The Liouville Theorem

It is required to find a set of independent (almost everywhere) integrals /i /r that their common level surfaces be sufficiently simple for instance, that all of them (in the case of general position) be diffeomorphic to a same simple manifold. Besides, it is also desirable that, being restricted to this common level surface, the initial system be transformed on it into a simply organized system, i.e., that the integral trajectories admit a simple description. 

It is remarkable that in many cases this program can be realized. 

For some Hamiltonian systems, there exists an extremely advisable partial integrability, for which the common level surface of integrab turns out to be a torus and the restriction to it of the initial system sets a conditionally-periodic motion along the torus. 

We will conditionally refer to this situation as the case of commutative integrability of a Hamiltonian system and will examine it in more detail. 

Theorem 1.2.5. Let a set ofn smooth functions /i. /n iii involution be given on a symplectic manifold M. We will denote by M the common level surface given by the system of equations fi x) =. .., /n( ) = in Suppose that on this 

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Thus these integrators are measure preserving and give trajectories that satisfy the Liouville theorem. [12] This is an important property of symplectic integrators, and, as mentioned before, it is this property that makes these integrators more stable than non-symplectic integrators. [30, 33]... 

Liouville theorem and related forms The Helmholtz-Lagrange relation given in equ. (4.46) is related to many other forms which all state certain conservation laws (the Clausius theorem, Abbe s relation, the Liouville theorem). The most important one in the present context is the Liouville theorem [Lio38] which describes the invariance of the volume in phase space. The content of this theorem will be discussed and represented finally in a slightly different form which allows a new access to the luminosity introduced in equ. (4.14). 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

In particular, the evolution of a system in time also represents a canonical transformation which implies that the volume in phase space is conserved as it evolves in time. This is known as the Liouville theorem (1809-1882). 

The Boltzmann equation can be derived using a procedure founded on the Liouville theorem . In this case the balance principle is applied to a control volume following a trajectory in phase space, expressed as... 

The model derivation given above using the Liouville theorem is in many ways equivalent to the Lagrangian balance formulation [83]. Of course, a consistent Eulerian balance formulation would give the same result, but includes some more manipulations of the terms in the number balance. However, the Eulerian formulation is of special interest as we have adopted this framework in the preceding discussion of the governing equations of classical fluid dynamics, chap 1. 

This relation is known as the Liouville theorem as discussed in sect. 2.2.3. Introducing a generalized form of the substantial derivative, measuring the rate of change as the observer moves along with the system points in... 

In this section, an analysis based on the Boltzmann equation will be given. Before we proceed it is essential to recall that the translational terms on the LHS of the Boltzmann equation can be derived adopting two slightly different frameworks, i.e., considering either a fixed control volume (i.e., in which r and c are fixed and independent of time t) or a control volume that is allowed to move following a trajectory in phase space (i.e., in which r(t) and c t) are dependent of time t) both, of course, in accordance with the Liouville theorem. The pertinent moment equations can be derived based on any of these two frameworks, but we adopt the fixed control volume approach since it is normally simplest mathematically and most commonly used. The alternative derivation based on the moving control volume framework is described by de Groot and Mazur [22] (pp. 167-170). 

This means that the determinant of the Jacobian of the flow in phase space is equal to one. Consequently, the volume in phase space is conserved (the Liouville theorem). 

On the other hand, it follows from the Liouville theorem (Appendix A2.1) that the attractor volume must be zero. Indeed, the divergence of the vector field F constructed from the right-hand sides of the Lorenz equation, F = (— ax + ay, rx — y — xz, —bz + xy), is equal to... 

Secondly, for any closed line L the area Qt for the canonical system (A 10) is constant. Indeed, substituting in the integrand in (A9) for P = = —8H/8y, Q = +8H/8x we obtain d/dt Qt = 0. This result is known as the Liouville theorem. A consequence of the Liouville theorem is non--existence of stationary points of a node or focus type for a canonical... 

As we have concluded in Chapter 5, in the case of the Lorenz system a trajectory always remains within a confined region of the phase space, being non-periodical. For t - oo, the trajectory approaches a certain limit set the Lorenz attractor. It follows from the Liouville theorem that the Lorenz attractor has a zero volume (since divF < 0). This implies that, apparently, the Lorenz attractor is a point (dimension zero), a line (dimension one), or a plane (dimension two). Then, however, the trajectory for t -> oo would have remained within a confined region on the plane and, by virtue of the Poincare-Bendixon theorem. Hence, a conclusion follows that the Lorenz attractor has a fractional dimension, larger than two. 

Differential equations of pure mechanical systems generate transformation groups for which the Lebesgue measure is invariant this statement is called the Liouville theorem. Major results of the modern theory of dynamic systems are connected with physical sciences, mostly with mechanics. Differential equations of physics may refer to particle or planetary motions described by ordinary differential equations, or to wave motion described by partial differential equations. Dissipative effects are neglected in all these systems, and so the emphasis is on conservative or Hamiltonian systems. 

More specifically, the evolution equations of conservative systems can be derived from a time-independent Hamiltonian, but there are systems originating from a time-dependent Hamilton function where the Liouville theorem is valid. Another important viewpoint of classification may be whether a dynamic system is integrable or non-integrable. 

In Lemma 2.1.2, we have dealt with a critical level surface L which is, generally, not homeomorphic to a union of nonsingular tori for it may have singularities (although far from P ). The nonsingular level surfaces of the integral / and Q are compact and by the Liouville theorem are unions of tori. 

The Liouville theorem implies that r realizes the nonzero element of the group Xi(Ti ), since it is obtained through the small closure of an almost-periodic trajectory which has a made a revolution along the torus and returned to the point close to the initial one (the first return). 

In Ch. 1, we got acquainted with the Liouville theorem which makes it possible to describe the behaviour of integral trajectories of systems possessing a complete set of integrals in involution. In this chapter, we deal with modern methods of integration, a particular case of which is the method of integration by means of the Liouville theorem. We develop here, in particular, the fundamental investigations of E. Cartan, Marsden, Weinstein, Moser, Bernat, Conze, Duflo, and Vergne. 

We will formulate the theorem proved by Fomenko and Mishchenko (see [90], [93], [143]), which generalizes the Liouville theorem and the theorems of E. Cartan, Marsden, and Weinstein. 

The Liouville theorem asserts that each invariant compact submanifold in (for an integrable system) is a torus. Nekhoroshev and Kozlov have noticed that one may single out the case where a single isolated invariant submanifold I = (/i =. ../ +fc = 0)inM2 is a torus. To this end, one should require that on the submanifold L we have the relations ... 

The next theorem is closely connected without generalizations of the Liouville theorem on completely integrable systems, see, for instance [143]. 

The assertion of item (3) follows from the holomorphic version of the Liouville theorem on integrable systems, see 49 of the book [3j. The proof of the holomorphic version is quite similar to the standard one. 

PROOF By the Liouville theorem, F b a torus if F O Qf = 0. By virtue of convexity of dAf, there exists such a neighbourhood U of the boundary dQ in Q that each trajectory of the geodesic flow on Q coming into U leaves Qf transversally to dQj. For this reason F is a torus if and only if F fl 17 = 0. FVom this and from the implicit function theorem it follows that the set of points x eV, such that Fx is a torus, is open and closed in V, Show that this set is non-empty. 

Then Txo i torus. By the Liouville theorem it suffices to show that the mapping... 

Imagine that we start with a set of identical systems, whose states are distributed in phase space according to a density distribution p(P) at time / = 0, and let the systems move according to their equations of motion. The ensemble constituted by the systems (points in phase space) evolves in time. As the systems evolve, the density distribution p(P) should, in general, change with time. However, systems just move, no new systems are created, and none of the systems is destroyed. Therefore, there should be a conservation law for the probability density, similar to the continuity equation (mass conservation) of hydrodynamics. The conservation law in phase space is called the Liouville theorem. It states that the total time... 

This statement is predicated on the Liouville theorem that allows us to connect from the transition state back to the reactants. Our mental image is that we start with an ensemble of... 

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