Read e-book online Methods of Differential Geometry in Analytical Mechanics PDF

By M. de León, P.R. Rodrigues

ISBN-10: 0444880178

ISBN-13: 9780444880178

The differential geometric formula of analytical mechanics not just bargains a brand new perception into Mechanics, but in addition offers a extra rigorous formula of its actual content material from a mathematical perspective. issues lined during this quantity comprise differential types, the differential geometry of tangent and cotangent bundles, nearly tangent geometry, symplectic and pre-symplectic Lagrangian and Hamiltonian formalisms, tensors and connections on manifolds, and geometrical elements of variational and constraint theories. The ebook can be regarded as a self-contained textual content and in basic terms presupposes that readers are familiar with linear and multilinear algebra in addition to complicated calculus.

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By M. de León, P.R. Rodrigues

ISBN-10: 0444880178

ISBN-13: 9780444880178

The differential geometric formula of analytical mechanics not just bargains a brand new perception into Mechanics, but in addition offers a extra rigorous formula of its actual content material from a mathematical perspective. issues lined during this quantity comprise differential types, the differential geometry of tangent and cotangent bundles, nearly tangent geometry, symplectic and pre-symplectic Lagrangian and Hamiltonian formalisms, tensors and connections on manifolds, and geometrical elements of variational and constraint theories. The ebook can be regarded as a self-contained textual content and in basic terms presupposes that readers are familiar with linear and multilinear algebra in addition to complicated calculus.

Show description

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Then, if K E T'V (resp. L E T d V )we have Chapter 1. ir(resp. j,) are the components of K (resp. L). We define the (mixed) tensor space of type (r,s), or tensor space of contravariant degree r and covariant degree s as the tensor product T,'V = T'V @ T,V = V 8 . @ V 8 V * @ . . @ V * (V r-times and V* s-times). In particular, we have T,'V = T'V, T,OV = T,V, T,OV = TQV= TQV= R. It is obvious that the set {eil 8 . . @ ei, 21 . . ;1 5 i l , . . ,i,,jI,.. ,j,5 m) is a basis for T,'V. Then dim T,'V = mr+'.

Differential forms. The exterior algebra 1 Acp = s! 43 C€aY3,. Now, let w E APW, r E AqW. q! 6 F* : AW AV is an algebra homomorphism. Differential forms We introduce the following terminology. 7 A skew-symmetric couariant tensor field of degree p on a manifold M is called a differential form of degree p (or sometimes simply p-form). Chapter 1. Differential Geometry 44 The set APM of all such forms is a subspace of T;(M) (in fact, a C"(M)submodule). If w E ApM and r E A ~ Mwe , define the exterior product w A r E Ap+qM by ( w A T ) ( x ) = ~ ( xA)r ( x ) , x E M .

A : ) ( b l @ . . @ b a ) =< bl, > . . < ba,a: > . Now, from the universal factorization property, it follows that (T"V)*is isomorphic to the space of s-linear mappings of V x . . x V into R. j,ejl @ . . @ ej, E T,V, then K corresponds to an s-linear mapping of V x . . 6 TiV i s canonically isomorphic to the vector space of s-linear mappings ojV x . . x V into V . Proof We have TiV = V @ T,V. 2, V @ T,V @ V . 3. By the universal factorization property, Hom(T"V,V ) can be identified with the space of s-linear mappings of V x .

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Methods of Differential Geometry in Analytical Mechanics by M. de León, P.R. Rodrigues


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