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

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

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Among the principal governing equations, we have included the

Note that this equation also may be written in the form,

By replacing the Lagrangian time derivative D in the first expression [I.B.1] by its Eulerian counterpart, we directly obtain what is commonly referred to as the

We should explain what is meant by this "conservative form" label because the ideas associated with it carry over in an important way to other conservation equations.

By definition, the continuity equation (written in any form) should be recognized as a statement of mass conservation. Hence, the above equation [I.B.3] shows how, in an inertial Eulerian reference frame, the time-rate-of-change of the volume density of the mass must relate to the local divergence of the volume density of the mass in order to ensure that mass is conserved. Because there is nothing particularly special about the mass as a physical variable in this context, it must be true that when an equation governing the time variation of any physical variable "X" can be cast into the form

we will know that the variable "X" identifies the volume density of a physical quantity that is conserved during an evolution equally as well as the mass is conseved.

Similarly, from the Lagrangian representation of the continuity equation [I.B.2], we may conclude that when an equation governing the time variation of any physical variable "X" can be cast into the following generic

we will know that the variable "X" identifies the volume density of a conserved physical quantity.