In [1], the problem of the geodesic curves on a closed two-dimensional surface and some generalizations related to the addition of gyroscopic forces were considered. V. V. Kozlov and N. V. Denisova studied the one-parameter groups of symmetries in the four-dimensional phase space that are generated by the vector fields commuting with the original Hamiltonian vector field whose Hamiltonian is quadratic in the momenta. Because of homogeneity, it is possible to restrict the consideration to polynomial fields of symmetries: their components are polynomials in the momenta. From this point of view, Noetherian fields of symmetries are the simplest: they have the first degree in regards to the momenta. It is necessary to notice that, in the papers [1, 4, 5], a more general and more difficult problem on the structure of the symmetry groups is considered. These groups operate directly in the phase space and (unlike those introduced by E. Noether) are not an extension of the groups initially defined to act on the configuration space.
It was earlier established by V. V. Kozlov that if the genus of the configuration space surface is greater than one, then there exist no non-trivial symmetries. As proven by V. V. Kozlov and N. V. Denisova, for a surface of genus one (a two-dimensional torus), the first degree fields of symmetries are always Hamiltonian. Moreover, the latter fields are necessarily Noetherian and therefore a hidden cyclic coordinate exists. The second degree fields of symmetries are Hamiltonian only if the Gaussian curvature of the metric defined by the kinetic energy is not equal to zero. In this case, there is a quadratic integral, and, in the case of two degrees of freedom, the resulting equations are solved by using the method of separated variables.
In the papers [4, 5], the structure of the symmetry fields of degree 3 and 4 is studied for Hamiltonian dynamical systems whose configuration space is a two-dimensional torus. It was proved by N. V. Denisova that if a dynamical system admits a non-Hamiltonian symmetry field of degree 3, then there exists a linear integral. If a dynamical system admits a non-Hamiltonian symmetry fields of degree 4, then this system has a quadratic integral independent of the energy integral.
In addition, in [1], the structure of symmetries for dynamical systems on a two-dimensional torus with additional gyroscopic forces is analyzed.
Another research direction connected with the investigation of the symmetry groups is a study of the first integrals that are polynomials in the momenta. The geodesic curves of a Riemannian metric on a surface are described by a Hamiltonian system with two degrees of freedom and whose Hamiltonian is a quadratic function in the momenta. In mechanics, these equations describe the inertial motion. Because of homogeneity, every integral of the geodesic problem is a function of integrals polynomial in the momenta. As known from the classical results of Birkhoff, Darboux, and Whittaker, linear integrals are always Noetherian, whereas presence of quadratic integrals is due to the method of separated variables. A smallest degree polynomial integral independent of the energy integral is said to be irreducible.
For the first time in [1], a surprising connection between the degree of an irreducible additional integral and the topology of the configuration space of a mechanical system was discovered. The following hypotheses were stated. For the case of a two-dimensional sphere (its genus is equal to 0), the degree of an irreducible integral does not exceed 4. The integral of degree 3 corresponds to the Goryachev - Chaplygin case, and the integral of degree 4 is the Kowalevskaya integral, from the rigid body dynamics. For the case of a two-dimensional torus (its genus is equal to 1), the degree of an irreducible integral does not exceed 2. Notice that, as was earlier established by V. V. Kozlov, if the two-dimensional surface genus is greater than 1, then the mechanical system does not generally admit an additional non-constant integral.
In [2], the hypothesis that, for a surface of the genus of 1, the degree of an irreducible integral does not exceed 2 was proven for the metrics that can arbitrarily closely approximate any metric on a two-dimensional torus. These metrics have a trigonometric polynomial as their conformal factor.
In [6], N. V. Denisova obtained constructive criteria for the existence of the conditional linear and quadratic integrals on a two-dimensional torus.
The problem considered in the paper [8] is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in the momenta. The kinetic energy is a zero-curvature Riemannian metric, and the potential is a smooth function on a two-dimensional torus. The following hypothesis is also well known: if there exists an integral of degree n independent of the energy integral, then there exists an additional integral of degree 1 or 2. In [8], this result is established for n = 3 (which generalizes the theorem of M. L. Byaly), whereas, for n = 4, 5, or 6, this has also been proven under some additional assumptions about the spectrum of the potential.
In [3], the connection between the solution branching in the complex-time plane and the number of polynomial integrals is investigated for reversible systems on a sphere as the configuration space.
In the paper [7], the chaotic behavior of a system with coupled pendula is discovered. The conclusions are made on the basis of the perturbation theory of Hamiltonian systems and are based on a careful analysis of the set of resonant invariant tora, which are destroyed upon addition of perturbation.
In [9], N. V. Denisova and V. V. Kozlov develop a constructive method which is then used to introduce the new concepts of chaos and chaotic behavior in regards to stationary currents of an abstract continuous medium. The technique consists in expanding the movement equations of the medium as a power series in degrees of a small parameter and applying the resonant tora destruction conditions, as perturbation gets imposed on the system. As an application, the authors reveal that the velocity field obtained as a solution of the Burgers equations generates, most typically, a chaotic dynamical system.
For an ideal barotropic liquid enclosed in a potential field of forces, the presented method leads to the known necessary chaotization condition: the velocity field is collinear with its own rotor.
Particular attention is given to chaotic behavior of the typical stationary currents of a perfect heat conducting gas. Some integral relations for stationary currents of a perfect gas were derived. As proven by N. V. Denisova and V. V. Kozlov, if no heat influx through the walls of the vessel occurs, then the temperature of the gas in movement is constant.
In the paper [10], two models were considered: the barotropic compressible viscous fluid and the incompressible ideal fluid with the viscous friction. It was proved, with the help of the Poincaré - Kozlov theorem, that the typical stationary flows of the fluid (in the above-mentioned models) have no non-trivial integrals, fields of symmetries and integral invariants. In other words, these flows are chaotic.