Foundations of celestial mechanics
Author(s)
Bibliographic Information
Foundations of celestial mechanics
（Graduate texts in physics）
Springer, c2022
Available at 2 libraries
Note
Includes bibliographical references and index
Description and Table of Contents
Description
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.
Table of Contents
- 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.9.1 Jacobi equation: special cases . . . . . . . . . . . . . . . 1824.9.2 2{body problem with Hamilton{Jacoby . . . . . . . . . . 1864.10 Element variation . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.10.1 Constant variation method: an example . . . . . . . . . 1944.11 Apsidal precession . . . . . . . . . . . . . . . . . . . . . . . . . 1974.12 Orbits in General Relativity . . . . . . . . . . . . . . . . . . . . 2005 Gravitational potential 2075.1 Gauss theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085.2 Theorens of Poisson and Laplace . . . . . . . . . . . . . . . . . 2105.3 Potential of a massive point . . . . . . . . . . . . . . . . . . . . 2125.4 Spherical bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155.5 Legendre equation . . . . . . . . . . . . . . . . . . . . . . . . . 2215.5.1 Spherical harmonics . . . . . . . . . . . . . . . . . . . . 2215.5.2 Legendre equation and spherical harmonics . . . . . . . . 2235.5.3 Associated Legendre function . . . . . . . . . . . . . . . 2255.5.4 Spherical harmonics of integer degree . . . . . . . . . . . 2275.6 Expansion of the potential . . . . . . . . . . . . . . . . . . . . . 2305.7 Thin layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2335.8 Homogeneous spheroid . . . . . . . . . . . . . . . . . . . . . . . 2355.9 Potential of a homogeneus ellipsoid . . . . . . . . . . . . . . . . 2385.10 Ellipsoid: outer point potential . . . . . . . . . . . . . . . . . . 2425.11 Potential: explicit form . . . . . . . . . . . . . . . . . . . . . . . 2445.12 Earth distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2475.13 Potential with dominating body . . . . . . . . . . . . . . . . . . 2495.14 Torus potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 251A Spherical trigonometry elements 261B Transformation formulas 267C Vector operators 271D The mirror theorem 275E Kepler's equation 277E.1 Lagrange's theorem . . . . . . . . . . . . . . . . . . . . . . . . . 277E.2 Fourier's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 279E.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . 280F Hydrogen atom 283F.1 Bohr's atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284F.2 Quantum approach . . . . . . . . . . . . . . . . . . . . . . . . . 285G Variation of constants 287H Lagrange multipliers 291H.1 Variation of constants . . . . . . . . . . . . . . . . . . . . . . . 292I Visual binary orbits 295J Three bodies: planarity 301K Gravitational impact 305L Poisson and Lagrange brackets 309L.1 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 309L.2 Lagrange brackets . . . . . . . . . . . . . . . . . . . . . . . . . . 311L.3 Brackets of Poisson and Lagrange . . . . . . . . . . . . . . . . . 313M Special functions 315M.1 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . 315M.2 Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317M.3 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 319M.3.1 First kind Bessel functions . . . . . . . . . . . . . . . . . 319M.3.2 Second kind Bessel functions . . . . . . . . . . . . . . . . 323M.3.3 Hankel functions . . . . . . . . . . . . . . . . . . . . . . 324M.3.4 Modi ed Bessel functions . . . . . . . . . . . . . . . . . 324M.3.5 Spherical Bessel functions . . . . . . . . . . . . . . . . . 325M.4 Hypergeometric function . . . . . . . . . . . . . . . . . . . . . . 327M.5 Error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329N Orthogonal functions 331N.1 Least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331N.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . 334N.3 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . 335N.4 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 343N.5 Application of spherical harmonics . . . . . . . . . . . . . . . . 348N.6 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . 350N.7 Application of Hermite polynomials . . . . . . . . . . . . . . . . 352N.8 Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . 352N.9 Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . 355O Harmonic functions 357O.1 Special problems . . . . . . . . . . . . . . . . . . . . . . . . . . 361P Principles of mechanics 363P.1 Variational formulation of motion . . . . . . . . . . . . . . . . . 363P.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 365P.3 Maupertuis's principle . . . . . . . . . . . . . . . . . . . . . . . 368P.4 Geodesic lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369Q Invariance and conservation 373Q.1 Continuous trajectories . . . . . . . . . . . . . . . . . . . . . . . 373Q.2 Time-invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Q.3 Invariance to translations . . . . . . . . . . . . . . . . . . . . . . 375Q.4 Rotational invariance . . . . . . . . . . . . . . . . . . . . . . . . 376R Numerical methods 377R.1 The Euler method . . . . . . . . . . . . . . . . . . . . . . . . . 377R.2 Implicit Runge-Kutta method . . . . . . . . . . . . . . . . . . . 378R.3 Runge-Kutta fourth-order method . . . . . . . . . . . . . . . . . 379
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