Dynamics of the rigid body

Typical motion in integrable Hamiltonian systems is quasi-periodic (if motions are bounded, which is always the case for rigid body dynamics). Geometrically speaking, this implies that the motion takes place on invariant tori in phase space. In the dynamics of the rigid body the toral angles have very intuitive meanings of rotation, precession and nutation. For very low energies there may be pendulum-like motions as well.

My publications on this subject

Heinz Hanßmann

Normal forms for perturbations of the Euler top

p. 151-173 in :

Normal forms and homoclinic chaos, Waterloo 1992 (eds. W.F. Langford and W. Nagata)

Fields Institute Communications 4 (1995)

The abstract is reprinted both in Mathematical Reviews 97b:70008 and the Zentralblatt der Mathematik 831.70006.

Heinz Hanßmann

Quasi-periodic Motions of a Rigid Body

A case study on perturbations of superintegrable systems

Proefschrift, Rijksuniversiteit Groningen (1995)

Heinz Hanßmann

Equivariant perturbations of the Euler top

p. 227-253 in :

Nonlinear Dynamical Systems and Chaos, Groningen 1995 (eds. H.W. Broer et al.)

Progress in Nonlinear Partial Differential Equations and Their Applications 19, Birkhäuser (1996)

The abstract is reprinted in the Zentralblatt der Mathematik 847.70009 and there is a review in Mathematical Reviews 98c:58149.

Heinz Hanßmann

Quasi-periodic Motions of a Rigid Body I

Quadratic Hamiltonians on the Sphere with a Distinguished Parameter

Regular and Chaotic Dynamics 2(2), p. 41-57 (1997)

The abstract is available from the publisher and appeared slightly changed in the Zentralblatt der Mathematik 935.70006. Furthermore there is a review in Mathematical Reviews 2000h:70009. A preprint version can be downloaded as PostScript file (3.9 M) or in a gzipped version (244 K).

Heinz Hanßmann

Quasi-periodic Motion of a Rigid Body under Weak Forces

p. 398-402 in :

Hamiltonian Systems with Three or More Degrees of Freedom, S'Agaro 1995 (ed. C. Simó)

NATO ASI series C 533, Kluwer (1999)

The abstract appeared slightly changed in the Zentralblatt der Mathematik 970.70008. A preprint version can be downloaded as PostScript file (787 K) or in a gzipped version (89 K).

Heinz Hanßmann

Quasi-periodic motions of the perturbed Euler top

p. 1161-1166 in :

Equadiff 99, Berlin 1999 (eds. B. Fiedler, K. Gröger, J. Sprekels)

World Scientific (2000)

A preprint version can be downloaded as PostScript file (1.3 M) or in a gzipped version (100 K).

Heinz Hanßmann and Philip Holmes

On the global dynamics of Kirchhoff's equations

Rigid body models for underwater vehicles

p. 353-371 in :

Global Analysis of Dynamical Systems, Leiden 2001 (eds. H.W. Broer, B. Krauskopf, G. Vegter)

IoP publishing (2001)

The abstract is reprinted in Mathematical Reviews 2002h:70008 and there is a review in the Zentralblatt der Mathematik 1015.37043. A preprint version can be downloaded as PostScript file (7.4 M) or in a gzipped version (973 K).

Troy R. Smith, Heinz Hanßmann and Naomi Ehrich Leonard

Orientation control of multiple underwater vehicles with symmetry-breaking potentials

p. 4598-4603 in :

Proceedings of the 40th IEEE Conference on Decision and Control, Orlando 2001 (eds. D.W. Repperger et al.)

IEEE (2001)

The abstract is available from the publisher. A preprint version can be downloaded as PostScript file (456 K) or in a gzipped version (98 K).

Heinz Hanßmann and Jan-Cees van der Meer

On non-degenerate Hamiltonian Hopf bifurcations in 3DOF systems

p. 476-481 in :

Equadiff 2003, Hasselt 2003 (eds. F. Dumortier, H.W. Broer, J. Mawhin, A. Vanderbauwhede and S. Verdyun Lunel)

World Scientific (2005)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik 1102:37037. A preprint version can be downloaded as PostScript file (143 K) or in a gzipped version (60 K).

Heinz Hanßmann

Perturbations of integrable and superintegrable Hamiltonian systems

p. 1527-1536 in :

Fifth Euromech Nonlinear Dynamics Conference, Eindhoven 2005 (eds. D.H. van Campen, M.D. Lazurko and W.P.J.M. van den Oever)

Technische Universiteit Eindhoven (2005)

This publication can be downloaded as PostScript file (3.0 M) or in pdf (321 K).

Heinz Hanßmann, Naomi Ehrich Leonard and Troy R. Smith

Symmetry and Reduction for Coordinated Rigid Bodies

European Journal of Control 12(2), p. 176-194 (2006)

The abstract is available here and there is a review in Mathematical Reviews 2007d:37001. A preprint version can be downloaded as PostScript file (5.6 M) or in a gzipped version (1.2 M).

Henk W. Broer, Heinz Hanßmann, Jun Hoo and Vincent Naudot

Nearly-integrable perturbations of the Lagrange top:

applications to KAM-theory

p. 286-303 in :

Dynamics & Stochastics: Festschrift in Honor of M.S. Keane (eds. D. Denteneer, F. den Hollander and E. Verbitskiy)

Lecture Notes 48, Inst. of Math. Statistics (2006)

The abstract is available here and is reprinted in the Zentralblatt der Mathematik 1125:70003. Furthermore there is a review in Mathematical Reviews 2009h:70009. This publication can be downloaded from the ArXiv as PostScript file (1.3 M) or in pdf (894 K).

Heinz Hanßmann

A monkey saddle in rigid body dynamics

p. 92-99 in :

SPT 2007: Symmetry and Perturbation Theory, Otranto 2007 (eds. G. Gaeta, R. Vitolo and S. Walcher)

World Scientific (2008)

The abstract is reprinted in Mathematical Reviews 2009h:70016 and appeared slightly changed in the Zentralblatt der Mathematik 1142.70003. A preprint version can be downloaded as PostScript file (1.5 M) or in pdf (317 K).

Heinz Hanßmann

Quasi-periodic Motions of a Rigid Body II

Implications for the Original System

Preprint, Inst. Reine & Angew. Math., RWTH Aachen (1999)

This 14 pages preprint can be downloaded as PostScript file (1.3 M) or in a gzipped version (154 K).

Heinz Hanßmann

Perturbations superintegrable systems

Preprint, Universiteit Utrecht (2014)

This 19 pages preprint can be downloaded as PostScript file (1.0 M) or in pdf (215 K).