Preston S. Non-commuting Variations in Mathematics and Physics: A Survey

Springer, 2016. — 235 p. — ISBN: 3319283219.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.Content by Chapters:
Basics of the Lagrangian Field Theory
Lagrangian Field Theory with the non-commuting (NC) variations
Vertical connections in the Configurational bundle and the NC variations
K-twisted prolongations and μ-symmetries (by works of Muriel, Romero, Gaeta, Morando, etc.)
Applications: Holonomic and non-Holonomic Mechanics, H.Kleinert Action principle, Uniform Materials, and the Dissipative potentials
Material time, NC-variations and the Material Aging
Appendixes
Fiber bundles and their geometrical structures, absolute parallelism
Jet bundles, contact structures and connections on Jet bundles
Symmetry groups of systems of Differential Equations and the Noether balance laws