On First Integrals and Invariant Manifolds in the Generalized Problem of the Motion of a Rigid Body in a Magnetic Field

2021 ◽  
pp. 157-173
Author(s):  
Valentin Irtegov ◽  
Tatiana Titorenko
Keyword(s):  
Magnetic Field ◽  
Rigid Body ◽  
Invariant Manifolds ◽  
First Integrals

scholarly journals Conformal geodesics on gravitational instantons

2021 ◽  
pp. 1-32
Author(s):  
MACIEJ DUNAJSKI ◽  
PAUL TOD
Keyword(s):  
Magnetic Field ◽  
Geodesic Flow ◽  
Isometry Group ◽  
Three Dimensional ◽  
First Integrals ◽  
Complete Integrability ◽  
Conformal Killing ◽  
Gravitational Instantons ◽  
Geodesic Equations ◽  
Jacobi Equations

Abstract We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

On the first integrals of the problem of jumping a massive rigid body on a smooth horizontal plane

2004 ◽  
Vol 49 (6) ◽  
pp. 366-368
Author(s):  
A. A. Burov ◽  
D. P. Chevallier
Keyword(s):  
Rigid Body ◽  
Horizontal Plane ◽  
First Integrals

ON A NORMAL-FORM ANALYSIS FOR A CLASS OF PASSIVE BIPEDAL WALKERS

2001 ◽  
Vol 11 (09) ◽  
pp. 2411-2425 ◽  
Author(s):  
PETRI T. PIIROINEN ◽  
HARRY J. DANKOWICZ ◽  
ARNE B. NORDMARK
Keyword(s):  
Normal Form ◽  
Rigid Body ◽  
Center Manifold ◽  
Invariant Manifolds ◽  
Symmetric Body ◽  
Numerical Schemes ◽  
Form Analysis ◽  
Natural Dynamics

This paper implements a center-manifold technique to arrive at a normal-form for the natural dynamics of a passive, bipedal rigid-body mechanism in the vicinity of infinite foot width and near-symmetric body geometry. In particular, numerical schemes are developed for finding approximate forms of the relevant invariant manifolds and the near-singular dynamics on these manifolds. The normal-form approximations are found to be highly accurate for relatively large foot widths with a range of validity extending to widths on the order of the mechanisms' height.

E-layer precession in a plasma

1971 ◽  
Vol 6 (2) ◽  
pp. 413-424 ◽  
Author(s):  
H. L. Berk ◽  
R. N. Sudan
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Rigid Body ◽  
Plasma Density ◽  
Uniform Magnetic Field ◽  
Alfven Wave ◽  
Background Plasma ◽  
Low Density ◽  
High Background ◽  
Plasma Background

A weak E layer in a non-uniform magnetic field will tend to precess as a rigid body in response to the radial focusing of external magnetic fields and fields due to wall currents. We study the interaction of this precessional mode with a background plasma, and we explicitly include dissipation mechanisms in the plasma, walls and external resistors. When the plasma background is treated in the MHD approximation, we find that the mode changes character from a precessional mode at low density to a compressional Alfvén wave at high density. For a very weak E layer, instability is found, even without dissipation, when a sufficiently high background plasma density is present. However, for moderate E-layer strengths, the modes are found to be stable, even with dissipation.

scholarly journals The Routh theorem for mechanical systems with unknown first integrals

2017 ◽  
Vol 44 (2) ◽  
pp. 169-180
Author(s):  
Alexander Karapetyan ◽  
Alexander Kuleshov
Keyword(s):  
Rigid Body ◽  
Mechanical Systems ◽  
First Integrals ◽  
Rotationally Symmetric ◽  
Rough Plane ◽  
Symmetric Rigid Body

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.

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