By G. Giachetta, G. Sardanashvily, L. Mangiarotti
ISBN-10: 9810215878
ISBN-13: 9789810215873
This paintings accommodates 3 glossy points of mathematical physics: the jet tools in differntial geometry, the Lagrangian formalism on jet manifolds and the multimomentum method of the Hamiltonian formalism. a number of modern box versions are investigated intimately. this isn't a booklet on differential geometry, even supposing glossy recommendations of differential geometry are used through the ebook. Quadratic Lagrangians and Hamiltonians are studied at the normal point together with the remedy of Hamiltonian formalism on composite fibre manifolds. The ebook provides new geometric tools and leads to box concept.
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Additional resources for New Lagrangian and Hamiltonian Methods in Field Theory
Sample text
2. For the sake of simplicity, we will denote the pull-backs YxTX and x simply by TX and T'X. YxT'X x Vertical tangent bundles of many fibred manifolds are pull-back bundles. 1. 7) CHAPTER 1. 28 FIBRED MANIFOLDS where V -' X is a vector bundle. 7) if the induced coordinates of the vertical tangent bundle VY take the form y`=Vea where (zµ, V) are bundle coordinates of V. In this case, coordinate transformations of the coordinates y' are independent of the coordinates y'. 3. A vector bundle Y IVY X has the canonical vertical splitting Tr'Y.
If Y -- X is a bundle endowed with an algebraic structure, this algebraic structure may be inherited by the fibred jet manifold J'Y -. X due to the jet prolongations of the corresponding morphisms. 4. For instance, if Y -. X be a vector bundle, so is J'Y X. 11) where Y' -e X and WY)' X are the bundles dual of Y X and J'Y -' X, respectively. 5. 1. FIRST ORDER JET MANIFOLDS 47 where y' and yi are dual bundle coordinates on Y and Y', respectively, and r is the canonical coordinate of R. The jet prolongation of () is the linear bundle morphism J'():J'YxJ'Y'-'T'X x R, x xµ O J' () = yµyi + y'y,,i.
A vector bundle Y IVY X has the canonical vertical splitting Tr'Y. The linear bundle coordinates of Y are adapted to this vertical splitting so that = y`. Also an affine bundle Y modelled over a vector bundle V has the canonical vertical splitting The affine bundle coordinates of Y are adapted to this splitting so that y' = y'. Note that the tangent functor T preserves algebraic structures of fibred mani- folds. In particular, if Y - X is a vector bundle, so is TY - X. If Y - X is an affine bundle modelled over a vector bundle V -y X, then TY X.
New Lagrangian and Hamiltonian Methods in Field Theory by G. Giachetta, G. Sardanashvily, L. Mangiarotti
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