Volume Preserving Flows Liouvilles Theorem

Consider a set of points f) in phase space with evolution associated to a differential equation z = /(z) described by the flow map f(S(0)) = t). Liouville s theorem [ 16] states that the volume of such a set is invariant with respect to t if the divergence of / vanishes, i.e. 

It is a simple exercise to show that for a Hamiltonian system the divergence vanishes, since 

To understand where Liouville s theorem comes from, recall that the variational equations of the last chapter are a system of ordinary differential equations for W t) = 

Example 2.3 A 1-d oscillator with Lennard-Jones potential is described by the equations 

As a consequence of energy conservation, any bounded individual trajectory of this system will be a periodic orbit. Consider the propagation of a small disk of initial conditions, 

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A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

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