Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance

The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application.     

The orbital stability of periodic motions of a Hamiltonian system with two degrees of freedom in the case of 3:1 resonance

The problem of the orbital stability of periodic motions, produced from an equilibrium position of an autonomous Hamiltonian system with two degrees of freedom is considered. The Hamiltonian function is assumed to be analytic and alternating in a certain neighbourhood of the equilibrium position, the eigenvalues of the matrix of the linearized system are pure imaginary, and the frequencies of the linear oscillations satisfy a 3:1 ratio. The problem of the orbital stability of periodic motions is solved in a rigorous non-linear formulation. It is shown that short-period motions are orbitally stable with the sole exception of the case corresponding to bifurcation of short-period and long-period motions. In this particular case there is an unstable short-period orbit. It is established that, if the equilibrium position is stable, then, depending on the values of the system parameters, there is only one family of orbitally stable long-period motions, or two families of orbitally stable and one family of unstable long-period motions. If the equilibrium position is unstable, there is only one family of unstable long-period motions or one family of orbitally stable and two families of unstable long-period motions. Special cases, corresponding to bifurcation of long-period motions or degeneration in the problem of stability, when an additional analysis is necessary, may be exceptions. The problem of the orbital stability of the periodic motions of a dynamically symmetrical satellite close to its steady rotation is considered as an application.     

Non-linear oscillations of a Hamiltonian system with 2:1 resonance

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.     

Non-linear oscillations of a Hamiltonian system in the case of 3:1 resonance

The motion of an autonomous Hamiltonian system with two degrees of freedom near its equilibrium position is considered. It is assumed that, in a certain region of the equilibrium position, the Hamiltonian is an analytic and sign-definite function, while the frequencies of linear oscillations satisfy a 3:1 ratio. A detailed analysis of the truncated system, corresponding to the normalized Hamiltonian is given, in which terms of higher than the fourth order are dropped. It is shown that the truncated system can be integrated in terms of Jacobi elliptic functions, and its solutions describe either periodic motions or motions that are asymptotic to periodic motions, or conventionally periodic motions. It is established, using the KAM-theory methods, that the majority of conventionally periodic motions are also preserved in the complete system. Moreover, in a fairly small neighbourhood of the equilibrium position, the trajectories of the complete system, which are not conventionally periodic, form a set of exponentially small measure. The results of the investigation are used in the problem of the motion of a dynamically symmetrical satellite in the region of its cylindrical precession.     

Non-linear oscillations of a 1:1 resonance Hamiltonian system

The non-linear oscillations of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its stable equilibrium position are considered. It is assumed that the Hamilton function is sign-definite in the neighbourhood of the equilibrium position and that the values of the frequencies of its linear oscillations are equal or close to one another (1:1 resonance). The investigation is carried out using the example of the problem of the motion of a dynamically symmetrical rigid body (satellite) about its centre of mass in a circular orbit in a central Newtonian gravitational field. In this problem there is relative equilibrium of the rigid body in the orbital system of coordinates, for which its axis of dynamic symmetry is directed along the velocity vector of the centre of mass. Resonance occurs when the ratio of the polar and equatorial principal central moments of inertia is equal to 4/3 or is close to it. The problem of the existence, bifurcation and orbital stability of the periodic motions of a rigid body generated from its relative equilibrium is solved. Some aspects of the existence of quasiperiodic motions are also considered.     

The periodic motions of a non-autonomous Hamiltonian system with two degrees of freedom at parametric resonance of the fundamental type

The motion of an almost autonomous Hamiltonian system with two degrees of freedom, 2π-periodic in time, is considered. It is assumed that the origin is an equilibrium position of the system, the linearized unperturbed system is stable, and its characteristic exponents ±iωj (j = 1,2) are pure imaginary. In addition, it is assumed that the number 2ω₁ is approximately an integer, that is, the system exhibits parametric resonance of the fundamental type. Using Poincaré's theory of periodic motion and KAM-theory, it is shown that 4π-periodic motions of the system exist in a fairly small neighbourhood of the origin, and their bifurcation and stability are investigated. As applications, periodic motions are constructed in cases of parametric resonance of the fundamental type in the following problems: the plane elliptical restricted three-body problem near triangular libration points, and the problem of the motion of a dynamically symmetrical artificial satellite near its cylindrical precession in an elliptical orbit of small eccentricity.     

Non-linear oscillations of a hamiltonian system with one degree of freedom and fourth-order resonance

Non-linear oscillations of a 2π-periodic Hamiltonian system with one degree of freedom are considered . It is assumed that the origin of coordinates is an equilibrium position, the linearized system is assumed to be stable, its characteristic exponents ±iv are pure imaginary, and the value of 4v is close to an integer. When the methods of classical perturbation theory are used, the investigation reduces to an analysis of a model system which can be described by the typical Hamiltonian of problems on the motion of Hamiltonian systems with one degree of freedom in the case of fourth-order resonance. The system is analysed in detail. The results for the model system are applied to the total system using Poincaré's theory of periodic motion and the KAM-theory. The existence, number and stability of 8π-periodic motions of the initial system are investigated. Trajectories of motion which start in a fairly small neighbourhood of the origin of coordinates are bounded. An estimate of the size of that neighbourhood is given. The examples considered are of a point mass above a curve in the shape of an ellipse which collides with the curve, and plane non-linear oscillations of a satellite in an elliptical orbit in the case of fourth-order resonance.     

Resonant periodic motions of Hamiltonian systems with one degree of freedom when the Hamiltonian is degenerate

The motions of a non-autonomous Hamiltonian system with one degree of freedom which is periodic in time and where the Hamiltonian contains a small parameter is considered. The origin of coordinates of the phase space is the equilibrium position of the unperturbed or complete system, which is stable in the linear approximation. It is assumed that there is degeneracy in the unperturbed Hamiltonian when account is taken of terms no higher than the fourth degree (the frequency of the small linear oscillations depends on the amplitude) and, in this case, one of the resonances of up to the fourth order inclusive is realized in the system. Model Hamiltonians are constructed for each case of resonance and a qualitative investigation of the motions of the model system is carried out. Using Poincaré's theory of periodic motions and KAM-theory, a rigorous solution is given of the problem of the existence, bifurcations and stability of the periodic motions of the initial system, which are analytic with respect to fractional powers of the small parameter. The resonant periodic motions (in the case of the degeneracy being considered) of a spherical pendulum with an oscillating suspension point are investigated as an application.     

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.     

The non-linear oscillations of a pendulum of variable length on a vibrating base

A generalized scheme for averaging a system with several small independent parameters is described: equations of the first and second approximations are obtained, and an estimate is made of the accuracy of the approximation and the value of the asymptotically long time interval. The problem of the oscillations of a pendulum of variable length on a vibrating base for high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and its suspension point is considered. Averaged equations of the first and second approximations are obtained, and the bifurcations of the steady motions in the equations of the first approximation, and also in the second approximation for 1:2 resonance, are obtained. One of the possible bifurcations of the phase portrait in the neighbourhood of 1:2 resonance is described based on a numerical investigation. It is shown that a change in the resonance detuning parameter from zero to a value of the first order of infinitesimals in the small parameter leads to stabilization of the upper equilibrium position through a splitting of the separatrices for the resonance case; the splitting of separatrices is accompanied by the occurrence of a stochastic web in the neighbourhood of this equilibrium, its localization, and subsequent contraction to an equilibrium point and the formation of a new oscillation zone.     

The asymptotic properties of the motions of satellites in a central field due to internal dissipation

A satellite in the form of a system of bodies that does not have the property of a gyrostat in the general case is considered. An algorithm for determining all the equilibrium configurations of the system that correspond to steady motions in a central gravitational field and an algorithm for analysing their stability are given. A method based on Routh's first theorem is used to investigate the asymptotic stability of the steady motions in the unconstrained problem. Three effects caused by internal dissipation are established in a model example: stabilization of the satellites in a neighbourhood of rotations about a normal to the orbital plane, which is codirectional with the axis of the largest moment of inertia, evolution of elliptic orbits into circular orbits, and capture of the satellites in resonant oscillatory modes of motion.     

Non-linear oscillations of a satellite at 1:1:1 resonance

The motion of a satellite about its centre of mass in a central Newtonian gravitational field is investigated. The satellite is considered to be a dynamically symmetrical rigid body. It is assumed that the ratio of the polar and equatorial principal central moments of inertia of the satellite is 4/3, or close to this. The orbit of the centre of mass is elliptic, and the orbit eccentricity is assumed to be small. In the limit case, when the orbit of the centre of mass is circular, a steady motion exists (corresponding to relative equilibrium of the satellite in the orbital system of coordinates) in which the axis of dynamic symmetry is directed along the velocity vector of the centre of mass of the satellite; here, the frequencies of the small linear oscillations of the axis of symmetry are equal or close to one another. But in an elliptic orbit of small eccentricity, multiple 1:1:1 resonance occurs in this case, as the oscillation frequencies mentioned are equal or close to the frequency of motion of the centre of mass of the satellite in orbit. The non-linear problem of the existence, bifurcations and stability of periodic motions of the satellite with a period equal to the rotation period of its centre of mass in orbit is investigated.     

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.     

Forced oscillations of a non-linear system with a repulsive positional force

Non-linear systems with one degree of freedom, in which the positional force is directed away from the equilibrium position of the system, are considered. The existence of forced periodic oscillations, their Lyapunov stability, and the behaviour of amplitude-frequency characteristics are investigated. It is shown that stable periodic oscillations are possible in the case when the positional force has non-monotonic properties. Forced oscillations of a pendulum with respect to the upper equilibrium position are considered as an example.     

Periodic motions of a reversible second-order mechanical system: Application to the Sitnikov problem

A theory of the symmetric periodic motions (SPMs) of a reversible second-order system is presented which covers both oscillations and rotations. The structural stability property of the generating autonomous reversible system, which lies in the fact that the presence or absence of SPMs in a perturbed system is independent of the actual form of the “reversible” perturbations, is established. Both the case of the generation of SPMs from the family of SPMs of the generating system and birth cycle from the equilibrium state are investigated. Criteria of Lyapunov stability in a non-degenerate situation are obtained for the SPMs which are generated (in case of small values of the parameter). A method is proposed for constructing and investigating the Lyapunov stability of all the SPMs. The conditions for the existence of a cycle (symmetric and asymmetric) in the neighbourhood of a support “almost” resonance SPM are established for all cases of resonances. The theoretical results are applied to a study of the motion of a particle along a straight line which passes through the centre of mass of the system perpendicular to the plane of the identical attracting and simultaneously radiating main bodies (an extension of the Sitnikov problem) in the photogravitational version of the three-body problem. The circular problem is analysed and two different series of families of SPMs are found in the weakly elliptic problem. The instability of the equilibrium state is proved in the case of parametric resonance and the stability (and instability) domains are distinguished for arbitrary values of the eccentricity. All the SPMs with a period of 2π are constructed and the property of Lyapunov stability is investigated for these motions.     

The quasistatic motions of a three-body system on a plane

A controlled three-body system on a horizontal plane with dry friction is considered. The interaction forces between each pair of bodies are controls that are not subject to prior constraints but must be chosen in such a way that the motions of the system generated by them are quasistatic, that is, the total force acting on each of the bodies must be close to zero. All motions in which one body moves and the other two are fixed are found in the class of quasistatic motions. The problem of the optimal displacement of a moving body between two specified positions on a plane such that the absolute magnitude of the work of the friction forces along the trajectory is a minimum is solved. The quasistatic controllability of a three-body system is demonstrated and algorithms for bringing it into a specified position are discussed. The system considered simulates a mobile robot consisting of three bodies between which control forces act that can be realized by linear motors. The sizes of the bodies are assumed to be negligibly small compared with the distances between them so that the bodies are treated as particles.     

The solution of a problem in contact fracture mechanics on the nucleation and development of a bridged crack in the hub of a friction pair

The contact deformation of the hub of a plunger pair is considered. It is assumed that, during the repeated reciprocating motion of the plunger, a crack is initiated and fracture of the materials of the elements of the contact pair occurs. The problem of the equilibrium of the hub of a friction pair with a crack nucleus reduces to solving a system of non-linear singular integrodifferential equations with a Cauchy-type kernel. The normal and shear forces in the zone where the crack originates are found from the solution of this system of equations. The condition for the appearance of a crack is formulated, taking account of the criterion of the limit traction of the bonds in the material. A problem for the plunger of a friction pair as applied to a borehole sucker rod pump is considered as an example. In conclusion, the case when there are several arbitrarily distributed rectilinear bridged cracks, with bonds between the crack faces in the end zone, close to the contact surface of the hub is investigated.     

On nonlinear motions of Hamiltonian system in case of fourth order resonance

We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin, which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3: 1. We study nonlinear conditionally periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally periodic. By using the KAM theory methods we show that most of the conditionally periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that are not conditionally periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.     

Continuation of periodic motions of a reversible system in non-structurally stable cases. Application to the N-planet problem

The problem of continuing symmetric periodic solutions of an autonomous or periodic system with respect to a parameter is solved. Non-structurally stable cases, when the generating system does not guarantee that the solution can be continued, are considered. Three approaches are proposed to solving the problem: (a) particular consideration of terms that depend on the small parameter and the selection of generating solutions; (b) the selection of a generating system depending on the small parameter; (c) reduction to a quasi-linear system which is then analysed using the first approach. Within the framework of the third approach the existence of a periodic motion is also established that differs from the generating one by a quantity whose order is a fractional power of the small parameter. The theoretical results are used to prove the existence of two families of periodic three-dimensional orbits in the N-planet problem. The orbit of each planet is nearly elliptical and situated in the neighbourhood of its fixed plane; the angle between the planes is arbitrary. The average motions of the planets in these orbits relate to one another as natural numbers (the resonance property), and at instants of time that are multiples of the half-period the planes are either aligned in a straight line—the line of nodes (the first family), or cross the same fixed plane (the second family). The phenomenon of a parade of planets is observed. The planets' directions of motion in their orbit are independent.     

Reverse motions of mechanical systems

The possibility of the occurrence of sections of reverse motions in natural mechanical systems, when, in the second half of a time interval, the motion in the first half of the interval is repeated in the reverse order and the opposite velocity with a specified accuracy, is investigated. It is shown that such motions are characteristic of natural mechanical systems in the neighbourhood of a non-degenerate equilibrium position if the natural frequencies are independent. Systems with gyroscopic and dissipative forces are also considered. It is shown that, in these systems, sections of reverse motion can be observed in a special system of coordinates. Examples are presented.