twisted Prolongations and -symmetries (by Works of Muriel,Romero.- Applications: Holonomic and Non-Holonomic Mechanics,H.KleinertAction Principle, Uniform Materials,and the Dissipative Potentials.- Material Time, NC-variations and the Material Aging.- Fiber Bundles and Their Geometrical Structures, Absolute Parallelism.- Jet Bundles, Contact Structures and Connections on Jet Bundles.- Lie Groups Actions on the Jet Bundles and the Systems of Differential Equations.

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