Non-commuting Variations in Mathematics and Physics: A Survey (Interaction of Mechanics and Mathematics) by Serge Preston (PDF)

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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.

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