TY - JOUR
T1 - Contact complete integrability
AU - Khesin, B.
AU - Tabachnikov, S.
N1 - Funding Information:
We are grateful to M. Audin, A. Banyaga, E. Lerman, Yu. Manin, I. Miklashevskii, R. Montgomery and V. Ovsienko for stimulating discussions. B.K. is grateful to the MSRI and the Weizmann Institute for kind hospitality during the work on this paper. The authors were partially supported by NSERC and NSF grants, respectively.
PY - 2010
Y1 - 2010
N2 - Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical ℝ ⋉ ℝn-1 structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.
AB - Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical ℝ ⋉ ℝn-1 structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.
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U2 - 10.1134/S1560354710040076
DO - 10.1134/S1560354710040076
M3 - Article
AN - SCOPUS:85027928313
SN - 1560-3547
VL - 15
SP - 504
EP - 520
JO - Regular and Chaotic Dynamics
JF - Regular and Chaotic Dynamics
IS - 4
ER -