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

The Geometry of Higher-Order Hamilton Spaces

Applications to Hamiltonian Mechanics

By: R. Miron

Hardcover Published: 31st October 2003
ISBN: 9781402015748
Number Of Pages: 247

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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.

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From the reviews:

"The book is devoted to an extensive study of formal-geometric properties of higher-order nondegenerate one-dimensional variational integrals. ... The author's approach is useful for the construction of geometric models ... . The book is precisely written, very clear, in principle self-contained and can be understood by non-specialists." (Jan Chrastina, Zentralblatt MATH, Vol. 1044 (19), 2004)

Geometry of the [kappa]-Tangent Bundle T[superscript [kappa]]Mp. 1
Lagrange Spaces of Higher Orderp. 27
Finsler Spaces of Order [kappa]p. 43
The Geometry of the Dual of [kappa]-Tangent Bundlep. 59
The Variational Problem for the Hamiltonians of Order [kappa]p. 77
Dual Semispray, Nonlinear Connectionsp. 97
Linear Connections on the Manifold T[superscript [kappa]]Mp. 125
Hamilton Spaces of Order [actual symbol not reproducible]p. 151
Subspaces in Hamilton Spaces of Order [kappa]p. 177
The Cartan Spaces of Order [kappa] as Dual of Finsler Spaces of Order [kappa]p. 199
Generalized Hamilton and Cartan Spaces of Order [kappa]. Applications to Hamiltonian Relativistic Opticsp. 221
Referencesp. 235
Indexp. 246
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9781402015748
ISBN-10: 1402015747
Series: Fundamental Theories of Physics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 247
Published: 31st October 2003
Publisher: Springer-Verlag New York Inc.
Country of Publication: NL
Dimensions (cm): 24.74 x 16.76  x 1.96
Weight (kg): 0.54

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