Proof of the Smoothness Theorem

This is primarily a technical chapter introducing another algebraic tool. We will use it at once to complete the proof of the smoothness theorem (11.6) and then draw on it throughout the rest of the book. To begin, we call a ring homomorphism A - B flat if, whenever M - JV is an injection of /1-modules, then MaB- N<8>aB is also an injection. For example, any localization A->S 1A is flat. Indeed, an element m a/s in M S = S lM is zero... 

In fact, no common upper bound exists on the number of the periodic orbits which can be generated from a fixed point of a smooth map through the given bifurcation. If the smoothness r of the map is finite, the absence of this upper estimate is obvious because it follows from the proof of the last theorem that to estimate the number of the periodic orbits within the resonant zone 1/ = M/N the map must be brought to the normal form containing terms up to order (AT — 1). In this case the smoothness of the map must not be less than (iV — 1). Hence, we can estimate only a finite number of resonant zones if the smoothness is finite. 

In her proof of the above theorem, Leontovich had assumed C -smoothness for the system, where r > 4n + 6. First of all, she proved that when the first saddle value is close to zero, a system near the saddle can be transformed into... 

When the curve in question is non-trigonal, and not a smooth plane quintic, this gives a new proof of Torelli s theorem. Green s proof relies on a subtle theorem by Kempf [K] which asserts that, when C is not hyperrelliptic, the Kodaira-Spencer map ... 

On first acquaintance, [Del appears to offer a neat way to cut through the complexity—a direct abstract proof of the existence of / , with indications about how to derive the concrete special situations (which, after all, motivate and enliven the abstract formalism). Such an impression is bolstered by Verdier s paper [V j. Verdier gives a reasonably short proof of the flat base change theorem, sketches some corollaries (for example, the flnite tor-dimension case is treated in half a page [ibid., p. 396], as is the smooth case [ibid., pp. 397-398]), and states in conclusion that all the results of [H], except the theory of dualizing and residual complexes, are easy consequences of the existence theorem. In short, Verdier s concise summary of the main features, together with some background from [H] and a little patience, should suffice for most users of the duality machine. 

The proof of this theorem is based on the reduction of the problem to a study of some map ( the essential map below) of a circle. In fact, this reduction works independently of the value of m or of the smoothness of and our two next sections are based on it. 

The proof of this theorem relies on showing that if trajectories are started from a point sufficiently close to z they cannot wander away to infinity. Although the result holds in greater generality, it is easy to show under assumptions of local smoothness of U (which we are normally happy to make in molecular dynamics). For more discussion see the text [216]. If the potential is C, the linearized version of this system has kinetic plus potential form with Hamiltonian... 

We will proceed to the proof of Theorem 3.3.2. At the same time, we investigate some important properties of momentum mapping, which are also of interest independent of Theorem 3.3.2. Let a maximal linear Lie algebra G of smooth functions be given on Consider the corresponding momentum mapping... 

We will present in Sec. 12.2 a summary of results for the case where the unstable manifold of the saddle-node is homeomorphic to a torus along with the proof of a theorem on the persistence of the invariant torus in the smooth case. There, we will also develop a general theory for an effective reduction of the problem to a study of some family of endomorphisms (smooth non-invertible maps) of a circle. 

The bifurcations of periodic orbits from a homoclinic loop of a multidimensional saddle equilibrium state are considered in Sec. 13.4. First, the conditions for the birth of a stable periodic orbit are found. These conditions stipulate that the unstable manifold of the equilibrium state must be one-dimensional and the saddle value must be negative. In fact, the precise theorem (Theorem 13.6) is a direct generalization of the Andronov-Leontovich theorem to the multi-dimensional case. We emphasize again that in comparison with the original proof due to Shilnikov [130], our proof here requires only the -smoothness of the vector field. 

It follows from the annulus principle (see the proof of Theorem 11.4) that the function depends smoothly on lj for p > 0. Moreover, as we have done with the derivatives of Tp with respect to (p we can verify that the derivatives of with respect to u tend to zero as 0. This implies that is a smooth function of a for all p>0. 

In its full generality, this lemma is proven in [140] and it implies almost immediately the basic Theorem 12.4 below. The proof is based on a lengthy calculations and we omit them here. A simple proof of an analogous statement is given in Sec. 12.5 under some additional assumptions. Namely, it is assumed there that the system is sufficiently smooth with respect to all variables and and that the saddle-node L is simple. Moreover, instead of proving that all of the derivatives tend to zero, the vanishing of only a sufficiently large number of derivatives is established. Of course, all this does not represent a severe restriction. 

Originally, this theorem was proved for C -smooth systems. We stress here that our proof includes the Crease which allows for a direct use of this theorem in the situation where the system is defined on a -smooth invariant manifold (see Theorem 13.9). 

PROOF An extension X/G = admits a morphism-section this follows using the criterion of GEOTHENDIECK for smooth morphisms (cf, SGA, III, theorem 3 1) from the fact that G is reduced (and hence X -... 

Proof The analogue to this Lemma in the ordinary Morse theory is well known, but in our case the proof is more delicate, since here we deal with the integral f (and not merely with a smooth function), and therefore we should essentially use the conditons of Theorem 2.1.3. An arbitrary smooth perturbation of an integral... 

If the surface and the Hamiltonian F are analytic, then both conditions 1 and 2 of Theorem 5.2.3 are automatically fulfilled (the property 1 requires special proof), and therefore in an analytic case Theorem 5.2.3 immediately implies Theorem 5.2.1. More generally, if a compact orient able surface M is nonhomeo-morphic to a sphere and to a torus then the above-mentioned equations of system motion do not have a new integral which is a smooth function on T M analytic for fixed x M on cotangent two-dimensional planes T M and having only a finite number of distinct critical values. The number of critical points is not necessarily finite. Functions polynomial in momenta are an example of integrals analytic in the momenta... 

---