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.

Klappentext

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.

Zusammenfassung

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.

Inhalt

Basics of the Lagrangian Field Theory.- Lagrangian Field Theory with the Non-commuting (NC) Variations.- Vertical Connections in the Congurational Bundle and the NCvariations.- K-twisted Prolongations and -symmetries (by Works of Muriel,Romero.- Applications: Holonomic and Non-Holonomic Mechanics,H.KleinertAction Principle, Uniform Materials,and the Dissipative Potentials.- Material Time, NC-variations and the Material Aging.- Fiber Bundles and Their Geometrical Structures, Absolute Parallelism.- Jet Bundles, Contact Structures and Connections on Jet Bundles.- Lie Groups Actions on the Jet Bundles and the Systems of Differential Equations.