This book provides an introduction to classical celestial mechanics. It is based on lectures delivered by the authors over many years at both Padua University (MC) and V.N. Karazin Kharkiv National University (EB). The book aims to provide a mathematical description of the gravitational interaction of celestial bodies. The approach to the problem is purely formal. It allows the authors to write equations of motion and solve them to the greatest degree possible, either exactly or by approximate techniques, when there is no other way. The results obtained provide predictions that can be compared with the observations. Five chapters are supplemented by appendices that review certain mathematical tools, deepen some questions (so as not to interrupt the logic of the mainframe with heavy technicalities), give some examples, and provide an overview of special functions useful here, as well as in many other fields of physics. The authors also present the original investigation of torus potential. This book is aimed at senior undergraduate students of physics or astrophysics, as well as graduate students undertaking a master's degree or Ph.D.

Autorentext

Elena Bannikova graduated in astronomy from the V.N. Karazin Kharkiv National University. From 2002-2006, she was Scientific Researcher at the Institute of Astronomy of the University. Since 2006, she is Associate Professor of the Astronomy Department of the University and Scientific Researcher at the Institute of Radio Astronomy, National Academy of Ukraine; winner of the regional competition "The best young scientist of Kharkiv region - 2006" in the nomination "physics and astronomy;" Fellow of the young scientist's grants by President of Ukraine (2009-2010); Principal Investigator of the young scientists' grants "Interaction of supernova remnants with molecular clouds" by National Academy of Ukraine (2009-2010); Member of the International Astronomical Union and the European Astronomical Society. Her fields of research are active galactic nuclei, gravitational potential, and vortex hydrodynamics.

Massimo Capaccioli is Italian Astrophysicist. He has served as Professor of astronomy at the Universities of Padua and then of Naples Federico II, where he is currently Emeritus. The results of his studies, dealing mainly with the dynamics and evolution of stellar systems and the observational cosmology, are presented in over 550 scientific articles in international journals. For a long time director of the Capodimonte Astronomical Observatory in Naples, he has conceived and managed, in synergy with the European Southern Observatory (ESO), the construction of the VLT Survey Telescope (VST), one of the largest reflectors fully dedicated to astronomical surveys. He has chaired the Italian Astronomical Society (SAIt) for a decade and for a three-year turn the National Society of Sciences, Letters, and Arts in Naples. He has collaborated with various Italian newspapers and with the national public broadcasting company of Italy (RAI). He has authored both university manuals and popular books. The list of his honors includes the title of Commander of the Italian Republic for scientific merits (2005), the honorary professorship granted by the University of Moscow Lomonosov in 2010, the honorary doctor-degrees by the Universities of Dubna (Russia, 2015), Kharkiv (Ukraine, 2017), and Pyatigorsk (Russia, 2019), and the medals Struve (2010; Russian Academy of Sciences), Tacchini (2013; SAIt), Karazin (2019; Karazin University, Kharkiv, Ukraine), and Gamov (2019: University of Odessa, Ukraine). He is Member of some academies in Italy and of the Academia Europaea, and since 2021 foreign Member of the National Academy of Sciences of Ukraine.

Inhalt

1 N-body problem 111.1 Self-gravitating systems of massive points . . . . . . . . . . . . . 141.2 Fundamental rst integrals . . . . . . . . . . . . . . . . . . . . . 171.2.1 Conservation of momentum . . . . . . . . . . . . . . . . 181.2.2 Angular momentum conservation . . . . . . . . . . . . . 211.2.3 Energy conservation . . . . . . . . . . . . . . . . . . . . 231.3 Barycentric and relative systems . . . . . . . . . . . . . . . . . . 251.4 N-body problem solution . . . . . . . . . . . . . . . . . . . . . . 261.5 Virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 The two-body problem 312.1 Motion about center of mass . . . . . . . . . . . . . . . . . . . . 342.2 Reduction to the plane . . . . . . . . . . . . . . . . . . . . . . . 382.3 E ective potential energy . . . . . . . . . . . . . . . . . . . . . 402.4 The trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5 Laplace{Runge{Lenz vector . . . . . . . . . . . . . . . . . . . . 432.6 Geometry of conics . . . . . . . . . . . . . . . . . . . . . . . . . 462.6.1 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6.2 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.6.3 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . 522.7 Conic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.7.1 Elliptical orbit . . . . . . . . . . . . . . . . . . . . . . . . 562.7.2 Parabolic orbit . . . . . . . . . . . . . . . . . . . . . . . 612.7.3 Hyperbolic orbit . . . . . . . . . . . . . . . . . . . . . . 622.8 Keplerian elements . . . . . . . . . . . . . . . . . . . . . . . . . 632.9 Ephemerides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.10 The method of Laplace . . . . . . . . . . . . . . . . . . . . . . . 702.11 Ballistics and space ight . . . . . . . . . . . . . . . . . . . . . . 803 The three-body problem 853.1 Stationary solutions . . . . . . . . . . . . . . . . . . . . . . . . . 873.1.1 Collinear solutions . . . . . . . . . . . . . . . . . . . . . 923.1.2 Triangular solutions . . . . . . . . . . . . . . . . . . . . . 943.2 The restricted problem . . . . . . . . . . . . . . . . . . . . . . . 973.3 Zero{velocity curves . . . . . . . . . . . . . . . . . . . . . . . . 1013.3.1 The (x; y) plane . . . . . . . . . . . . . . . . . . . . . . 1023.3.2 The (x; z) plane . . . . . . . . . . . . . . . . . . . . . . . 1043.3.3 The (y; z) plane . . . . . . . . . . . . . . . . . . . . . . . 1053.4 About the Lagrangian points . . . . . . . . . . . . . . . . . . . . 1073.5 Stability of the Lagrangian points . . . . . . . . . . . . . . . . . 1083.5.1 The equilibrium conditions . . . . . . . . . . . . . . . . . 1083.5.2 Collinear solutions . . . . . . . . . . . . . . . . . . . . . 1103.5.3 Triangular solutions . . . . . . . . . . . . . . . . . . . . . 1113.6 Variation of the elements . . . . . . . . . . . . . . . . . . . . . . 1133.6.1 Variation of the orientation elements . . . . . . . . . . . 1163.6.2 Variation of the geometric elements . . . . . . . . . . . . 1184 Analytical mechanics 1254.1 Lagrange function . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.2 Generalized coordinates . . . . . . . . . . . . . . . . . . . . . . 1294.3 Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . . . 1314.4 Hamilton function . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.5 Canonical equations . . . . . . . . . . . . . . . . . . . . . . . . . 1374.6 Constants of motion . . . . . . . . . . . . . . . . . . . . . . . . 1384.7 Elliptical orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.8 Canonical transformations . . . . . . . . . . . . . . . . . . . . . 1504.8.1 Characteristic function . . . . . . . . . . . . . . . . . . . 1514.8.2 Forms of the characteristic function . . . . . . . . . . . . 1544.8.3 Canonicity conditions . . . . . . . . . . . . . . . . . . . . 1554.8.4 Canonical invariants . . . . . . . . . . . . . . . . . . . . 1614.8.5 In nitesimal canonical transformations . . . . . . . . . . 1634.8.6 Canonical systems of motion constants . . . . . . . . . . 1684.8.7 Canonical elements for elliptical orbit . . . . . . . . . . . 1754.9 Jacobi equation . . . . . . …