Non-commuting Variations in Mathematics and Physics : A Survey

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.