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TOPICS in CELESTIAL MECHANICS 1. the Newtonian N-Body Problem

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TOPICS in CELESTIAL MECHANICS 1. the Newtonian N-Body Problem TOPICS IN CELESTIAL MECHANICS RICHARD MOECKEL 1. The Newtonian n-body Problem Celestial mechanics can be defined as the study of the solution of Newton's differ- ential equations formulated by Isaac Newton in 1686 in his Philosophiae Naturalis Principia Mathematica. The setting for celestial mechanics is three-dimensional space: 3 R = fq = (x; y; z): x; y; z 2 Rg with the Euclidean norm: p jqj = x2 + y2 + z2: A point particle is characterized by a position q 2 R3 and a mass m 2 R+. A motion of such a particle is described by a curve q(t) where t runs over some interval in R; the mass is assumed to be constant. Some remarks will be made below about why it is reasonable to model a celestial body by a point particle. For every motion of a point particle one can define: velocity: v(t) =q _(t) momentum: p(t) = mv(t): Newton formulated the following laws of motion: Lex.I. Corpus omne perservare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare 1 Lex.II. Mutationem motus proportionem esse vi motrici impressae et fieri secundem lineam qua vis illa imprimitur. 2 Lex.III Actioni contrarium semper et aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. 3 The first law is statement of the principle of inertia. The second law asserts the existence of a force function F : R4 ! R3 such that: p_ = F (q; t) or mq¨ = F (q; t): In celestial mechanics, the dependence of F (q; t) on t is usually indirect; the force on one body depends on the positions of the other massive bodies which in turn depend on t. The third law postulates the symmetry of the mutual interaction of two bodies which will apply, in particular, to the gravitational interaction. Date: May 4, 2020. 1Every body continues in its quiescent state or moves uniformly in direction, unless it is compelled by impressed forces to change its state. 2The change of momentum is proportional to the motive force impressed and takes place along the line where this force is impressed. 3To every action there is always an equal and opposite reaction: the actions of two bodies on one another are always equal and aimed in opposite directions. 1 2 RICHARD MOECKEL The n-body problem is about the motion of n point particles under the influence of their mutual gravitational attraction. Each particle has a mass mi > 0 and 3 position, velocity and momentum vectors qi; vi; pi 2 R . The whole system can be described using the vectors q; v; p 2 R3n where q = (q1; q2; : : : ; qn) v = (v1; v2; : : : ; vn) p = (m1v1; m2v2; : : : ; mnvn): According to Newton, the gravitational force acting on particle i due to the presence of particle j is Gmimj(qj − qi) Fij = 3 jqi − qjj where G is a constant. Note that Fij acts along the line containing the masses. It's proportional to the product of the two masses and inversely proportional to the distance between them (see Figure 1). The force produced on mj by mi is Fji = −Fij by Newton's third law. By choosing the units of mass, one can arrange that G = 1 and this will be assumed from now on (see Exercise 1.1). 1.0 m2 F21 0.5 -2.0 -1.5 -1.0 -0.5 0.5 1.0 m1 F12 -0.5 Figure 1. Newtonian gravitational forces. The force on the i-th mass due to the other n − 1 masses is: X X mimj(qj − qi) Fi = Fij = 3 : jqj − qij j6=i j6=i This can be written: Fi(q) = riU(q) where X mimj (1) U(q) = jq − q j (i;j) i j i<j and ri is the partial gradient operator: @U @U @U 3 riU = ; ; 2 R : @xi @yi @zi The function U(q) will be called the Newtonian gravitational potential function. V (q) = −U(q) is the gravitational potential energy. Newton's second law becomes p_i = miq¨i = riU(q) i = 1,. ,n: or, more concisely (2)p _ = rU(q) or Mq¨ = rU(q) TOPICS IN CELESTIAL MECHANICS 3 where r is the gradient operator in R3n and M is the 3n × 3n mass matrix M = diag(m1; m1; m1; : : : ; mn; mn; mn): It is worth digressing at this point to note two important, special features of the Newtonian interparticle potential: m m i j : jqi − qjj First of all, the presence of the factor mimj has the effect that the equation for the acceleration of the i-th mass, 1 q¨i = Fi mi is independent of mi. This corresponds to the observation, notably by Galileo, that the trajectory of a falling body is independent of its mass. Figure 2. Masses falling from a tower. Second, the fact that the potential is inversely proportional to the distance be- tween the particles provides some justification for the modeling of celestial bodies by point particles. While such bodies are not even approximately pointlike, they are approximately spherically symmetric. It turns out that with the Newtonian poten- tial, spherically symmetric bodies behave as if their total mass were concentrated at their centers. To see this, consider a more general massive body, specified by giving a bounded subset B ⊂ R3 together with a continuous mass density functions ρ. The gravita- tional force exerted by such a mass distribution on a point mass m at position q is F = rU(q) where Z m U(q) = dm B jq − pj where the triple integral is over p = (x; y; z) 2 B and dm = ρ(x; y; z)dxdydz. 4 RICHARD MOECKEL 3 Proposition 1.1. Suppose B is a ball of radius R centered at q0 2 R and ρ is a spherically symmetric density function. Then the mutual Newtonian potential of the ball and a mass m at any point q with jqj > R is m m U(q) = 0 jq0 − qj where m0 is the total mass in the ball. Proof. Using the symmetry of the Euclidean distance under rotations and trans- lations, it is no loss of generality to assume q0 = (0; 0; 0) and q = (0; 0; z); z > R. Using spherical coordinates (x; y; z) = r(cos θ sin φ, sin θ sin φ, cos φ), the spherical symmetry means that the density function depends only on r and then Z R Z π Z 2π m ρ(r)r2 sin φ dθ dφ dr U(q) = i : p 2 2 0 0 0 r + z − 2rz cos φ It is an exercise to carry out the first two integrals to show Z R m0m 2 U(q) = m0 = 4πr ρ(r) dr: z 0 QED There is an alternative proof of this result, based on the fact that f(x; y; z) = 1=jq − pj is a harmonic function of p = (x; y; z), that is, fxx + fyy + fzz = 0. The well-known mean value theorem for harmonic functions states that the average value of a harmonic function over a sphere is equal to the value at the center of the sphere, which can be proved using the divergence theorem (see exercise 1.3). Fixing a value of r and applying this to the function f(x; y; z) = mρ(r)=jq − pj and the sphere Sr = fjq − q0j = rg gives Z 4πr2ρ(r) m f = : Sr jq0 − qj Then integration over 0 ≤ r ≤ R completes the proof. Although R3 is the natural home of celestial mechanics, it is useful and interesting to consider the point-mass n-body problem in Rd for any positive integer d. In this d case the position vectors are q1; : : : ; qn 2 R and the vectors q; v; p are elements of Rdn. Newton's equations (2) form a system of real-analytic, second order differential equations on the configuration space, X = Rdn n ∆, where ∆ = fq : qi = qj for some i 6= jg is the collision set. It can be transformed in the usual way into a first-order system in the phase space: dn dn TX = X × R = f(q; v): q 2 X and v 2 R g namely: q_ = v (3) v_ = M −1rU(q): The notation TX takes note of the fact that the phase space is the tangent bundle of X. The Newtonian n-body problem is to study the solutions of equations (3). A solution to (3) is a differentiable curve (q(t); v(t)) where the time t lies in some interval I. Since the differential equation is given by real-analytic functions on phase space, the solutions will be real-analytic functions of time and of their TOPICS IN CELESTIAL MECHANICS 5 initial conditions, that is, they are given locally by convergent power series. Since the phase space is not compact, it may not be possible to extend solutions for all time t 2 R. In general the maximal interval of existence will be of the form I = (a; b) with −∞ ≤ a < b ≤ 1. By the general theory of ordinary differential equations, if b < 1 then as t ! b−,(q(t); v(t)) must leave every compact subset of Rdn n ∆ × Rdn and similarly for a > −∞. For example, this can happen if q(t) converges to a collision configurationq ¯ 2 ∆ (see exercise 1.5). Exercise 1.1. Using units of kilograms for mass, meters for distance and seconds −11 m3 for time, the gravitational constant is G ' 6:674 × 10 kg·sec2 .
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