Celestial Mechanics

Celestial Mechanics

HARRY POLLARD
Volume: 18
Copyright Date: 1976
Edition: 1
Pages: 145
https://www.jstor.org/stable/10.4169/j.ctt5hh9bd
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  • Book Info
    Celestial Mechanics
    Book Description:

    The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook format. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. This monograph presents the basic mathematics underlying the subject of celestial mechanics. Chapter 1 formulates the central force problem and then deals with Kepler’s first and second laws, orbits under non-Newtonian attraction, elements of an orbit, the two-body system, the solar system, and disturbed motion. Chapter 2 introduces the n-body problem. Included in Chapter 2 are the Lagrange-Jacobi formula, Sundman’s theorem on total collapse, the three-body problem, and Lagrange’s and Euler’s solutions to the three-body problem. Chapter 3 is an introduction to Hamilton-Jacobi Theory.

    eISBN: 978-1-61444-018-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. PREFACE
    (pp. vii-viii)
    Harry Pollard
  3. Table of Contents
    (pp. ix-x)
  4. CHAPTER 1 THE CENTRAL FORCE PROBLEM
    (pp. 1-56)

    Celestial mechanics begins with the central force problem: to describe the motion of a particle of massmwhich is attracted to a fixed center0by a forcemf(r)which is proportional to the mass and depends only on the distancerbetween the particle and0.The functionfwill be called alaw of attraction.It is assumed to be continuous for$0 < r < \infty $

    Mathematically, the problem is easy to formulate. Indicate the position of the mass by the vector r directed from0.According to Newton’s second law, the motion of the particle is governed by the...

  5. CHAPTER 2 INTRODUCTION TO THE n-BODY PROBLEM
    (pp. 57-92)

    In then-body problem (better, then-particle problem) we are concerned with the motion ofnmass particles of masses${m_i}{\rm{ }}i = 1$, . . . ,nrespectively, attracting one another in pairs with the force$G{m_j}{m_k}r_{jk}^{ - 2}$where${r_{jk}}$is the distance between thekth andjth particle. We suppose that$n > 2.$

    Let0represent an origin fixed in space and let${r_i},{v_i}$denote the position and velocity vectors of theith particle. Then, by Newton’s second law, thekth particle satisfies the equation

    ${m_k}{{\ddot r}_k} = \sum\limits_{\scriptstyle j = 1 \hfill \atop\scriptstyle j \ne k \hfill}^n {\frac{{G{m_j}{m_k}}}{{r_{jk}^2}}\frac{{{r_j} - {r_k}}}{{{r_{jk}}}}}

    where the right-hand side represents the total force exerted on the kth particle by the remaining...

  6. CHAPTER 3 INTRODUCTION TO HAMILTON-JACOBI THEORY
    (pp. 93-132)

    We begin by recalling some basic facts from advanced calculus. Let the functions

    ${y_k} = {y_k}\left( {{x_1}{\rm{, }}{\rm{. }}{\rm{. }}{\rm{. , }}{x_m}} \right),{\rm{ }}k = 1,{\rm{ }}{\rm{. }}{\rm{. }}{\rm{. , }}m$(1.1)

    denote a transformation of variables in anm-dimensional region. It will be supposed that each of the partial derivatives

    $\partial {y_k}/\partial {x_l}$exists and is continuous. The matrix m with entries$\partial {y_k}/\partial {x_l}$(k= row index,l= column index) is known as theJacobian matrixof the transformation; in more detail it is

    $\mathfrak = \left( {\begin{array}{*{20}{c}} {\frac{{\partial {y_1}}}{{\partial {x_1}}}} & {\frac{{\partial {y_1}}}{{\partial {x_2}}}} & \cdots & {\frac{{\partial {y_1}}}{{\partial {x_m}}}} \\ \vdots & {} & {} & {} \\ {\frac{{\partial {y_m}}}{{\partial {x_1}}}} & {\frac{{\partial {y_m}}}{{\partial {x_2}}}} & \cdots & {\frac{{\partial {y_m}}}{{\partial {x_m}}}} \\ \end{array}} \right)$

    The determinant of$\mathfrak{W}$, written$\left| \mathfrak{W} \right|$, is called the Jacobian of the transformation (1.1). It is known that if the transformation (1.1) carries a particular point$(x_1^0,{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }},{\text{ }}x_m^0)$into the point$(y_1^0,{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }},{\text{ }}y_m^0)$and if the Jacobian does not vanish...

  7. INDEX
    (pp. 133-134)