| The Geometry of Higher-Order Hamilton Spaces: Applications to Hamiltonian Mechanics 2003 Edition Contributor(s): Miron, R. (Author) |
|
 |
ISBN: 1402015747 ISBN-13: 9781402015748 Publisher: Springer OUR PRICE: $104.49 Product Type: Hardcover - Other Formats Published: October 2003 Annotation: 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; 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. |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Differential - Mathematics | Applied |
| Dewey: 516.36 |
| LCCN: 2003061855 |
| Series: Fundamental Theories of Physics |
| Physical Information: 0.77" H x 6.6" W x 9.74" (1.19 lbs) 247 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |