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.
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