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.