5 1 0 2 n a J Symmetries of Differential equations and Applications in Relativistic 1 2 Physics. ] c q - r g [ Andronikos Paliathanasis 1 v 9 Athens, 2014 2 1 5 0 . 1 0 5 1 : v i X r a ii iii iv Preface This thesis is part of the PhD program of the Department of Astronomy, Astrophysics and Mechanics of the Faculty of Physics of the University of Athens, Greece. In memory of my grandmother Amalia v vi Abstract Inthisthesis,westudytheoneparameterpointtransformationswhichleaveinvariantthedifferentialequations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new geometric method which relates the point symmetries of the differential equations with the collineations of the underlying manifold where the motion occurs. This geometric method is applied in order thetwoandthreedimensionalNewtoniandynamicalsystemstobeclassifiedinrelationtothepointsymmetries; to generalize the Newtonian Kepler-Ermakov system in Riemannian spaces; to study the symmetries between classical and quantum systems and to investigate the geometric origin of the Type II hidden symmetries for the homogeneous heat equation and for the Laplace equation in Riemannian spaces. At last but not least, we apply this geometric approachin order to determine the dark energy models by use the Noether symmetries as a geometric criterion in modified theories of gravity. vii Acknowledgement This doctoral thesis would not have been possible without the guidance and support of numerous people and I wish to express my gratitude here. Professor Peter G.L. Leach deserves a special mention. It was your research that inspired me to wade into the symmetriesofdifferentialequationsandyourencouragementwhenI firstsetaboutmyownresearchproved invaluable. IamgratefulformycollaborationwithmySupervisor,ProfessorMichaelTsamparlis. Yourfaithinmyabilities and your advice, on both my research and my career, throughout the years, has allowed me to grow to the scientist I am today. You introduced me to the subject of differential geometry and taught me the geometric thought process. I would especially like to thank both my advisors Dr. Spyros Basilakos, who opened up to me the subject of cosmology, and Dr. Christos Efthimiopoulos; the discussions we had on the topic have been priceless. Also, I feel fortunate and grateful for my examination committee, Prof. S. Capozziello, Prof. P. J. Ioannou, Prof T. Apostolatos and Prof A. H. Kara. Without your important comments I could not have completed this thesis. Above all, I feel grateful for my friends and family. Stelios, Sifis, Venia and Thanos your friendship and encouragement all these years have been my most valuable support mechanism. Last but not least, I would not be who I am today without my family. My parents Maria and Sotiris, and my grandfather Ioannis, youhave always believed in and youhave supported me in all the important decisions ofmylife. Marianthi,yourpatience,yourkindnessandyourhelponalllevels,ismorethananybrotherdeserves. viii Contents I Introduction 1 1 Introduction 3 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Summary of Part I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Summary of Part II: Symmetries of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Summary of Part III: Symmetries of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Summary of Part IV: Noether symmetries and theories of gravity . . . . . . . . . . . . . . 6 2 Point transformations and Invariant functions 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Point Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Infinitesimal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Invariant Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.3 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Lie symmetries of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Prolongationof point transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Lie symmetries of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Lie symmetries of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Lie B¨acklund symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Noether point symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.1 Noether symmetries of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.2 Noether point symmetries of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Collineations of Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7.1 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7.2 Motions of Riemannian spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7.3 Symmetries of the connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ix x CONTENTS II Symmetries of ODEs 39 3 Lie symmetries of geodesic equations 41 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 The Lie symmetry conditions in an affine space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Lie symmetries of autoparallelequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 Lie and Noether symmetries of geodesic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4.1 Noether symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5.1 The geodesic symmetries of Einstein spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5.2 The geodesic symmetries of G¨odel spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5.3 The geodesic symmetries of Taub spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5.4 The geodesic symmetries of a 1+3 decomposable spacetime metric . . . . . . . . . . . . . 56 3.5.5 The geodesic symmetries of the FRW metrics . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.A The determining equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4 Motion on a curved space 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Lie symmetries of a dynamical system in a Riemannian space . . . . . . . . . . . . . . . . . . . . 68 4.3 Noether symmetries of a dynamical system in a Riemannian space . . . . . . . . . . . . . . . . . 71 4.4 2D autonomous systems which admit Lie/Noether point symmetries . . . . . . . . . . . . . . . . 73 4.4.1 2D autonomous Newtonian systems which admit Noether symmetries . . . . . . . . . . . 75 4.5 3D autonomous Newtonian systems admit Lie/Noether point symmetries . . . . . . . . . . . . . 77 4.5.1 3D autonomous Newtonian systems admit Lie point symmetries . . . . . . . . . . . . . . 78 4.5.2 3D autonomous Newtonian systems which admit Noether point symmetries . . . . . . . . 80 4.6 Motion on the two dimensional sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.6.1 Noether Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.7.1 Lie point symmetries of the Kepler-Ermakovsystem. . . . . . . . . . . . . . . . . . . . . . 85 4.7.2 Point symmetries of the H`enon - Heiles potential . . . . . . . . . . . . . . . . . . . . . . . 86 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.A Proof of main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.B Tables of Newtonian systems admit Lie and Noether symmetries . . . . . . . . . . . . . . . . . . 94