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MATHEMATICS RESEARCH DEVELOPMENTS V C ARIATIONAL ALCULUS T S ON IME CALES No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services. M R ATHEMATICS ESEARCH D EVELOPMENTS Additional books and e-books in this series can be found on Nova’s website under the Series tab. MATHEMATICS RESEARCH DEVELOPMENTS V C ARIATIONAL ALCULUS T S ON IME CALES SVETLIN G. GEORGIEV Copyright © 2018 by Nova Science Publishers, Inc. All rights reserved. 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Library of Congress Cataloging-in-Publication Data ISBN: (cid:28)(cid:26)(cid:27)(cid:16)(cid:20)(cid:16)(cid:24)(cid:22)(cid:25)(cid:20)(cid:23)(cid:16)(cid:22)(cid:26)(cid:25)(cid:16)(cid:27)(cid:3)(cid:11)(cid:72)(cid:37)(cid:82)(cid:82)(cid:78)(cid:12) Published by Nova Science Publishers, Inc. † New York Contents Preface ix 1 ElementsoftheTimeScaleCalculus 1 1.1. ForwardandBackwardJumpOperators, GraininessFunction . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3. Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4. TheExponentialFunction. . . . . . . . . . . . . . . . . . . . . 19 1.5. HyperbolicandTrigonometricFunctions . . . . . . . . . . . . . 29 1.6. TheMultidimensionalTimeScaleCalculus . . . . . . . . . . . 30 1.7. LineIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 1.8. Green’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . 72 1.9. AdvancedPracticalProblems . . . . . . . . . . . . . . . . . . . 75 2 DynamicSystemsonTimeScales 79 2.1. StructureofDynamicSystemsonTimeScales . . . . . . . . . . 79 2.2. ConstantCoefficients . . . . . . . . . . . . . . . . . . . . . . . 115 2.3. AdvancedPracticalProblems . . . . . . . . . . . . . . . . . . . 127 3 Functionals 131 3.1. DefinitionforFunctionals. . . . . . . . . . . . . . . . . . . . . 131 3.2. Self-AdjointSecondOrderMatrixEquations . . . . . . . . . . 133 3.3. Jacobi’sCondition . . . . . . . . . . . . . . . . . . . . . . . . 144 3.4. SturmianTheory . . . . . . . . . . . . . . . . . . . . . . . . . 153 vi Contents 4 LinearHamiltonianDynamicSystems 157 4.1. LinearSymplecticDynamicSystems . . . . . . . . . . . . . . . 157 4.2. HamiltonianSystems . . . . . . . . . . . . . . . . . . . . . . . 163 4.3. ConjoinedBases . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.4. RiccatiEquations . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.5. Picone’sIdentity . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.6. “Big”LinearHamiltonianSystems . . . . . . . . . . . . . . . . 198 4.7. PositivityofQuadraticFunctionals . . . . . . . . . . . . . . . . 211 5 TheFirstVariation 219 5.1. TheDubois-ReymondLemma . . . . . . . . . . . . . . . . . . 219 5.2. TheVariationalProblem . . . . . . . . . . . . . . . . . . . . . 225 5.3. TheEuler-LagrangeEquation. . . . . . . . . . . . . . . . . . . 233 5.4. Legendre’sCondition . . . . . . . . . . . . . . . . . . . . . . . 241 5.5. Jacobi’sCondition . . . . . . . . . . . . . . . . . . . . . . . . 245 5.6. AdvancedPracticalProblems . . . . . . . . . . . . . . . . . . . 248 6 HigherOrderCalculusofVariations 251 6.1. StatementoftheVariationalProblem . . . . . . . . . . . . . . . 251 6.2. Euler’sEquation. . . . . . . . . . . . . . . . . . . . . . . . . . 254 6.3. AdvancedPracticalProblems . . . . . . . . . . . . . . . . . . . 260 7 DoubleIntegralCalculusofVariations 261 7.1. StatementoftheVariationalProblem . . . . . . . . . . . . . . . 262 7.2. FirstandSecondVariation . . . . . . . . . . . . . . . . . . . . 263 7.3. Euler’sCondition . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.4. AdvancedPracticalProblems . . . . . . . . . . . . . . . . . . . 272 8 Noether’sSecondTheorem 275 8.1. InvarianceunderTransformations . . . . . . . . . . . . . . . . 275 8.2. Noether’sSecondTheoremwithout TransformationsofTime . . . . . . . . . . . . . . . . . . . . . 279 8.3. Noether’sSecondTheoremwithTransformationsofTime . . . 281 8.4. Noether’sSecondTheorem-DoubleDelta IntegralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Contents vii References 291 AuthorContactInformation 293 Index 295 Preface This book encompasses recent developments of variational calculus on time scales. It is intended for use in the field of variational calculus and dynamic calculus on time scales. It is also suitable for graduate courses in the above fields. The book contains eight chapters. The chapters in the book are ped- agogically organized. This book is specially designed for those who wish to understandvariationalcalculusontimescaleswithouthavingextensivemathe- maticalbackground. The basic definitions of forward and backward jump operators are due to Hilger. InChapter1aregivenexamplesofjumpoperatorsonsometimescales. The graininessfunction,which isthe distance from a pointtothe closed point on the right, is introduced in this chapter. They are given the definitions for delta derivative and delta integral and they are deducted some of their proper- ties. They are introduced the exponential function and the trigonometric and hyperbolicfunctions. Inthischapter isgivenan expositionof themultidimen- sional dynamic calculus on time scales. They are introduced line integrals on timescalesandGreen’sformula. Thebasicresultsinthischaptercanbefound in[4] and[5]. Chapter2introducesdynamicsystemsontimescales. Itiscon- sidered the case of constant coefficients. Chapter 3 deals with functionalsand self-adjointsecondordermatrixequations. ItisformulatedandprovedJacobi’s condition. ItisintroducedSturmiantheory. Chapter4 isconcernedwithlinear Hamiltoniandynamicsystems. Theyaredeductedsomeofthebasicproperties of the symplectic dynamic systems and Hamiltonian dynamic systems. They are introduced Riccati equations and it is proved Picconi’s identity. They are given some criterionsfor positivedefinitenessof quadratic functionals. Chap- ter 5 is devoted on the first and second variation. It is formulated and proved