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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 *k*th particle are given by the real vector [x.sub.*k*] =
([*x.sub.k*1], [*x.sub.k*2], [*x.sub.k*3]) [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 *k*th and *j*th particles, and . [equivalent to] *d/dt*. Equation (1.1)
expresses the fact that the acceleration of the *k*th 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 3*n* 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.k*1], [*v.sub.k*]2, [*v.sub.k*3]) =
([*[??].sub.k*1], [*[??].sub.k*2], [*[??].sub.k*1]) [member of] **[R.sup.3** are the velocity
vectors of the *k*th particle,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

*U = U* ([*x*.sub.1],..., [x.sub.*n*]) is a real-valued function of 3*n* 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 6*n* first order differential
equations for the 6*n* 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 3*n* 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.6*n*]) 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,..., 6*n*, in a domain |**y - [y**.sub.0]| < *p*, then

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

(see [204]).

A system of integrals exist for (1.1) which can be used to reduce the
dimension of the (6*n*+1)-dimensional coordinate space (**x, y**, *t*). An *integral*
is a real-valued function of the 6*n* + 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 6*n*-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 6*n*+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 6*n* remaining coordinates, at least implicitly. For notation we refer
to the 6*n*-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 (6*n* + 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 6*n* 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...)*

Excerpted fromCapture Dynamics and Chaotic Motions in Celestial MechanicsbyEdward BelbrunoCopyright © 2004 by Princeton University Press. Excerpted by permission.

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