Phase topology of one system with separated variables and singularities of the symplectic structure

We consider an example of a system with two degrees of freedom admitting separation of variables but having a subset of codimension 1 on which the 2-form defining the symplectic structure degenerates. We show how to use separation of variables to calculate the exact topological invariant of non-degenerate singularities and singularities appearing due to the symplectic structure degeneration. New types of non-orientable 3-atoms are found.Comment: Corrected according to the version accepted in J. of Geometry and Physics, On-line July 2014, LaTex, 23 pp., 17 figures, 6 table

Reduction in mechanical systems with symmetry

The first part of the article is, in fact, the classical Routh method delivered in the language of contemporary theory of Lagrangian systems. But the Routh method deals only with concrete equations and, therefore, can be applied only in the case when the configuration spaces of the initial and the reduced systems are open submanifolds in Euclidean spaces. The global approach gives a possibility to find the structure of these manifolds in the general case and also to reveal some properties of the reduced system, first of all, the existence for this system of a global Lagrange function. We use the notion of a mechanical system introduced by S. Smale. The described method is applied to the global reduction in the problem of the motion of a rigid body having a fixed point in the potential force field with an axial symmetry. We present the complete proof of the theorem formulated by G.V. Kolosov on the equivalence of the reduced system in this case to the problem of the motion of a material point over an ellipsoid and also some corollaries of this theorem based on the results of L.A. Lyusternik and L.G. Shnirelman.Comment: LaTex, 15 pages, 4 figures, English versio

Some applications of differential geometry in the theory of mechanical systems

In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form of flows of a given geodesic curvature is proposed. For illustration, the problem of the motion of a rigid body about a fixed point in an axially symmetric force field is examined. The form of gyroscopic forces of the reduced system is calculated. It is shown that this form is a product of the momentum constant, the volume form of the 2-sphere, and an explicitly written everywhere positive function on the sphere.Comment: LaTex, 15 pages, English translation of Russian publicatio

Regions of possible motion in mechanical systems

A method to study the topology of the integral manifolds basing on their projections to some other manifold of lower dimension is proposed. These projections are called the regions of possible motion and in mechanical systems arise in a natural way as the regions on a space of configuration variables. To classify such regions we introduce the notion of a generalized boundary of a region of possible motion and give the equation to find the generalized boundaries. The inertial motion of a gyrostat (the Euler--Zhukovsky case) is considered as an example. Explicit parametric equations of generalized boundaries are obtained. The investigation gives the main types of connected components of the regions of possible motion (including the sets of the admissible velocities over each point of the region). From this information, the phase topology of the case is established.Comment: LaTex, 4 pages, 2 figure

Bifurcation of common levels of first integrals of the Kovalevskaya problem

The structure of integral manifolds in the Kovalevskaya problem of the motion of a heavy rigid body about a fixed point is considered. An analytic description of a bifurcation set is obtained, and bifurcation diagrams are constructed. The number of two-dimensional tori is indicated for each connected component of the supplement to the bifurcation set in the space of the first integrals constants. The main topological bifurcations of the regular tori are described.Comment: LaTex, 5 figure

Topological analysis of classical integrable systems in the dynamics of the rigid body

The general integrability cases in the rigid-body dynamics are the solutions of Lagrange, Euler, Kovalevskaya, and Goryachev-Chaplygin. The first two can be included in Smale's scheme for studying the phase topology of natural systems with symmetries. We modify Smale's program to suit the most complicated last two cases with non-linear first integrals. The bifurcation sets are found and all transformations of the integral tori are described and classified. New non-trivial bifurcation of a torus is established in the Kovalevskaya and Goraychev-Chaplygin cases.Comment: LaTex, 4 pages, 4 figure

Integral manifolds of the reduced system in the problem of inertial motion of a rigid body about a fixed point

The reduced system in the problem of the inertial motion of a rigid body with a fixed point (the Euler case) is equivalent, by the Maupertuis principle, to some geodesic flow on the 2-sphere. We describe the phase topology of this case including the types of the bifurcations of the integral tori. We establish the topology of the singular integral surfaces. Using the geometrical interpretation of $SO(3)$ as a 3-ball with opposite points of the boundary sphere identified we show how the singular integral surfaces and the families of 2-tori are settled in an iso-energy level.Using the contemporary language, one can say that this is the first description of the bifurcation of the type $C_2$ ($2T^2 \to 2T^2$).Comment: LaTex, 5 pages, 5 figures, English versio

Symmetry in systems with gyroscopic forces

We consider a generalization of the notion of a natural mechanical system to the case of additional forces of gyroscopic type. Such forces appear, for example, as a result of global reduction of a natural system with symmetry. We study symmetries in the systems with gyroscopic forces to find out when these systems admit a global analogue of a cyclic integral. The results are applied to the problem of the motion of a rigid body about a fixed point in potential and gyroscopic forces to find the most general form of such forces admitting the global area type integral.Comment: LaTex, 9 pages, English translation of the paper previuosly published only in Russia

Characteristic class of a bundle and the existence of a global Routh function

The possibility of the global Lagrangian reduction of a mechanical system with symmetry is shown to be connected with the characteristic class of a principal fiber bundle of the configuration space over the factor manifold. It is proved that the reduced system is globally Lagrangian if and only if the product of the momentum constant with this characteristic class is zero. In the case of a rigid body rotating about a fixed point in an axially symmetric force field the bundle over a 2-sphere is non-trivial, therefore the reduced system admits a global Routh function if and only if the momentum constant is zero.Comment: LaTeX versio

Smale-Fomenko diagrams and rough topological invariants of the Kowalevski-Yehia case

We present the complete analytical classification of the atoms arising at the critical points of rank 1 of the Kowalevski-Yehia gyrostat. To classify the Smale-Fomenko diagrams, all separating values of the gyrostatic momentum are found. We present a kind of constructor of the Fomenko graphs; its application gives the complete description of the rough topology of this integrable case. It is proved that there exists exactly nine groups of identical molecules (not considering the marks). These groups contain 22 stable types of graphs and 6 unstable ones with respect to the number of critical circles on the critical levels.Comment: LaTex, 20 page