Scientific Interests
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Geometry of the momentum map: The momentum map is the main tool in the theory of dynamical systems with symmetries. The works of Atiyah and of Guillemin and Sternberg drastically improved the understanding of the bifurcation diagram associated to momentum maps. Under the guide of Sergei Petrovich Novikov, I worked on an extension of the results of Atiyah, Guillemin and Sternberg to the case of non-Hamiltonian group actions (see [1] and [7]). One of the two main results in these works has been obtained independently by Y. Benoist.
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Non-holonomic "completely integrable" systems: Completely integrable systems are special dynamical systems whose phase space fibrates in tori and whose flow is a winding line in such tori. Classical completely integrable systems are Hamiltonian, but completely integrable behaviour is common in non-Hamiltonian systems. A fundamental class of such systems are the non-holonomic ones. Francesco Fassò and I did investigate the geometry of a possible, broader definition of complete integrability (see [3]), first introduced by Bogoyavlenskij (Lie?). In [8] we studied, with Francesco and Nicola Sansonetto, the geometry of a torus fibration and the appearence of a Poisson structure in a very specific non-holonomic system: the homogeneous ball rolling on a surface of revolution. In [9] we described the existence of a flower bifibration, typical in superintegrable systems, also in non-Hamiltonian systems with a compact symmetry group and periodic reduced dynamics.
It is well known that symmetries of a dynamical system are related to conserved functions. In [12] we gave a precise meaning to such relation outside of the Hamiltonian setting. In [13] we gave an estimate on the number of Weakly-Nöetherian integrals of motion for a non-holonomic systems. Weakly-Nöetherian integrals of motion are those functions that turn out to be integrals of motion for every non-holonomic system with a give symmetry group and given kinetic form.
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Monodromy and integrable systems: Obstruction theory plays a main role in completely integrable systems. In [2] I described the obstruction theory `a la Duistermaat' of a completely integrable system borrowed form geometry, the Gelfand-Cetlin system, and showed that the integrals in involution that define this system can be obtained via the introduction of a dynamical system admitting a Lax pair representation.
My post-doc under the guide of Richard Cushman made me become involved in dynamical systems with monodromy. With Holger Dullin and Richard we studied a special dynamical system, the "swing spring" (or elastic pendulum), whose non-trivial monodromy is somewhat related to a remarkable behaviour called "precession of the swing plane". This cooperation produced the article [4]
Boris Zhilinskií and Dimitrií Sadovskií guided me through the process of quantizing the swing spring and we analyzed its quantum spectrum. Its description can be found in [6]. The interactions of Dimitrii with chemists (among which Marc Joyeux) brought into light a relationships betweem the swing spring and the CO2 molecule. This started a collaboration among the authors of [5].
My collaboration with Boris and Dimitrii moved towards the investigation of Boris's idea of "fractional monodromy", a phenomena that was first geometrically described in a joint work of Nikolaí Nekhoroshev with Boris and Dimitrii, and later analyzed analytically by Konstantinos Efstathiou, Dimitrii and Richard. I gave a small ccontribution discussing the differnces between the homological and the homotopical approach in the case of fractional monodromy[9], which is the first case in which some differences happen. The investigation of peculiar topologies for completely integrable systems continues with the introduction of the idea of "bidromy" (again Boris's). This concept is strictly related to cuspidal singularities. In order to give a list of important singularities of 2 degerees of freedom completely integrable systems i wrote [11].
Publications and Preprints
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[8] Periodic flows,
Poisson structures, and nonholonomic mechanics,
Regular and
Chaotic Dynamics 10/3 267--284 (2005).
(with F. Fassò and N. Sansonetto)
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[7] Convexity of
multi-valued momentum maps,
Geometria Dedicata 111, 1--22 (2005).
for an older version of the work (2000): arXiv.
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[6] The CO2 molecule as
a quantum realization of the 1:1:2 resonant swing spring with
monodromy,
Physical Review Letters
93/2, 024302 (2004).
(with R. Cushman, H. Dullin, D. Holm, M. Joyeaux, P. Linch, D. Sadovskii and B. Zhilinskii)
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[5] Monodromy of the
quantum 1:1:2 resonant swing spring,
Journal of Mathematical
Physics 45/12, 5076--5100 (2004).
(with R. Cushman, D. Sadovskii and B. Zhilinskii)
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[4] Monodromy in the
resonant swing spring,
Physica
D 190/1-2, 15--37 (2004).
(with R. Cushman and H. Dullin)
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arXiv
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