Lagrange Case Research Articles

On Bivariate Hermite Interpolation with Minimal Degree Polynomials

A Newton-type approach is used to deal with bivariate polynomial Hermite interpolation problems when the data are distributed in the intersections of two families of straight lines, as a generalization of regular grids. The interpolation operator is degree-reducing and the interpolation space is a minimal degree space. Integral remainder formulas are given for the Lagrange case, then extended to the Hermite case, and finally used to obtain error estimates.

A new solution to the lagrange's case of the three‐body problem

The motion of three particles, interacting by gravitational forces, is studied in a new coordinate system given by the principal axes of inertia, as determined by Euler angles, and using the inertia principal moments and an auxiliar angle as coordinates. The solution to the particular Lagrange case of the three‐body problem is reviewed and solved in these new coordinates.

Orbital classification of the integrable problems of Lagrange and Goryachev-Chaplygin by the methods of computer analysis

In an earlier paper Orel computed the trajectory (or orbital) invariant for the Goryachev-Chaplygin problem. She noted, however, that the investigation of this invariant is beset with insurmountable analytic difficulties. The present paper completes the construction of the trajectory invariant for the Goryachev-Chaplygin problem, as well as for the Lagrange problem, by the methods of computer analysis. Thus the orbital classification question as been solved for these problems. In the paper we also formulate conjectures relating to the Lagrange case, which can serve as a basis for further investigations in this area. The computational algorithms themselves are due to Takahashi.

Perturbed rotational motions of a rigid body similar to regular precession

Perturbed rotational motions of a rigid body, similar to regular precession in the Lagrange case, when the restoring torque depends on the angle of nutation, are investigated. It is assumed that the angular velocity of the body is fairly large, its direction are close to the dynamic axis of symmetry of the body and the perturbing torques are small compared with the restoring torques. A small parameter is introduced in a special way and the method of averaging is used. The averaged equations of motion are derived in the first and second approximations. Specific mechanical models of the perturbations are considered. Lagrange-like perturbed motions of a rigid body were also investigated in /1/. Almost regular perturbed rotational motions of a rigid body have been studied /2, 3/, and some attention has been given to pseudoregular precession in the Lagrange case; in the former case it was assumed that a constant restoring torque is applied to the body.

Periodic motions of a heavy, rigid, dynamically almost symmetric body with a fixed point

A motion is studied, about a fixed point, of a heavy rigid body, the mass distribution in which resembles the mass distribution encountered in the Lagrange's case. The equations of motion are written in the action-angle variables for the Lagrange's case, after which the method developed by Poincaré /1/ for such systems is used to prove the existence of new families of periodic motions represented by series in powers of a small parameter introduced into the process.

ON THE MOTION OF A MULTIDIMENSIONAL BODY WITH FIXED POINT IN A GRAVITATIONAL FIELD

The problem of the motion of an n-dimensional solid body with a fixed point in a gravitational field is considered. More precisely, the integrable case of such motion determined by certain symmetry conditions of the body is considered. These conditions are obtained as a generalization of the conditions for the Lagrange case of the motion of a three-dimensional heavy gyroscope. For the n-dimensional Lagrange case the collection of first integrals presented in the paper is sufficient for complete integrability. The fact that the case considered provides an example of a completely integrable Hamiltonian system with a noncommutative algebra of first integrals is of interest.Bibliography: 7 titles.

Perturbed motions of a rigid body, close to the Lagrange case: PMM vol. 43, no. 5, 1979, pp. 771–778

The perturbed motions of a rigid body, close to the Lagrange case, are investigated. Conditions are presented for the possibility of averaging the equations of motion with respect to the nutation angle and the averaged system of equations is obtained. An actual mechanical model of the perturbations,corresponding to the body's motion in a medium with linear dissipation, is considered. A numerical solution of the averaged system is constructed.

Stability of permanent rotations of a symmetric solid: PMM vol. 40, n≗1, 1976, pp. 171–173

We establish that a heavy solid with one fixed point can execute, in the Lagrange case, steady rotations about an axis situated arbitrarily within the body, in addition to a rotation about the dynamic symmetry axis. By combining the integrals of perturbed motion, we find sufficient conditions for the stability of the permanent rotation under consideration. We also indicate the necessary conditions of stability, using the system of the first approximation equations. The stability of the rotation of a heavy solid with one fixed point about the dynamic symmetry axis situated vertically, was investigated in [1] for the Lagrange case. The necessary and sufficient condition for this rotation in a more general force field was obtained in [2].

On the justification of the principle of potential energy minimum in problems of equilibrium stability of nonlinearly elastic membranes: PMM vol. 35, n≗1, 1971, pp. 168–171

Stability conditions for the vertical position of a heavy symmetrical gyroscope (Lagrange's case) were derived by Chetaev [1], who used the second method of Liapunov without any simplifying assumptions (as it is used for example in the small vibrations method), and also by Rumiantsev [2], who found stability conditions for Kovalevska's case. This note supplements the above-mentioned results, presenting the derivation of stability conditions for the vertical position of a heavy symmetrical gyroscope on gimbals [in a Cardan mounting]. The solution of this problem can be regarded as a generalization of the Lagrange case, and it shows the existence of certain new effects noticed previously by Nikolai [3], when he investigated a special case of a rapidly rotating astatic gyroscope.

Sur un nouveau cas particulier du probleme des trois corps

The plane of the three bodies M 0, M 1, M 2, is assumed to envelop a cone of revolution C , of which the vertex is M 0 and the axis direction is fixed in space. Referring M 1 and 2 to M 0 by means of an inertial triaxial rectangular Cartesian coordinate system of fixed directions and with the axis of C as one of the axes, it is shown that the motions of M 1 and M 2 are given by the integration of a differential equation D of the third order with respect to an angular variable φ. If C is reduced to its axis (case 1), D can actually be written out and the only two cases possible are then Lagrange's case and the case where two of the three bodies M 0, M 1, and M 2 collide. If one of the two bodies M 1, M 2 has zero mass (case 2), D becomes a second-order differential equation which can actually be written out, and the elliptic motion of the body of nonzero mass permits integration of D . The motion of the body of zero mass is given by the integration of a differential equation of second order with respect to an auxiliary variable v which is actually written out. The Cartesian coordinates of M 1 and M 2 are algebraic functions of the masses of M 1 and M 2, of the vertex angle of C , and of sin φ, cos φ, dφ dt , d 2φ dt 2 , d 3φ dt 3 . In case 1, these can actually be written out. In case 2, the Cartesian coordinates of the body on nonzero mass are algebraic functions of its mass and of sin φ, cos φ, dφ dt which are actually written out.

On the stability of 4 heavy symmetrical gyroscope on gimbals

Stability conditions for the vertical position of a heavy symmetrical gyroscope (Lagrange's case) were derived by Chetaev [1], who used the second method of Liapunov without any simplifying assumptions (as it is used for example in the small vibrations method), and also by Rumiantsev [2], who found stability conditions for Kovalevska's case. This note supplements the above-mentioned results, presenting the derivation of stability conditions for the vertical position of a heavy symmetrical gyroscope on gimbals [in a Cardan mounting]. The solution of this problem can be regarded as a generalization of the Lagrange case, and it shows the existence of certain new effects noticed previously by Nikolai [3], when he investigated a special case of a rapidly rotating astatic gyroscope.