Taylor series and twisting-index invariants of coupled spin–oscillators

Abstract

About six years ago, semitoric systems on 4-dimensional manifolds were classified by Pelayo & Vũ Ngo . c by means of five invariants. A standard example of such a system is the coupled spin-oscillator on S 2 × R 2 . Calculations of three of the five semitoric invariants of this system (namely the number of focus-focus singularities, the generalised semitoric polygon, and the height invariant) already appeared in the literature, but the so-called twisting index was not yet computed and, of the so-called Taylor series invariant, only the linear terms were known.In the present paper, we complete the list of invariants for the coupled spin-oscillator by calculating higher order terms of the Taylor series invariant and by computing the twisting index. Moreover, we prove that the Taylor series invariant has certain symmetry properties that make the even powers in one of the variables vanish and allow us to show superintegrability of the coupled spin-oscillator on the zero energy level.Semitoric systems, which in addition allow for so-called focus-focus singularities, have arXiv:1712.06402v2 [math.SG]

Symplectic classification of coupled angular momenta

“…To our knowledge this is the first time that higher order terms of the Taylor series invariant have been computed for a compact semitoric system. The authors calculated the Taylor series invariant for the coupled spin-oscillator in [ADH17], which is a non-compact semitoric system, and the second author also calculated the Taylor series invariant of the spherical pendulum in [Dul13], which is in fact not a semitoric system in the usual sense of the word, since the angular momentum integral is not proper (more details on the different types of properness in semitoric systems can be found in Pelayo & Ratiu & Vũ Ngo . c [PRVuN17]).…”

“…Since the system (1.1) depends on three parameters, the computation of the Taylor series invariant becomes quite involved at some points. To deal with this situation, a different approach is used than in [ADH17]. More precisely, the invariant is not obtained directly from the computation of the expansion of the action of the system, but from the period of the reduced system and the rotation number instead.…”

“…To our knowledge, this invariant has never been calculated for a compact semitoric system before. For the non-compact case, the authors computed it for the coupled spin-oscillator in [ADH17]. For the coupled angular momenta we have the following result:…”

A family of compact semitoric systems with two focus-focus singularities

“…The next theorem follows from Theorem 4.4 in Section 4, in which we take R 1 = 1 and R 2 = 2 for simplicity. Then (J (1,2) , H (s1,s2) ) has the following properties: 1) it is an integrable system for all (s 1 , s 2 ) ∈ [0, 1] 2 ; 2) it is a semitoric system when (s 1 , s 2 ) ∈ [0, 1] 2 \ γ where γ ⊂ [0, 1] 2 is the union of four smooth curves; 3) the points (N, S), (S, N ) ∈ S 2 × S 2 transition between being elliptic-elliptic, focus-focus, and degenerate depending on the value of (s 1 , s 2 ); 4) it is semitoric with exactly two focus-focus points for all (s 1 , s 2 ) in an open neighborhood of 1 2 , 1 2 ; 5) it is semitoric with no focus-focus point if (s 1 , s 2 ) ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}.…”

“…An image of the momentum map (J (1,2) , H (s1,s2) ) with the rank 0 points marked for varying values of s 1 , s 2 ∈ [0, 1]. Notice that the coupled angular momenta system shown in Figure 3 is the bottom row of the system shown in this figure since the coupled angular momenta is the special case for which s 2 = 0.…”