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Full Description
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.
Contents
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.
