Non-commuting Variations in Mathematics and Physics

This text presentsand studies the method of so –called noncommuting variations in VariationalCalculus. This methodwas pioneered by Vito Volterra  whonoticed that the conventional Euler-Lagrange (EL-)  equations  are not applicable in Non-Holonomic Mechanicsand  suggested to modify the basic ruleused in Variational Calculus. This book presents a survey of   VariationalCalculus with non-commutative variations and shows  that most basic properties of conventional  Euler-LagrangeEquations  are, with somemodifications,  preserved for  EL-equations with  K-twisted (defined by K)-variations.     Most of thebook can be understood by readers without strong mathematical preparation (someknowledge of Differential Geometry is necessary).  In order to make the text more accessible thedefinitions and several necessary results in Geometry are presented separatelyin Appendices  I and II Furthermore inAppendix III  a  short presentation of the Noether Theoremdescribing the relation  between thesymmetries of  the differential equationswith dissipation   and  corresponding s balance laws is presented.