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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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