Geometric Derivation of the Generalized Liouville Equation

APPENDIX 2 Geometric Derivation of the Generalized Liouville Equation 

A set of points M is said to be a -dimensional manifold if each point of M has an open neighborhood, which has a continuous 1 1 map onto an open set of of R , the set of all w-tuples of real numbers. Consider an w-dimensional Riemannian manifold with metric G. In an arbitrary coordinate system x, .. . , x , the volume -form is generally given by u = dx a a dx . Here, g is the determinant of the metric in this basis, and a denotes the wedge or antisymmetric tensor product. For a flow field on the manifold prescribed by x = x) with density f x, t), a continuity equation for f x, t) can be obtained by considering the number of ensemble members T t) within a volume Q of phase space given by 

The continuity condition states that the rate of change of the number of ensemble members within Q must be balanced by the flux of members through the surface, a condition that is stated mathematically as 

Since the volume element is not zero, the term in brackets must vanish, yielding 

Equation [253] is a general form of the Liouville equation, valid on a curved manifold. 

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