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.
