The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline
tohline@rouge.phys.lsu.edu

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The time-evolution of dynamical systems is usually much easier to understand, or at least the future behavior of such systems is much easier to predict, once we realize that certain constants or "integrals" of the motion are preserved in time. The most familiar integrals of motion are energy and angular momentum. It is important to realize, however, that when particles (or fluid elements) move under the influence of an external potential, these familiar integrals may not be constant in time, but other integrals of the motion may exist. We present here a general discussion of integrals of motion for systems of noninteracting particles (i.e., effects of the gas pressure will be ignored) that move in time-invarient, external potentials.

Referring back to the principal governing equations, if the effects of pressure are ignored Euler's equation takes the form

Our objective is to find one or more "integrals" of the motion I_i for which the mathematical statement,

is both true and derivable from the equation of motion.

We'll focus, first, on the right-hand-side of the equation of motion. From the general (curvilinear form) of the gradient operator, we can write

Note, also, that from the definition of the Lagrangian time derivative of any scalar function, in general we can write,

Here, physically, the Lagrangian (total) time-derivative DF identifies the time-variation of the gravitational potential as seen by a moving particle (fluid element) while the Eulerian (partial) time-derivative identifies the actual time-variation of the potential as viewed by a stationary observer. Our discussions will be confined to time-invarient potentials, as desired, by simply demanding that

Hence, for our present discussions we may adopt the following relationship between the time- and spatial-variation of the gravitational potential, as seen by a moving particle:

Now we turn our attention to the left-hand-side of the equation of motion. Realizing that the general (curvilinear coordinate) form of the velocity is

we can, quite generally, write the time-derivative of v as,

Utilizing the general expression for the time-derivative of any unit vector, derive the above expression [VI.I.7] for the acceleration (Dv) from the definition of v.

As a shorthand, it frequently is useful to adopt the variables,

to denote the three components of the particle velocity. In terms of these variables, the above expressions for the velocity vector [VI.M.18] and acceleration vector [VI.I.7] take the following, simpler forms:

Combining the above expressions for the left-hand-side [VI.I.10] and the right-hand-side [VI.I.3] of the equation of motion, we can write in the following very general form the

Building upon this presentation of the equation of motion written in curvilinear coordinates, the accompanying pages present derivations and discussions of:

• Energy as a valid integral of the motion in any time-invariant potential;
• Components of the angular momentum vector as integrals of the motion in spherically symmetric potentials;
• Components of the Laplace-Runge-Lenz vector as an integral of the motion in 1/r potentials;
• Integrals of motion in Kuzmin-like potentials;