Capture Dynamics and Chaotic Motions in Celestial Mechanics

With Applications to the Construction of Low Energy Transfers
By Edward Belbruno

Princeton University Press

Copyright © 2004 Princeton University Press
All right reserved.

ISBN: 978-0-691-09480-9


Chapter One

Introduction to the N-Body Problem

The basic differential equations are defined that we will use throughout this book. These include the Newtonian n-body problem in section 1.1, and the planar three-body problem using Jacobi coordinates in section 1.2. In section 1.3, we derive the classical solutions for the two-body problem. In section 1.4 regularization is defined and collision is regularized via the classical Levi-Civita transformation and the Kustaanheimo-Stiefel transformation. Section 1.5 introduces the equations of motion for the restricted three-body problem in different variations and coordinate systems. This problem is the main focus of subsequent chapters. Also, we discuss briefly in section 1.1 the global behavior of solutions in the n-body problem having collision and noncollision singularities. Key results are stated, including Sundman's basic theorems and the Painlevé conjecture proven by Xia. This material serves as background introductory material and provides an historical perspective. The integrals of motion for the n-body problem are also derived. In section 1.6 geo desic equivalent flows on spaces of constant curvature are derived using the Euler-Lagrange differential equations, and their equivalence with the flow of the two-body problem is described. A new proof is given for this equivalence which is substantially shorter than previous proofs. The geodesic flows give rise to n-dimensional regularizations.

1.1 THE N-BODY PROBLEM

We consider the n-body problem, n [greater than or equal to] 2. Of particular interest will be the case n = 3 for the Newtonian three-body problem. Before this problem and variations of it are defined, we define the general n-body problem and discuss existence of solutions. The basic conservation laws are derived.

It is defined by the motion of n [greater than or equal to] 2 mass particles [P.sub.k] of masses [m.sub.k] > 0, k = 1, 2,..., n, moving in three-dimensional space [x.sub.1], [x.sub.2], [x.sub.3] under the classical Newtonian inverse square gravitational force law. We assume the Cartesian coordinates of the kth particle are given by the real vector [x.sub.k] = ([x.sub.k1], [x.sub.k2], [x.sub.k3]) [member of] [R.sup.3]. The differential equations defining the motion of the particles are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)

k = 1, 2,..., n, where [r.sub.jk] = |[x.sub.j] - [x.sub.k]| = [square root of [summation].sup.3.sub.i=1][([x.sub.ji] - [x.sub.ki]).sup.2]] is the Euclidean distance between the kth and jth particles, and . [equivalent to] d/dt. Equation (1.1) expresses the fact that the acceleration of the kth particle [P.sub.k] is due to the sum of the forces of the n - 1 particles [P.sub.i], i = 1,..., n, i [not equal to] k. The time variable t [member of] [R.sup.1]. Equation (1.1) represents 3n second order differential equations. This equation can be put into a simpler form by first dividing both sides through by [m.sub.k], and expressing it as a first order system,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2)

where [v.sub.k] = ([v.sub.k1], [v.sub.k]2, [v.sub.k3]) = ([[??].sub.k1], [[??].sub.k2], [[??].sub.k1]) [member of] [R.sup.3 are the velocity vectors of the kth particle,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

U = U ([x.sub.1],..., [x.sub.n]) is a real-valued function of 3n variables [x.sub.kj], j = 1, 2, 3, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

k = 1, 2,..., n. Equation (1.2) represents a system of 6n first order differential equations for the 6n variables [x.sub.kl], [v.sub.kl], k = 1, 2,..., n; l = 1, 2, 3. U is the potential energy. G is the universal gravitational constant.

If we assume [r.sub.jk] > 0, then U is a well-defined function and is a smooth function in the 3n variables [x.sub.jk], where smooth means that U has continuous partial derivatives of all orders in the variables [x.sub.jk] and is real analytic. For notation, we set x = ([x.sub.1], [x.sub.2],..., [x.sub.n]). Then x [member of] [R.sup.3n]. Similarly, v = ([v.sub.1],..., [v.sub.n]) [member of] [R.sup.3n]. With this notation, U = U (x).

System (1.2) is of the form

[??] = f (y), (1.3)

where y = (x, v) [member of] [R.sup.6n], and also where f = (v, [m.sup.-1.sub.1] [partial derivative]U/[partial derivative][x.sub.1],..., [m.sup.-1.sub.n] [partail derivative]U/[partial derivative][x.sub.n]) [member of] [R.sup.6n]. Thus, the standard existence and uniqueness theorems of ordinary differential equations can be applied to (1.3), and hence (1.2).

Since f = ([f.sub.1],..., [f.sub.6n]) is a smooth vector function of y, then these theorems guarantee that through any initial point y([t.sub.0]) = [y.sub.0] at initial time [t.sub.0] there exists a locally unique solution for |t - [t.sub.0]| < [delta], where [delta] is sufficiently small. This can be made more precise: If the real functions [f.sub.k] satisfy |[f.sub.k]| < M, k = 1, 2,..., 6n, in a domain |y - [y.sub.0]| < p, then

[delta] = p/(1 + 6n)M

(see [204]).

A system of integrals exist for (1.1) which can be used to reduce the dimension of the (6n+1)-dimensional coordinate space (x, y, t). An integral is a real-valued function of the 6n + 1 variables [x.sub.kj], [v.sub.kj], t which is constant when evaluated along a solution of (1.1). Let x(t), v(t) represent a solution of (1.1).

Definition 1.1 A integral of (1.1) is a real-valued function I(x, v, t) such that

d/dt I (x(t), v(t), t) = 0,

where the solution x(t), v(t) is defined.

This definition implies that I = c = constant along the given solution. This defines a 6n-dimensional integral manifold,

[I.sup.-1](0) = {(x, v, t) [member of] [R.sup.6n+1]|I = c},

on which the solutions will lie.

Thus, an integral constrains the motion of the mass particles and can be used to reduce the dimension of the space of 6n+1 coordinates, [x.sub.kl], [v.sub.kl], t, k = 1, 2,..., n; l = 1, 2, 3 by 1, by solving for one of the coordinates as a function of the 6n remaining coordinates, at least implicitly. For notation we refer to the 6n-dimensional real space of coordinates (x, v) [member of] [R.sup.3n x [R.sup.3n, as the phase space, and (x, v, t) [member of] [R.sup.3n] x [R.sup.3n] x [R.sup.1 as the extended phase space.

When two or more integrals [I.sub.1](x, v, t), [I.sub.2](x, v, t) exist for (1.1), they are called independent if the gradient vectors [[partial derivative]x.sub.,v,t] = ([[partial derivative].sub.[x.sub.1]],..., [[partial derivative].sub.[x.sub.n]], [[partial derivative].sub.[v.sub.1]],..., [[partial derivative].sub.[v.sub.n]], [[partial derivative].sub.t]) of [I.sub.1] and [I.sub.2] are independent. This implies that the rank of the 2 x (6n + 1) matrix

[partial derivative]([I.sub.1], [I.sub.2])[partial derivative](x, v, t)

is in general 2.

Equation (1.1) has a set of 10 independent algebraic integrals. These are given by the three classical conservation laws of linear momentum, energy, and angular momentum. We will derive these now.

First, we derive the conservation of linear momentum. To do this, we add up the right side of (1.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

j [not equal to] k. S = 0 is verified, since each term [x.sub.j] - [x.sub.k] occurs with its negative, and mutual cancellations occur for all the terms. This implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Setting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [rho] = ([[rho].sub.1], [[rho].sub.2], [[rho].sub.3]) [member of] [R.sup.3] is the center of mass vector of the particles, then [??] = 0. This yields

[rho] = [c.sub.1]t + [c.sub.2], (1.4)

|t| < [delta], where [c.sub.1], [c.sub.2] yield six constants which are uniquely determined from the initial conditions [x.sub.k]([t.sub.0]), [v.sub.k]([t.sub.0]). Equation (1.4) expresses the law of the conservation of linear momentum: The center of mass moves uniformly in a straight line.

The origin of the Cartesian coordinate system [x.sub.1], [x.sub.2], [x.sub.3] for the motion of [P.sub.k] can be shifted to the center of mass by setting [[??].sub.j] = [x.sub.j] - [[rho].sub.j]. This does not alter the form of (1.1) since [[??].sub.j] = 0, and we can replace [x.sub.j] by [[??].sub.j]. Thus, without loss of generality, we can assume [rho] = 0, which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.5)

and also by differentiation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.6)

It is verified that (1.5), (1.6) represent six independent algebraic integrals [I.sub.k], k = 1, 2,..., 6.

Another independent algebraic integral is given by the conservation of energy H,

H = T - U, (1.7)

where H is the total energy of the system of n particles, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

is the kinetic energy. Thus, H is the sum of the potential and kinetic energies. It is an integral since one verifies by direct computation that d/dt (T -U) = 0 using (1.2). The law of conservation of energy states that the energy is constant along solutions.

The remaining three integrals are given by the conservation of angular momentum. This is derived by forming the vector cross product [x.sub.k] x [[??].sub.k] using (1.1) and summing over k, where it is verified that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.9)

where j [not equal to] k and where we used the fact [x.sub.k x [x.sub.k] = 0. The double sum is zero since [x.sub.j x [x.sub.k] = -[x.sub.k x [x.sub.j]. Integrating the left-hand side of (1.9) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

where c = ([c.sub.1], [c.sub.2, [c.sub.3]) [member of] [R.sup.3 is the vector constant of angular momentum. Equation (1.10) expresses the law of conservation of angular momentum.

The angular momentum can be viewed as a measure of the rotational motion of (1.1). This measure of the rotation is illustrated in an important theorem of Sundman.

Theorem 1.2 (Sundman) If at time t = [t.sub.1] all the particles [P.sub.k] collide at one point, then c = 0.

This is called total collapse. The fact c = 0 means that the particles are able to all collapse to a single location. In a sense, this is enabled because with c = 0, the rotation has been taken away from the motion of the particles. For the two-body problem for n = 2, collision between [P.sub.1], [P.sub.2], where [r.sub.12] = 0, can occur only if c = 0. Theorem 1.2 is not proven here. (See [204].)

We conclude our introduction to the n-body problem with a brief summary of the extension of a solution x(t), v(t) of (1.2) which has initial values x([t.sub.0]) = [x.sub.0], v([t.sub.0]) = [v.sub.0] at t = [t.sub.0]. We extend this general solution for t > [t.sub.0]. Now, either the 6n coordinates remain smooth for all time t > [t.sub.0], or else there is a first time t = [t.sub.1] where there is a singularity for at least one of the coordinates, where all coordinates are smooth for [t.sub.0] [less than or equal to] t < [t.sub.1]. The extent to which the solution can be continued in t beyond [t.sub.1] depends on whether or not, during the course of the motion of the [P.sub.k], the right hand- side of (1.2) remains smooth. Let [r.sub.min](t) = min{[r.sub.jk](t)}, j < k. [r.sub.min](t) is the minimum of the n(n - 1)/2 distances [r.sub.jk]. It can be proved that if [t.sub.1] is finite, then [r.sub.min](t) [right arrow] 0 as t [right arrow] [t.sub.1]. This implies U [right arrow] [infinity] as t [right arrow] [t.sub.1]. (See [204].)

In this case we say that there is a singularity of the solution at t = [t.sub.1]. Surprisingly, this does not necessarily imply that a collision between the particles has to take place. This is called a noncollision singularity. The particles can get very close to each other and move in a complicated way so that the potential increases without bound. This question is a subtle one and is not considered in this book, as it is not the focus. However, we briefly summarize some key results on the nature of the singularity if [r.sub.min] [right arrow] 0 as t [right arrow] [t.sub.1].

If n = 2, then as t [right arrow] [t.sub.1] a collision must occur between [m.sub.1] and [m.sub.2]. From Theorem 1.2, the condition that c = 0 implies that the particles [m.sub.1], [m.sub.2] lie along a line, and as [r.sub.12] [right arrow] 0 the collision can be regularized. This means that the solution can be smoothly continued to t [greater than or equal to] [t.sub.1] by a change of coordinates and time t. This is carried out in detail in sections 1.4 and 1.6. This means physically that [m.sub.1], [m.sub.2] perform a smooth bounce at collision and their motion for [t [greater than or equal to] [t.sub.1] falls back along the same line on which they collided. Since c = 0, three dimensions can be eliminated from the six-dimensional phase space. This means that the set of all collision orbits has a lower dimension than that of the phase space. The fact they have a lower dimension means that the total volume they make up is actually a set of relative zero volume in the full phase space. For example, in the two-dimensional Euclidean plane, the volume is area and all one-dimensional curves have zero area. The generalized volume of the phase space we use is called measure, which will mean Lebesgue measure. Thus, the set of collision orbits in the two-body problem is a set of measure zero in the phase space. There is a natural way to assign a measure µ to the phase space ([x.sub.1], [x.sub.2], [[??].sub.1], [[??].sub.2]) [member of] R.sup.12 of the three-dimensional two-body problem by setting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This defines a twelve-dimensional volume element and generalizes in the natural way to the n-body problem, n > 2.

(Continues...)



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