The Geometry of Higher-Order Hamilton Spaces: Applications to Hamiltonian Mechanics

Asisknown,theLagrangeandHamiltongeometrieshaveappearedrelatively recently [76, 86]. Since 1980thesegeometrieshave beenintensivelystudied bymathematiciansandphysicistsfromRomania,Canada,Germany,Japan, Russia, Hungary,e.S.A. etc. PrestigiousscientificmeetingsdevotedtoLagrangeandHamiltongeome- tries and their applications have been organized in the above mentioned countries and a number ofbooks and monographs have been published by specialists in the field: R. Miron [94, 95], R. Mironand M. Anastasiei [99, 100], R. Miron, D. Hrimiuc, H. Shimadaand S.Sabau [115], P.L. Antonelli, R. Ingardenand M.Matsumoto [7]. Finslerspaces,whichformasubclassof theclassofLagrangespaces, havebeenthesubjectofsomeexcellentbooks, forexampleby:Yl.Matsumoto[76], M.AbateandG.Patrizio[1],D.Bao,S.S. Chernand Z.Shen [17]andA.BejancuandH.R.Farran [20]. Also, wewould liketopointoutthemonographsofM. Crampin [34], O.Krupkova [72] and D.Opri~,I.Butulescu [125],D.Saunders [144],whichcontainpertinentappli- cationsinanalyticalmechanicsandinthetheoryofpartialdifferentialequa- tions. Applicationsinmechanics, cosmology,theoreticalphysicsandbiology can be found in the well known books ofP.L. Antonelliand T.Zawstaniak [11], G. S. Asanov [14]' S. Ikeda [59], :VI. de LeoneandP.Rodrigues [73]. TheimportanceofLagrangeandHamiltongeometriesconsistsofthefact that variational problems for important Lagrangiansor Hamiltonians have numerous applicationsinvariousfields, such asmathematics, thetheoryof dynamicalsystems, optimalcontrol, biology,andeconomy. Inthisrespect, P.L. Antonelli'sremark isinteresting: "ThereisnowstrongevidencethatthesymplecticgeometryofHamilto- niandynamicalsystemsisdeeplyconnectedtoCartangeometry,thedualof Finslergeometry", (seeV.I.Arnold,I.M.GelfandandV.S.Retach [13]). The above mentioned applications have also imposed the introduction x RaduMiron ofthe notionsofhigherorder Lagrangespacesand, ofcourse, higherorder Hamilton spaces. The base manifolds ofthese spaces are bundles ofaccel- erations ofsuperior order. The methods used in the construction ofthese geometries are the natural extensions ofthe classical methods used in the edification ofLagrange and Hamilton geometries. These methods allow us to solvean old problemofdifferentialgeometryformulated by Bianchiand Bompiani [94]morethan 100yearsago,namelytheproblemofprolongation ofaRiemannianstructure gdefinedonthebasemanifoldM,tothetangent k bundleT M, k> 1. Bymeansofthissolutionofthe previousproblem, we canconstruct, for thefirst time,goodexamplesofregularLagrangiansand Hamiltoniansofhigherorder.

Table of 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 Algebraof 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 SpaceC(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.