Bled Workshops in Physics Vol. 11, No. 1 p. 27
Some classical motions of three quarks tethered to the Torricelli string*
V. Dmitrasinovica, M. Suvakovb, and K. Nagatac
a Vinca Institute for Nuclear Sciences, Belgrade University, Physics lab 010, P.O.Box 522, 11001 Beograd, Serbia; Permanent address after October 1st 2010: Institute of Physics, Belgrade University, Pregrevica 118, Zemun, P.O.Box 57,11080 Beograd, Serbia b Institute of Physics, Belgrade University, Pregrevica 118, Zemun, P.O.Box 57, 11080 Beograd, Serbia
c Research Institute for Information Science and Education, Hiroshima University, Higashi-Hiroshima 739-8521, Japan
In this talk we report some soon to be published results [1] of our studies of figure-eight orbits of three-bodies in three potentials: 1) the Newtonian gravity, i.e., the pairwise sum of —1 /r two-body potentials; 2) the pairwise sum of linearly rising r two-body potentials (a.k.a. the A string potential); 3) the Y-junction string potential [2] that contains both a genuine three-body part, as well as two-body contributions (this is the first time that the figure-eight has been found in these string potentials, to our knowledge). These three potentials share two common features, viz. they are attractive and symmetric under permutations of any two, or three particles. We were led to do this study after recognizing the existence of a dynamical symmetry underlying the remarkable regularity in the Y- and A string energy spectra [3].
We have found that a set of variables that consists of the "hyper-radius" R = v^TA^, the "rescaled area of the triangle" ^rlp x A|) and the ("braiding") hyper-angle cj> = arctan ^ Ai^p2) makes this permutation symmetry manifest; we use them to plot the motion of the numerically calculated figure-eight orbit. According to Ref. [4], H. Hopf was the first one to introduce these variables, Ref. [5]. As there are three independent three-body variables, and there are two independent permutation-symmetric three-body variables, R and the area the third variable cannot be permutation-symmetric. Moreover, it must be a continuous variable and not be restricted only to a discrete set of points, as is natural for permutations. We identify here the third independent variable as cj> = arctan (A2^p2) and show that it grows /descends (almost) linearly with the time t spent on the figure-eight trajectory and reaches ±2n after one period T. Thus, on the figure-eight orbit ^ is, for most practical purposes, interchangeable with the time variable t. The hyper-angle ^ is the continuous braiding variable that interpolates smoothly be-
* Talk delivered by V. Dmitrasinovic
tween permutations and thus plays a fundamental role in the braiding symmetry of the figure-eight orbits [6,8].
Then we constructed the hyper-angular momentum G3 = j (pp A pA p) conjugate to the two forming an (approximate) pair of action-angle variables for this periodic motion. Here we calculate numerically and plot the temporal variation of as well as the hyper-angular momentum G3(t), the hyper-radius R and r. We show that the hyper-radius R(t) oscillates about its average value R with the same angular frequency and phase, as the new ("reduced area") variable r(t). Thus, we show that ^(t) is, for most practical purposes, interchangeable with the time variable t, in agreement with the tacit assumption(s) made in Refs. [7], [4], though the degree of linearity of this relationship depends on the precise functional form of the three-body potential.
As stated above, ^ is not exactly proportional to time t, but contains some non-linearities that depend on the specifics of the three-body potential; consequently the hyper-angular momentum G3 is not an exact constant of this motion, but oscillates about the average value G3, with the same basic frequency 3t]3. Thus, the time-averaged hyper-angular momentum G3 is the action variable conjugate to the linearized hyper-angle ^ .
We used these variables to characterize two new planar periodic, but non-choreographic three-body motions with vanishing total angular momentum. One of these orbits corresponds to a modification of the figure-eight orbit with ^(t) that also grows more or less linearly in time, but has a more complicated periodicity pattern defined by the zeros of the area of the triangle formed by the three particles (also known as "eclipses", "conjunctions" or "syzygies"). Another new orbit has ^(t) that grows in time up to a point, then stops and "swings back". We show that this motion, and the other two, can be understood in view of the analogy between the three-body hyper-angular ("shape space") Hamilto-nian on one hand and a variable-length pendulum in an azimuthally periodic in-homogeneous gravitational field, on the other.
References
1.	M. Suvakov, V. Dmitrasinovic, "Approximate action-angle variables for the figure-eight and new periodic three-body orbits", submitted to Phys. Rev. E, (2010).
2.	V. Dmitrasinovic, T. Sato and M. Suvakov, Eur. Phys. J. C 62, 383 (2009)
3.	V. Dmitrasinovic, T. Sato and M. Suvakov, Phys. Rev. D 80, 054501 (2009).
4.	A. Chenciner, R. Montgomery, Ann. Math. 152, 881 (2000).
5.	H. Hopf, Math. Annalen 104, 637 (1931).
6.	C. Moore, Phys. Rev. Lett. 70, 3675 (1993).
7.	T. Fujiwara, H. Fukuda and H. Ozaki, J. Phys. A 36, 2791 (2003).
8.	G.C. Rota, Mathematical Snapshots, Killian Faculty Achievement Award Lecture (1997).