The Geometry of Higher-Order Hamilton Spaces: Applications to Hamiltonian Mechanics: 132 (Fundamental Theories of Physics, 132) - Hardcover

Miron, R.

Hardcover

ISBN 10: 1402015747 ISBN 13: 9781402015748

Publisher: Springer, 2003

Synopsis

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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Review

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)

Synopsis

This book is the first to present an overview of higher-order Hamilton geometry with applications to higher-order Hamiltonian mechanics. It is a direct continuation of the book "The Geometry of Hamilton and Lagrange Spaces", (Kluwer Academic Publishers, 2001). It contains the general theory of higher order Hamilton spaces H(k)n, k>=1, semisprays, the canonical nonlinear connection, the N-linear metrical connection and their structure equations, and the Riemannian almost contact metrical model of these spaces. In addition, the volume also describes new developments such as variational principles for higher order Hamiltonians; Hamilton-Jacobi equations; higher order energies and law of conservation; Noether symmetries; and Hamilton subspaces of order k and their fundamental equations. The duality, via Legendre transformation, between Hamilton spaces of order k and Lagrange spaces of the same order is pointed out. Also, the geometry of Cartan spaces of order k =1 is investigated in detail. This theory is useful in the construction of geometrical models in theoretical physics, mechanics, dynamical systems, optimal control, biology, economy etc.

Edition details

• Publisher: Springer
• Publication date: 2003
• Language: English
• ISBN 10: 1402015747
• ISBN 13: 9781402015748
• Binding: Hardcover
• Number of pages: 263