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.

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.