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The Geometry of Higher-Order Hamilton Spaces [electronic resource] : Applications to Hamiltonian Mechanics / by R. Miron

Author
Miron, Radu
Published
Dordrecht : Springer Netherlands : Imprint: Springer, 2003.
Edition
1st ed. 2003.
Physical Description
XVI, 247 pages : online resource
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SpringerLink (Online service)
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Contents
1 Geometry of the k-Tangent Bundle TkM -- 1.1 The Category of k-Accelerations Bundles -- 1.2 Liouville Vector Fields. k-Semisprays -- 1.3 Nonlinear Connections -- 1.4 The Dual Coefficients of a Nonlinear Connection -- 1.5 The Determination of a Nonlinear Connection -- 1.6 d-Tensor Fields. N-Linear Connections -- 1.7 Torsion and Curvature -- 2 Lagrange Spaces of Higher Order -- 2.1 Lagrangians of Order k -- 2.2 Variational Problem -- 2.3 Higher Order Energies -- 2.4 Jacobi-Ostrogradski Momenta -- 2.5 Higher Order Lagrange Spaces -- 2.6 Canonical Metrical N-Connections -- 2.7 Generalized Lagrange Spaces of Order k -- 3 Finsler Spaces of Order k -- 3.1 Spaces F(k)n -- 3.2 Cartan Nonlinear Connection in F(k)n -- 3.3 The Cartan Metrical N-Linear Connection -- 4 The Geometry of the Dual of k-Tangent Bundle -- 4.1 The Dual Bundle (T*k M, ?*k, M) -- 4.2 Vertical Distributions. Liouville Vector Fields -- 4.3 The Structures J and J* -- 4.4 Canonical Poisson Structures on T*kM -- 4.5 Homogeneity -- 5 The Variational Problem for the Hamiltonians of Order k -- 5.1 The Hamilton-Jacobi Equations -- 5.2 Zermelo Conditions -- 5.3 Higher Order Energies. Conservation of Energy ?k ?1(H) -- 5.4 The Jacobi-Ostrogradski Momenta -- 5.5 Nöther Type Theorems -- 6 Dual Semispray. Nonlinear Connections -- 6.1 Dual Semispray -- 6.2 Nonlinear Connections -- 6.3 The Dual Coefficients of the Nonlinear Connection N -- 6.4 The Determination of the Nonlinear Connection by a Dual k-Semispray -- 6.5 Lie Brackets. Exterior Differential -- 6.6 The Almost Product Structure ?. The Almost Contact Structure $$ \mathbb{F} $$ -- 6.7 The Riemannian Structure G on T*kM -- 6.8 The Riemannian Almost Contact Structure $$(\mathop \mathbb{G}\limits^ \vee ,\mathop \mathbb{F}\limits^ \vee )$$ -- 7 Linear Connections on the Manifold T*kM -- 7.1 The Algebra of Distinguished Tensor Fields -- 7.2 N-Linear Connections -- 7.3 The Torsion and Curvature of an N-Linear Connection -- 7.4 The Coefficients of a N-Linear Connection -- 7.5 The h-,??- and ?k-Covariant Derivatives in Local Adapted Basis -- 7.6 Ricci Identities. Local Expressions of d-Tensor of Curvature and Torsion. Bianchi Identities -- 7.7 Parallelism of the Vector Fields on the Manifold T*kM -- 7.8 Structure Equations of a N-Linear Connection -- 8 Hamilton Spaces of Order k ? 1 -- 8.1 The Spaces H(k)n -- 8.2 The k-Tangent Structure J and the Adjoint k-Tangent Structure J* -- 8.3 The Canonical Poisson Structure of the Hamilton Space H(k)n -- 8.4 Legendre Mapping Determined by a Lagrange Space L(k)n= (M, L) -- 8.5 Legendre Mapping Determined by a Hamilton Space of Order k -- 8.6 The Canonical Nonlinear Connection of the Space H(k)n -- 8.7 Canonical Metrical N-Linear Connection of the Space H(k)n -- 8.8 The Hamilton Space H(k)n of Electrodynamics -- 8.9 The Riemannian Almost Contact Structure Determined by the Hamilton Space H(k)n -- 9 Subspaces in Hamilton Spaces of Order k -- 9.1 Submanifolds $${T^{*k}}\mathop M\limits^ \vee$$ in the Manifold T*kM -- 9.2 Hamilton Subspaces $${{\mathop H\limits^ \vee} ^{(k)m}}$$ in H(k)n. Darboux Frames -- 9.3 Induced Nonlinear Connection -- 9.4 The Relative Covariant Derivative -- 9.5 The Gauss-Weingarten Formula -- 9.6 The Gauss-Codazzi Equations -- 10 The Cartan Spaces of Order k as Dual of Finsler Spaces of Order k -- 10.1 C(k)n-Spaces -- 10.2 Geometrical Properties of the Cartan Spaces of Order k -- 10.3 Canonical Presymplectic Structures, Variational Problem of the Space C(kn) -- 10.4 The Cartan Spaces C(k)n as Dual of Finsler Spaces F(k)n -- 10.5 Canonical Nonlinear Connection. N-Linear Connections -- 10.6 Parallelism of Vector Fields in Cartan Space C(kn) -- 10.7 Structure Equations of Metrical Canonical N-Connection -- 10.8 Riemannian Almost Contact Structure of the Space C(kn) -- 11 Generalized Hamilton and Cartan Spaces of Order k. Applications to Hamiltonian Relativistic Optics -- 11.1 The Space GH(kn) -- 11.2 Metrical N-Linear Connections -- 11.3 Hamiltonian Relativistic Optics -- 11.4 The Metrical Almost Contact Structure of the Space GH(kn) -- 11.5 Generalized Cartan Space of Order k -- References.
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ISBN
9789401000703
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