Kinematic interpretation of the motion of a body with a fixed point

Kharlamov, P. V.

doi:10.1016/0021-8928(64)90102-9

Cited by 16 publications

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“…On the (x 1 , x 2 )-plane the motion is represented as a "sum" of two independent partial motions (x 1 (ψ 1 (τ )), x 0 2 ) and (x 0 1 , x 2 (ψ 2 (τ ))). For the orientation matrix and the angular velocity vectors we obtain according to (8) ω = ω (1) + ω (2) , Ω = Ω (1) + Ω (2) ,…”

Section: Decomposition Of Motionsmentioning

confidence: 99%

“…P.V. Kharlamov [2] proposed a natural way to find the fixed hodograph and to investigate its properties for all values of the existing parameters. This method is known as the hodographs method of the kinematic interpretation of motion and is based on applying some non-holonomic kinematic characteristics.…”

Section: Introductionmentioning

confidence: 99%

“…Kharlamov has shown that this surface is a surface of rotation with the meridian completely defined by the initial periodic solution by means of explicit functions and the missing angular coordinate of the fixed hodograph can be found by integrating the known function of time. The equations obtained in [2] gave rise to geometric interpretations built for numerous cases of partial integrability (see reviews in [3,4] and the contemporary state of investigations in [5]).…”

Section: Introductionmentioning

confidence: 99%

“…Gashenenko investigated the hodographs properties for quasi-periodic solutions in the classical cases of Goryachev -Chaplygin and Kovalevskaya. Using the equations of the paper [2] and the ideas of the number theory and Fourier analysis, Gashenenko described those classes of non-resonant motions in the integrable reducible rigid body systems for which the components of the angular velocity in the inertial space also are two-periodic functions of time or, in other words, onevalued functions on the corresponding regular Liouville torus (on the connected component of a regular integral manifold of the Euler -Poisson equations) bearing trajectories with irrational rotation number. The motion is represented by rolling of one surface "through" another in such a way that a curve dense in the first surface rolls without slipping over a similar curve dense in the second surface.…”

Section: Introductionmentioning

confidence: 99%

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On the geometry of motions in one integrable problem of the rigid body dynamics

Kharlamova

Savushkin

2015

Journal of Geometry and Physics

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Due to Poinsot's theorem, the motion of a rigid body about a fixed point is represented as rolling without slipping of the moving hodograph of the angular velocity over the fixed one. If the moving hodograph is a closed curve, visualization of motion is obtained by the method of P.V. Kharlamov. For an arbitrary motion in an integrable problem with an axially symmetric force field the moving hodograph densely fills some two-dimensional surface and the fixed one fills a three-dimensional surface. In this paper, we consider the irreducible integrable case in which both hodographs are two-frequency curves. We obtain the equations of bearing surfaces, illustrate the main types of the surfaces. We propose a method of the so-called non-straight geometric interpretation representing the motion of a body as a superposition of two periodic motions.

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Section: Decomposition Of Motionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the geometry of motions in one integrable problem of the rigid body dynamics

Kharlamova

Savushkin

2015

Journal of Geometry and Physics

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“…En el capítulo dos se hace un estudio breve del planteamiento del problema del cuerpo rígido con un punto fijo en un campo de gravedad constante y se estudian los casos integrables conocidos [5,17,18] .…”

Section: Capítulo 1 Introducciónunclassified

Caos en la dinámica del cuerpo rígido

Ramírez

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Capítulo 2. Breve estudio de la cinemática y dinámica del cuerpo rígido La matriz inversa de la matriz de rotación es su matriz traspuesta. Estas transformaciones lineales que mantienen invariante la norma se llaman transformaciones ortogonales. Las rotaciones son trasformaciones lineales ortogonales que además dejan invariante la propiedad de la base de vectores. Un conjunto ordenado de vectores ortonormales b 1 , b 2 y b 3 , se dice que forman una base cuando el determinante de la matriz que tienen por columnas primera, segunda y tercera, respectivamente, a los vectores b 1 , b 2 y b 3 es igual a 1: |(b 1 b 2 b 3)| = 1, (2.7) donde las barras verticales en laúltima ecuación significan que hay que realizar la operación de tomar el determinante. Para el caso cuando este toma el valor de-1, la base se llamara izquierda. El producto escalar es invariante ante una rotación. Por esta razón una base ortonormal se transformará mediante alguna rotación en otra base ortonormal. Veremos a continuación la consecuencia de que una base derecha se rote en una base derecha. Primeramente consideramos el determinante de la matriz de los tres vectores rotados. |(Rb 1 Rb 2 Rb 3)| = 1, (2.8) el cual tiene el valor uno debido a que R es una rotación. De estaúltima ecuación se puede sacar la matriz de rotación como factor común de la matriz de vectores (b 1 b 2 b 3): (Rb 1 Rb 2 Rb 3) = R(b 1 b 2 b 3) (2.9) ahora usamos la propiedad de que el determinante de un producto matricial es igual al producto de los determinantes de las matrices factores |(Rb 1 Rb 2 Rb 3)| = |R(b 1 b 2 b 3)| = |R||(b 1 b 2 b 3)| = 1.

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Application of the Kovacic algorithm for the investigation of motion of a heavy rigid body with a fixed point in the Hess case

Bardin¹,

Kuleshov²

2022

Z Angew Math Mech

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In 1890, Hess found new special case of integrability of Euler-Poisson equations of motion of a heavy rigid body with a fixed point. In 1892, Nekrasov proved that the solution of the problem of motion of a heavy rigid body with a fixed point under Hess conditions reduces to integrating the second-order linear differential equation. In this paper, the corresponding linear differential equation is derived and its coefficients are presented in the rational form. Using the Kovacic algorithm, we proved that the Liouvillian solutions of the corresponding secondorder linear differential equation exist only in the case, when the moving rigid body is the Lagrange top, or in the case, when the constant of the area integral is zero.

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Kinematic interpretation of the motion of a body with a fixed point

Cited by 16 publications

References 0 publications

On the geometry of motions in one integrable problem of the rigid body dynamics

On the geometry of motions in one integrable problem of the rigid body dynamics

Caos en la dinámica del cuerpo rígido

Application of the Kovacic algorithm for the investigation of motion of a heavy rigid body with a fixed point in the Hess case

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