Chaplygin ball over a fixed sphere: an explicit integration

Borisov, Alexey V.; Fedorov, Yu. N.; Mamaev, Ivan S.

doi:10.1134/s1560354708060063

Cited by 53 publications

(73 citation statements)

References 16 publications

(13 reference statements)

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“…рис. 1), ω и M -угловая ско-рость тела и угловой момент относительно точки контакта, тогда уравнения движения, описывающие их эволюцию, представляются в форме [1,10] …”

Section: уравнения движения и первые интегралыunclassified

“…Оказалось, что с точки зрения явного интегрирования новая система пред-ставляет существенные трудности. На нулевом уровне одного из найденных интегралов, который линеен по импульсам (и является аналогом интеграла площадей), явное решение приведено в работах [1,9]. Случай ненулевой константы указанного интеграла до сих пор далек от разрешения.…”

Section: Introductionunclassified

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Topological analysis of one integrable system related to the rolling of a ball over a sphere

Borisov¹,

Mamaev²

2012

Nelin. Dinam.

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Рассматривается новая интегрируемая система, описывающая качение твердого тела со сферической полостью по шаровому основанию. Ранее авторами для данной системы было найдено разделение переменных на нулевом уровне линейного (по угловой скорости) пер-вого интеграла, в то время как в общем случае разделить переменные не удается. В данной работе показано, что слоение на инвариантные торы в этой задаче эквивалентно соответ-ствующему слоению в интегрируемой системе Клебша в динамике твердого тела (для ко-торой также не найдено вещественное разделение переменных). В частности, для данной системы возможна неподвижная точка типа фокус, что может служить топологическим препятствием для вещественного разделения переменных.

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Section: уравнения движения и первые интегралыunclassified

Section: Introductionunclassified

Topological analysis of one integrable system related to the rolling of a ball over a sphere

Borisov¹,

Mamaev²

2012

Nelin. Dinam.

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“…The classical problem of rolling motion of a dynamically non-symmetric balanced Chaplygin ball has been sufficiently well investigated [11]. A newer version of it, namely the rolling of a Chaplygin's ball over a sphere, has recently been discussed in a number of papers [1,5,6].…”

Section: Introductionmentioning

confidence: 99%

Two non-holonomic integrable problems tracing back to Chaplygin

Borisov

Mamaev

2012

Regul. Chaot. Dyn.

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The paper considers two new integrable systems which go back to Chaplygin. The systems consist of a spherical shell that rolls on a plane; within the shell there is a ball or Lagrange's gyroscope. All necessary first integrals and an invariant measure are found. The solutions are shown to be expressed in terms of quadratures.MSC2010 numbers: 76M23, 34A05 INTRODUCTIONRecent advances in the design of controlled devices that use one or several balls for their propulsion (see, e.g., [3,4,[7][8][9][10][12][13][14]) has recently evoked an increasing interest in various models (in particular, non-holonomic ones) for rolling motion of spherical shells, including the case where some of the shells contain intricate mechanisms inside. The classical problem of rolling motion of a dynamically non-symmetric balanced Chaplygin ball has been sufficiently well investigated [11]. A newer version of it, namely the rolling of a Chaplygin's ball over a sphere, has recently been discussed in a number of papers [1, 5, 6].Here we consider two new integrable systems that trace back to Chaplygin. His paper [2] is essentially concerned with generalized conditions for the existence of integrals linear in velocities for mechanical systems that consist of several spheres. These conditions are even now far from being completely appreciated. Chaplygin discusses in detail two problems. The first of them deals with a system that consists of a spherical, geometrically and dynamically symmetric shell with a homogeneous ball rolling inside; the shell itself rolls without slipping on a horizontal plane (Fig. 1). Chaplygin established the integrability of this system by expressing the solutions in terms of quadratures. Although his method for obtaining quadratures is quite natural (and also applies to the problem of rolling of a body of revolution on a plane), we believe that in solving this problem Chaplygin committed a few inaccuracies which resulted in enormously complicated and hardly verifiable formulas. Besides, Chaplygin missed one of the additional first integrals. The complexity of the results seems to discourage Chaplygin himself. Indeed, for each newly obtained analytical solution he always pursued clarification of its dynamical and geometrical aspects, but not in that case! Here we present a new approach to this system, a complete set of first integrals and reduce the problem to quadratures.The second problem is concerned with rolling of a shell (with a spherical pendulum inside) on a plane. For this problem the equations of motion were integrated by Chaplygin. We show that a more general system which consists of a Lagrange's gyroscope placed inside a rolling sphere is also integrable. The equations can be integrated using some generalized versions of the Chaplygin vector integrals and two additional linear integrals.

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“…Отметим, что динамика сферо-роботов, как правило, описывается неголономными уравнениями, выводу которых посвя-щено большое количество работ. Мы укажем здесь только работы последних лет, в которых можно найти подробный библиографический обзор [28][29][30][31]. Отметим также недавние рабо-ты, посвященные исследованию движения сферороботов (или систем близких к ним) в при-сутствии сил сопротивления различной природы, от сухого трения до движения в вязкой среде [32][33][34][35][36][37].…”

Section: Introductionunclassified

The dynamic of a spherical robot with an internal omniwheel platform

Karavaev¹,

Kilin²

2015

Nelin. Dinam.

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Chaplygin ball over a fixed sphere: an explicit integration

Cited by 53 publications

References 16 publications

Topological analysis of one integrable system related to the rolling of a ball over a sphere

Topological analysis of one integrable system related to the rolling of a ball over a sphere

Two non-holonomic integrable problems tracing back to Chaplygin

The dynamic of a spherical robot with an internal omniwheel platform

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