Poincare’s theorem

MSN. 112. T. Petrosky and II. Prigogine, Poincare s theorem and unitary transformations for classical and quantum systems, Physica, 147A, 439 60 (1987). 

The question stated above was formulated in two ways, each using an exact result from classical mechanics. One way, associated with the physicist Loschmidt, is fairly obvious. If classical mechanics provides a correct description of the gas, then associated with any physical motion of a gas, there is a time-reversed motion, which is also a solution of Newton s equations. Therefore if decreases in one of these motions, there ought to be a physical motion of the gas where H increases. This is contrary to the /f-theorem. The other objection is based on the recurrence theorem of Poincare [15], and is associated with the mathematician Zermelo. Poincare s theorem states that in a bounded mechanical system with finite energy, any initial state of the gas will eventually recur as a state of the gas, to within any preassigned accuracy. Thus, if H decreases during part of the motion, it must eventually increase so as to approach, arbitrarily closely, its initial value. 

While the Poincare-Bendixson theorem yields the existence of limit cycles, it is often important to know when limit cycles do not exist. For two-dimensional systems, a result in this direction which complements the Poincare-Bendixson theorem is called the Dulac criterion. Its proof is a direct application of the classical Green s theorem in the plane (after an assumption that the theorem is false) and will not be given here a good reference is [ALGM]. 

In this section, we compile some results from nonlinear analysis that are used in the text. The implicit function theorem and Sard s theorem are stated. A brief overview of degree theory is given and applied to prove some results stated in Chapters 5 and 6. The section ends with an outline of the construction of a Poincare map for a periodic solution of an autonomous system of ordinary differential equations and the calculation of its Jacobian (Lemma 6.2 of Chapter 3 is proved). 

We can t apply the Poincare-Bendixson theorem yet, because there s a fixed point... 

The classical and most common attempt, already discussed by Poincare, consists in trying to remove all dependencies on the angles from the Hamiltonian. This is usually called the normal form of Birkhoff. Such a normal form turns out to be particularly useful when the unperturbed Hamiltonian is linear, i.e., in (1) we have Ho(p) = (u),p) with some lo R". Therefore we illustrate the theory in the latter case. However, with some caveat, the method is useful also in the general case see Section 4.3 on Nekhoroshev s theorem. 

As we know from theorem 3, due to Poincare, in the general case of a non-degenerate unperturbed Hamiltonian Ho(p) the construction of the normal form can not be completely performed in a consistent manner. However, we are still allowed to perfom a suitable construction in domains where some resonances may be excluded, provided we perform a Fourier cutoff on the perturbation. We shall discuss this method in connection with Nekhoroshev s theorem. 

Figure 26 A schematic representation of the Poincare integral invariants for a system with three degrees of freedom. (A) The invariant is the sum of oriented areas projected onto the three possible (qi, pj) planes. (B) The invariant 2 is rhe sum of oriented volumes projected onto the three possible pj, q, pl) hyperplanes. Not shown (Liouville s theorem). Reprinted with permission from f. 119.

Similarly, it can be shown that Eq. [55] is a statement of the preservation of phase space volume under propagation by Hamilton s equations of motion, that is, Liouville s theorem. It is important to note that the Poincare integral invariants are also preserved under a canonical transformation of any kind and not just the propagation of Hamilton s equations. 

Eventually, aU of them are based on the methods of general qualitative theory of differential equations developed by Poincare more than a century ago [47]. This theory was essentially developed by Andronov in 1930s [48] and, finally, after Hopf s theorem on bifurcation appeared in 1942 [49] it became a self-consistent branch of mathematics. This subject is currently known luider several names Poincare-Andronov s general theory of dynamic systems theory of non-linear systems theory of bifurcation in dynamic systems. Although the first notion is, in our opinion, the most exact one, we will use the term bifurcation theory , or BT, for the sake of brevity. 

The complete set of equations with three variables, Eqs. (7.123)-(7.125), including the buffer step, may have a unique unstable solution. In view of the Poincare-Bendixson theorem, this is a necessary and sufficient condition for the occurrence of oscillations. For this set of equations, the solution is considered to be in the so-called reaction simplex S ... 

When additionally the complex A is pure, it follows from Theorem 12.3(2) that the reduced Betti number is nonzero only in the top dimension. Therefore, by the Euler-Poincare formula, in this case the cohomology groups can be computed simply by computing the Euler characteristic. In the even more special case that A is an order complex of a poset A = A P), by Hall s theorem, it suffices to compute the value of the Mobius function pp 0,1). 

Interlude 3.2 Poincare Recurrence Times We have seen that Boltzmann s entropy theorem leads not only to an expression for the equilibrium distribution function, but also to a specific direction of change with time or irreversibility for a system of particles or molecules. The entropy theorem states that the entropy of a closed system can never decrease so, whatever entropy state the system is in, it will always change to a higher entropy state. At that time, Boltzmann s entropy theorem was viewed to be contradictory to a well-known theorem in dynamics due to Poincare. This theorem states that... 

To complete the derivation, we multiply both sides of Eq. (828) by the phase factor el s>e lis> to obtain the B cyclic theorem. The latter is therefore equivalent to a commutator relation of the Poincare group between infinitesimal magnetic held generators. Similarly... 

When applying the Poincare-Ben-dixson theorem, it s easy to satisfy conditions (1)—(3) condition (4) is the tough one. How can we be sure that a confined trajectory C exists The standard trick is to construct a trapping region R, i.e., a closed connected set such that the vector field points inward everywhere on the boundary of R (Figure 7.3.2). Then all trajectories... 

Buslaev, V.I. Buslaeva, S.F. 2005. Poincare theorem for difference equations. Mathematical Notes 78 (5 6) 877-882. 

For the sake of the above application of the s-cobordism theorem we shall be primarily concerned with the s-triangulatiot theory of simple geometric Poincare complexes. Accordingly, we shall be dealing with the simple quadratic L-groups of group rings... 

The proof of the above theorem is based on the study of the Poincare map T == Ti oTq. As usual, the local map by the trajectories near O between some cross-sections 5o and S is denoted by To, and the global map from Si to by the trajectories close to the homoclinic loop T is denoted by Ti, respectively. 

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