Non-commuting Variations in Mathematics and Physics

This text presentsand studies the method of so -called noncommuting variations in VariationalCalculus. This methodwas pioneered by Vito Volterra whonoticed that the conventional Euler-Lagrange (EL-) equations are not applicable in Non-Holonomic Mechanicsand suggested to modify the basic ruleused in Variational Calculus. This book presents a survey of VariationalCalculus with non-commutative variations and shows that most basic properties of conventional Euler-LagrangeEquations are, with somemodifications, preserved for EL-equations with K-twisted (defined by K)-variations.

Most of thebook can be understood by readers without strong mathematical preparation (someknowledge of Differential Geometry is necessary). In order to make the text more accessible thedefinitions and several necessary results in Geometry are presented separatelyin Appendices I and II Furthermore inAppendix III a short presentation of the Noether Theoremdescribing the relation between thesymmetries of the differential equationswith 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.