Routh's Theorem Research Articles

On the Steiner–Routh Theorem for Simplices

It is shown in [28] that, using only tools of elementary geometry, the classical Steiner—Routh theorem for triangles can be fully extended to tetrahedra. In this article, we first give another proof of the Steiner—Routh theorem for tetrahedra, where methods of elementary geometry are combined with the inclusion—exclusion principle. Then we generalize this approach to (n — 1)-dimensional simplices. A comparison with the formula obtained using vector analysis yields an interesting algebraic identity.

The Routh theorem for mechanical systems with unknown first integrals

In this paper we discuss problems of stability of stationary motions of conservative and dissipative mechanical systems with first integrals. General results are illustrated by the problem of motion of a rotationally symmetric rigid body on a perfectly rough plane.

On Feynman's Triangle problem and the Routh Theorem

In this article, we give a brief history of the Feynman's Triangle problem and describe a simple method to solve a general version of this problem, which is called the Routh Theorem. This method could be found useful to school teachers, instructors or lecturers who are involved in teaching geometry.

On stability of stationary motion of a nonconservative nonholonomic rheonomic system

Abstract One considers, in this paper, the motion of a mechanical system in a nonstationary field of potential and positional forces, subject to the action of rheonomic holonomic and nonholonomic linear homogeneous constraints. Assuming that differential equations of motion of the system considered satisfy the conditions for the existence of Painleve's integral of energy, formulated in [Painleve, P., 1897. Lecons sur l'integration des equations de la Mecanique, Paris] and [Appell, P., 1911. Traite de mecanique rationnelle, T. II, Dynamique des systemes – Mecanique analitique, Gauthier-Villars, Paris] and generalized in [Covic, V., Veskovic, M., 2004. On stability of motion of a rheonomic system in the field of potential and positional forces, BAMM-1720/2004, No-2233, 93–100] and [Covic, V., Veskovic, M., 2005. Hagedorn's theorem in some special cases of rheonomic systems. Mechanics Research Communications 32 (3), 265–280], the original mechanical system is substituted by an equivalent one whose Lagrangian function, nontransformed with respect to nonholonomic constraints, does not depend on time explicitly. Using the properties of the equivalent system, which, in contrast to the original one, moves in a stationary field of potential forces and in a nonstationary field of gyroscopic forces, the definition of cyclic coordinates is generalized, as well as sufficient conditions for the existence of (cyclic) first integrals, corresponding to coordinates mentioned and linear in velocities are established. Further, the conditions for the existence of steady motion of the system considered are found. In the case of existence of such a motion of the system, the Theorem of Routh's type on stability of that motion, based on the minimum of reduced potential for which it is shown that, in contrast to known cases (see, for example, [Gantmacher, F., 1975. Lectures in Analytical Mechanics. Mir Publisher, Moscow; Neimark, J., Fufaev, N., 1972. Dynamics of Nonholonomic Systems. Amer. Math. Soc., Providence, RI; Pars, L., 1962. An Introduction to Calculus of Variations. Heinemann, London; Karapetyan, A., Rumyantsev, V., 1983. Stability of conservative and dissipative systems. In: Itogi Nauki I Tekhniki: Obschaya Mekh., vol. 6, VINITI, Moscow, pp. 3–128 (in Russian)]), it includes the influence of the positional forces field, is formulated. Thus, the Routh's Theorem on stability of steady motion of a conservative mechanical system is extended to the case of a nonconservative system.

Stabilization of the non-trivial relative equilibria of a gyrostat with an elastic element in a circular orbit

The motion of a gyrostat, regarded as a rigid body, in a circular Kepler orbit in a central Newtonian force field is investigated in a limited formulation. A uniformly rotating statically and dynamically balanced flywheel is situated in the rigid body. A uniform elastic element, which, during the motion of the system, is subjected to small deformations, is rigidly connected to the rigid body-gyrostat body. The problem is discretized without truncating the corresponding infinite series, based on a modal analysis or using a certain specified system of functions, for example, of the assumed forms of the oscillations, which depend on the spatial coordinates and which satisfy appropriate boundary-value problems of the linear theory of elasticity. The elastic element is specified in more detail (a rod, plate, etc.), as well as its mass and stiffness characteristics and the form of the fastening, and the choice of the system of functions is determined. Non-trivial relative equilibria of the system (the state of rest with respect to an orbital system of coordinates when the elastic element is deformed) is sought approximately on the basis of a converging iteration method, described previously. It is shown, using Routh's theorem, that by an appropriate choice of the gyrostatic moment and when certain conditions, imposed on the system parameters are satisfied, one can stabilize these equilibria (ensure that they are stable).

The existence and stability of the equilibria of mechanical systems with constraints produced by large potential forces

A modification of Routh's theorem is investigated for systems with unilateral constraints produced by large potential forces, which enables steady motions to be found and the sufficient conditions for their stability to be investigated. The problem of an orbital “monkey bridge” is considered as an example.

The stability of the steady motions of a symmetrical gyrostat on an absolutely rough horizontal plane

The problem of the motion, without slipping, of a dynamically symmetrical gyrostat on a fixed horizontal plane is investigated. It is shown that Chaplygin's equations express, in projections on the axes of a semimobile system of coordinates, the theorem on the angular momentum of the gyrostat about the point of contact with the plane. All possible steady motions of a heavy symmetrical gyrostat on an absolutely rough plane including the case when the nutation angle is equal to zero are investigated using the generalized Routh theorem. The effect of the rotor on the stability of steady motions is also examined. The case when the body of the gyroscope is a solid with a circular base, in particular, a disc with a rotor, is considered. It is shown that the rotating rotor has a stabilizing influence on equilibrium of the gyrostat and a destabilizing influence on the rolling of the gyrostat about a straight line, provided the axial angular momentum of the gyrostat is zero.

The stability of permanent rotations of a top with a cavity filled with liquid on a plane with friction

The problem of the motion of a heavy dynamically symmetrical sphere along a horizontal plane with friction is considered. Inside the sphere there is an axisymmetric ellipsoidal cavity, completely filled with an ideal incompressible fluid, which performs uniform rotational motion. It is shown that the system, in addition to an integral of the constancy of vortex intensity, allows of a Jellet type integral. In addition, the total mechanical energy of the system is a non-increasing function. The stability of permanent rotations of the system is investigated using a modified Routh theorem [1,2]. Special cases of a spherical cavity and a weightless envelope are considered.

Root distribution of a polynomial in subregions of complex plane

Modified processes are proposed for directly treating singularities in the Routh array, and procedures are developed for determining the respective orders of simple and/or repeated roots lying on the imaginary axis. The extended Sturm test is developed for determining the root distribution on some specified lines. Moreover, to enhance the Routh stability theorem, the extended Routh theorem for finding the numbers of roots of a polynomial lying within some specified regions is proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The positions of relative equilibrium in the circular orbit of an elastic artificial satellite and their stability

The problem of the stability of motion in a circular orbit in a central Newtonian field of force of an elastic artificial satellite, modelled as a free solid body with an arbitrary elastic section connected to it, is considered in a limited formulation. Using Routh's theorem and assuming that the vector of small deformations of the elastic section can be represented in the form of a finite series in the known eigenmodes of its free oscillations [1]and certain others, the positions of relative equilibrium of the spacecraft (numbering 24) are obtained. The positions of equilibrium in which the elastic section is deformed are found approximately. The sufficient conditions for the stability of the positions of equilibrium obtained are found, and the necessary and sufficient conditions that the elastic section should be undeformed in a position of equilibrium are indicated. An example of an artificial satellite modelled as a solid body with an arbitrary rectilinear elastic rod attached to it is considered.

The existence and stability of invariant sets of dynamical systems

The possibility of using Lyapunov functions to construct invariant sets of dynamical systems is discussed. The investigations presented herein are based on certain ideas known from the literature /1–11/ and culminate in a generalization of Routh's Theorem and its modification /1–6, 12, 13/.

Conditions for a sum of forms to be of fixed sign and for stability of motion on manifolds

Lyapunov's corollary of the Stability Theorem /1/, a special case of which is Routh's theorem on the stability of the steady motion of a system with cyclic coordinates, provides a point of departure for the investigation conducted in this paper of the stability of motion on manifolds, particularly those defined by the integrals of the equations of the perturbed motion. Sufficient conditions are obtained for a sum of forms to be positive- or negative-definite and for the motion of polynomial systems to be stable on these manifolds.

On the stability of motion of a gyroscope in gimbals and the evaluation of deflections

The problem of stability of gyroscope motion is considered in the formulation in /1/. Liapunov function is constructed in the form of a nonlinear bundle of integrals. The obtained sufficient conditions of regular precession stability of a gyroscope coincide with those obtained in /1/ using the Routh theorem. Deviations from stable motion of the gyroscope are evaluated on the basis of the method of /2/ with allowance for the findings of /3/.

Stability of dedekind ellipsoids

The stability of Dedekind ellipsoids, which are characterized by their being fixed in space and preserving their shape through internal motion of the fluid when there are ellipsoidal perturbations of the surface, has been considered by Riemann and Chandrasekhar [1–4]. On the basis of an energy criterion, that the potential energy be minimal, Riemann showed that these ellipsoids are stable but for perturbation of only some of the variables. Chandrasekhar studied the stability on the basis of linearized virial equations. He showed that the Dedekind ellipsoids are stable under the condition that restrictions in the form of certain equations are imposed on the perturbing frequency σ. In the present paper, the stability is studied exactly, without linearization of the equations, on the basis of Routh's theorem [5] with respect to all variables under the assumption that after the perturbation the ellipsoid must preserve its shape. It is shown that the Dedekind ellipsoids are stable for ellipsoidal perturbations with respect to all variables without any restrictions on the frequency σ.

A New Proof of Routh's Theorem

A New Proof of Routh's Theorem

A New Proof of Routh's Theorem

A New Proof of Routh's Theorem

On the stability of permanent rotation of a heavy solid body about a fixed point: PMM vol. 40, n≗ 3, 1976, pp. 408–416

The permanent rotation of a heavy solid body about its principal axis of inertia with a fixed point is considered. Stability is investigated with the use of the theorem on the stability of Hamiltonian systems with two degrees of freedom in the general elliptic case. It is shown that in the absence of certain resonance relationships in the region of necessary stability conditions, which does not coincide with the region of known sufficient conditions, the first approximation indicates the existence of stability, except possibly, in the case when the parameters of the problem lie on some specific manifolds of the parameter space. Subregions that are free of such exceptional manifolds are indicated in each region of necessary stability conditions. Necessary stability conditions for permanent rotation about principal axes of inertia of a solid body were investigated by Grammel [3]. Sufficient conditions that matched necessary conditions were obtained by Chetaev in the case of Lagrange integrability [4], and by Rumiantsev in that of Kowalewska integrability [5]. Permanent rotation of a body with arbitrary mass distribution about its principal axis of inertia was considered in [6–8], where sufficient stability conditions were established by Chetaev's method and on the basis of the Routh theorem. Bifurcation of permanent rotations and changes of stability were investigated in [9].

Stability of stationary rotation of a heavy solid body with two elastic rods: PMM vol. 40, n≗1, 1976, pp. 55–64

The example of the stability problem for stationary vertical rotation of a heavy “tuning fork” with a single point is used for showing that the method of stability investigation of a mechanical system stationary motions, based on the solution of the problem of minimum transformed potential energy of a system [1–6], is equally suitable for analyzing “trivial” and “nontrivial” stationary motions for which deformable elements of the system are, respectively, in the undeformed and the deformed state. Two methods for solving this problem are presented. Sufficient stability conditions that impose certain restrictions from below on the stiffness of rods and also from above and below on the angular velocity of the system uniform rotation are obtained and analyzed. The general statement of the problem of motion and stability of an elastic body with a cavity containing a fluid was presented by Rumiantsev [1]. The theorem on stability proved here is an extension of the Routh theorem to systems with distributed parameters, and reduces the problem of stability of stationary motion to that of minimum (transformed) potential energy W of the system. Solutions of a number of problems on the stability of stationary motions of a solid body with elastic rods and fluid in potential force fields appear in [2–5]. Two methods for establishing conditions for positive definiteness of the second variation δ 2 W in investigations of motion stability of mechanical systems consisting of absolutely rigid bodies and material points with attached deformable elastic and fluid bodies are presented and illustrated in [6] on the example of solutions of specific mechanical problems. The problem of stability of stationary motions of mechanical systems with distributed parameters were investigated in [7–10] (see the bibliography in [7–10]). The problem of stability of the “nontrivial” state of relative equilibrium of an absolutely rigid body with elastic rods is considered in [9], where the “trivial” and “nontrivial” equilibrium states of the system are characterized by the undeformed and deformed states, respectively. Only trivial states of relative equilibrium of rigid bodies with elastic rods were investigated in [2–6] and [7, 8]. It is noted in [9] that the previously used method based on the direct Liapunov method cannot be applied for solving problems of stability of the nontrivial state of relative equilibrium of the system, and another method of solution, based on the expansion of elastic displacements of rods in series of some complete system of functions is proposed. A finite number of first terms was retained in such series without any mathematical substantiation, and the system with distributed parameters is essentially reduced to some system with a finite number of degrees of freedom.

�ber die instabilit�t konservativer systeme mit gyroskopischen kr�ften

Routh's theorem states that a steady motion of a discrete, conservative mechanical system is stable if the dynamic potential W(q)=U(q)−T0(q) assumes a minimum. This is a generalized version of the theorem on the stability of equilibrium at a minimum of the potential energy, which is due to Dirichlet. It is well known that a steady motion may also be stable if W(q) assumes a maximum instead of a minimum. The stability is then due to the gyroscopic terms in the equations of motion, without which the steady motion would be unstable. Here it is shown that the steady motion is always unstable if not only W(q) but also H 0(q) assumes a maximum, H 0(q) being the part of the Hamiltonian that does not depend on the momenta. It is astonishing that this unexpectedly simple criterion was not found before now. In the proof, a variational formulation is used for the problem, and the instability is shown directly from the existence of certain motions which diverge from the trivial solution.

On the stability of stationary motions of systems with quasi-ignorable coordinates and of mechanical equilibrium under the action of a magnetic field: PMM vol. 37, n≗3, 1973, pp. 400–406

It has been shown that the stability of the steady-state motions of systems with quasi-ignorable coordinates can be judged from the stability of the equilibrium of a position subsystem with constant (invariable) quasi-ignorable generalized velocities. This allows us to disregard the degrees of freedom corresponding to the quasi-ignorable coordinates and to use Poincare's results on the change of stability at the bifurcations of equilibria by taking the quasi-ignorable velocities as parameters. Examples of systems of the class being considered are electromechanical systems not containing capacitances. The results mentioned above are valid for them also for a nonlinear connection between B and Hin magnetics. In the case of ignorable coordinates the judgement on the stability of a stationary motion from the stability of the equilibrium of a position subsystem is possible with the aid of Routh's theorem generalized and supplemented by Liapunov [1]. However, this case differs essentially from the one being considered in that it is the ignorable momenta and not the velocities that are taken as constants.