Arnold Math J. (2016) 2:511–578
DOI 10.1007/s40598-016-0054-6
RESEARCH EXPOSITION
Some Recent Generalizations of the Classical Rigid
Body Systems
Vladimir Dragovi´c
1,2
· Borislav Gaji´c
2
Received: 20 November 2014 / Revised: 13 July 2016 / Accepted: 25 August 2016 /
Published online: 19 September 2016
© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2016
Abstract Some recent generalizations of the classical rigid body systems are
reviewed. The cases presented include dynamics of a heavy rigid body fixed at a
point in three-dimensional space, the Kirchhoff equations of motion of a rigid body
in an ideal incompressible fluid as well as their higher-dimensional generalizations.
Keywords Rigid body dynamics · Lax representation · Euler–Arnold equations ·
Algebro-geometric integration procedure · Baker–Akhiezer function · Grioli
precession · Kirchhoff equations
Mathematics Subject Classification Primary 70E17 · 70E40 · 14H70; Secondary
70E45 · 70H06
Contents
1 Introduction .............................................512
2 The Hess–Appel’rot Case of Rigid Body Motion ..........................514
2.1 Basic Notions of Heavy Rigid Body Fixed at a Point ......................514
2.2 Integrable Cases .........................................517
2.3 Definition of the Hess–Appel’rot System ............................517
B
Vladimir Dragovi´c
vladimir.dragovic@utdallas.edu
Borislav Gaji´c
gajab@mi.sanu.ac.rs
1
The Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX,
USA
2
Mathematical Institute, Serbian Academy of Science and Art, Kneza Mihaila 36, 11000 Belgrade,
Serbia
123
512 V. D rag o vi ´c, B. Gaji´c
2.4 A Lax Representation for the Classical Hess–Appel’rot System:
An Algebro-Geometric Integration Procedure .........................518
2.5 Zhukovski’s Geometric Interpretation .............................521
3 Kowalevski Top, Discriminantly Separable Polynomials, and Two Valued Groups .........522
3.1 Discriminantly Separable Polynomials .............................522
3.2 Two-Valued Groups .......................................524
3.3 2-Valued Group Structure on CP
1
and the Kowalevski Fundamental Equation ........526
3.4 Fundamental Steps in the Kowalevski Integration Procedure ..................527
3.5 Systems of the Kowalevski Type: Definition ..........................531
3.6 An Example of Systems of the Kowalevski Type ........................532
3.7 Another Example of an Integrable System of the Kowalevski Type ..............535
3.8 Another Class of Systems of the Kowalevski Type .......................537
3.9 A Deformation of the Kowalevski Top .............................539
4 The Lagrange Bitop and the n-Dimensional Hess–Appel’rot Systems ...............544
4.1 Higher-Dimensional Generalizations of Rigid Body Dynamics ................544
4.2 The Heavy Rigid Body Equations on e(n) ...........................546
4.3 The Heavy Rigid Body Equations on s = so(n) ×
ad
so(n) ..................547
4.4 Four-Dimensional Rigid Body Motion .............................548
4.5 The Lagrange Bitop: Definition and a Lax Representation ...................549
4.5.1 Classical Integration ...................................551
4.5.2 Properties of the Spectral Curve .............................552
4.6 Four-Dimensional Hess–Appel’rot Systems ..........................554
4.7 The n-Dimensional Hess–Appel’rot Systems ..........................556
4.8 Classical Integration of the Four-Dimensional Hess–Appel’rot System ............559
5 Four-Dimensional Grioli-Type Precessions .............................562
5.1 The Classical Grioli Case ....................................562
5.2 Four-Dimensional Grioli Case .................................564
6 Motion of a Rigid Body in an Ideal Fluid: The Kirchhoff Equations ................567
6.1 Integrable Cases .........................................568
6.2 Three-Dimensional Chaplygin’s Second Case .........................570
6.2.1 Classical Integration Procedure ..............................571
6.2.2 Lax Representation for the Chaplygin Case .......................572
6.3 Four-Dimensional Kirchhoff and Chaplygin C ases .......................573
References ................................................575
1 Introduction
The rigid body dynamics is a core subject of the Classical Mechanics. Findings of
Mathematics of the last 40–50 years, shed a new light on Mechanics, and on rigid-
body dynamics in particular. Strong impetus to a novel approach and modernization
of Mechanics came from celebrated Arnold’s book (Arnold 1974). There are several
approaches and schools developing further those i deas. To mention a few: Kozlov,
Abraham, Marsden, Kharlamov, Manakov, Novikov, Dubrovin, Fomenko, Horozov,
Sokolov and their students and collaborators. Along that line, in 1993, the first author
initiated a new seminar in the Mathematical Institute of the Serbian Academy of
Sciences and Arts, inspired by and named after Arnold’s book. One of the most popular
research topics within the Seminar was rigid body dynamics, and several young people
got involved. The purpose of this article is to make a review of some of the results
obtained in almost quarter of the century of the Seminar’s activity. Let us mention that
the classical aspects of rigid body dynamics occupied attention of Serbian scientists
for a long time, see for example books and monographs (Bilimovi´c 1955; Dragovi´c
and Milinkovi´c 2003; Andjeli´c and Stojanovi´c 1966) and references therein.
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Some Recent Generalizations of the Classical... 513
In this paper we will focus mostly on the problems related to the integrability of
motion of a rigid body either in the case of a heavy body fixed at a point or a body
embedded in an ideal fluid and their higher dimensional generalizations. Although
the first higher dimensional generalizations of rigid body dynamics appeared in XIX
century (see Frahm 1874; Schottky 1891), a strong development of the subject came
after Arnold’s paper (1966).
Integrability or solvability, is one of the fundamental questions related to the system
of differential equations of motion of some mechanical system. The integrability is
closely related to the existence of enough number of independent first integrals, i.e.
functions that are constant along the solutions of the system. Early history was devel-
oped by classics (Euler, Lagrange, Hamilton, Jacobi, Liouville, Kowalevski, Poincaré,
E. Noether, and many others). The basic method of that time was the method of sep-
aration of variables and Noether’s theorem was the tool for finding first integrals
from the symmetries of the system. With the work of Kowalevski a more subtle alge-
braic geometry and more intensive theory of theta functions entered on the stage. The
final formulation of the principle theorem of the subject of classical integrability, the
Liouville–Arnold theorem, which gives a qualitative picture of the integrable finite-
dimensional Hamiltonian systems appeared in the Arnold’s paper (Arnold 1963)(see
also Arnold 1974).
In the classical history of integration in rigid body dynamics, the paper of
Kowalevski (1889) occupies a special place. For the previously known integrable
examples, the Euler and the Lagrange case, the solutions are meromorphic func-
tions. Starting from that observation, Kowalevski formulated the problem of finding
all cases of rigid body motion fixed at a point whose general solutions are single-
valued functions of complex time that admit only moving poles as singularities. She
proved that this was possible only in one additional case, named later the Kowalevski
case. She found an additional first integral of fourth degree and completely solved
the equations of motion in terms of genus two theta-functions. The importance of the
Kowalevski paper is reflected in the number and spectra of papers that are devoted to
the Kowalevski top. We will present here some recent progress in the geometric inter-
pretation of the Kowalevski integration and certain generalizations of the Kowalevski
top see (Dragovi´c 2010; Dragovi´c and Kuki´c 2011, 2014a, b).
A modern, algebro-geometric approach to integration of the equations of motion is
based on the existence of the so-called Lax representations. This method originated in
the 1960’s, with a significant breakthrough made in the theory of integrable nonlinear
partial differential equations (Korteveg-de Vries (KdV), Kadomtsev-Petviashvili (KP)
and others). These equations appear to be integrable infinite-dimensional Hamiltonian
systems. A system admits a Lax representation (or an L-A pair) with the spectral
parameter if there exists a pair of linear operators (matrices, for example) L(λ), A(λ)
such that the equations of the system can be written in the form:
d
dt
L(λ) =
L(λ), A(λ)
, (1.1)
123
514 V. D rag o vi ´c, B. Gaji´c
where λ is the complex spectral parameter. The first consequence of (1.1)isthatthe
spectrum of the matrix L(λ) is constant in time, i.e. the coefficients of the spectral
polynomial are first integrals. In the algebro-geometric integration procedure, the
so-called Baker–Akhiezer function plays a key role. This function is the common
eigenfunction of the operators
d
dt
+ A(λ) and L(λ), defined on the spectral curve
C, which is naturally associated to the L-A pair. The Baker–Akhiezer function is
meromorphic on C except in several isolated points where it has essential singularities.
For more detailed explanations of modern algebro-geometric integration methods, see
(Dubrovin 1977, 1981; Dubrovin et al. 2001; Adler and van Moerbeke 1980; Dragovi´c
2006; Belokolos et al. 1994; Gaji´c 2002). An important class of Lax presentations,
when L(λ) and A(λ) are matrix polynomials in λ, was studied by Dubrovin (1977)(see
also Dubrovin (1981); Dubrovin et al. (2001)). These methods have been successfully
applied to rigid body dynamics (see Manakov 1976; Bogoyavlensky 1984; Bobenko
et al. 1989; Ratiu and van Moerbeke 1982; Ratiu 1982; Gavrilov and Zhivkov 1998).
The Lax representations appear to be also a useful tool for constructing higher-
dimensional generalizations of a given system. We will review some of the results
obtained in Dragovi´c and Gaji´c(2001, 2004, 2006, 2009, 2012, 2014), Dragovi´c
(2010), Jovanovi´c (2007, 2008), Gaji´c (2013).
The paper is organized as follows. The basic facts about three-dimensional motion
of a rigid body are presented in Sect. 2. In the same Section, the basic steps of the
algebro-geometric integration procedure for the Hess–Appel’rot case of motion of
three-dimensional rigid body are given. A recent approach to the Kowalevski integra-
tion procedure is given in Sect. 3. The basic facts of higher-dimensional rigid body
dynamics are presented in Sect. 4. The same Section provides the definition of the iso-
holomorphic systems, such as the Lagrange bitop and n-dimensional Hess–Appel’rot
systems. The importance of the isoholmorphic systems has been underlined by Gru-
shevsky and Krichever (2010). In Sect. 5 we review the classical Grioli precessions
and present its quite recent higher-dimensional generalizations. The four-dimensional
generalizations of the Kirchhoff and Chaplygin cases of motion of a rigid body in an
ideal fluid are given in Sect. 6.
2 The Hess–Appel’rot Case of Rigid Body Motion
2.1 Basic Notions of Heavy Rigid Body Fixed at a Point
A three-dimensional rigid body is a system of material points in R
3
such that the
distance between each two points is a constant function of time. Important case of
motion is when rigid body moves with fixed point O. Then the configuration space
is the Lie group SO(3). In order to describe the motion, it is usual to introduce
two Euclidian frames associated to the system: the first one Oxyz is fixed in the
space, and the second, moving, OXYZ is fixed in the body. The capital letters will
denote elements of the moving reference frame, while the lowercase letters will denote
elements of the fixed reference frame. Let B(t) ∈ SO(3) is an orthogonal matrix which
maps OXYZ to Oxyz. The radius vector
Q of the arbitrary point in the moving
coordinate system maps to the radius vector in the fixed frame q(t) = B(t)
Q.The
123
Some Recent Generalizations of the Classical... 515
velocity of that point in the fixed reference frame is given by
v(t) =
˙
q(t) =
˙
B(t)
Q =
˙
B(t)B
−1
(t)q(t) = ω(t)q(t ),
where ω(t) =
˙
BB
−1
. The matrix ω is an skew-symmetric matrix. Using the iso-
morphism of (R
3
, ×), where × is the usual vector product, and (so(3), [, ]),given
by
a = (a
1
, a
2
, a
3
) → a =
⎡
⎣
0 −a
3
a
2
a
3
0 −a
1
−a
2
a
1
0
⎤
⎦
(2.1)
matrix ω(t) is corresponded to vector ω(t)—angular velocity of the body in the fixed
reference frame. Then v(t) =ω(t) ×q(t). One can easily see that ω(t) is the eigen-
vector of matrix ω(t) that corresponds to the zero eigenvalue.
In the moving reference frame,
V (t) = B(t)
−1
v(t),so
V (t) =
(t) ×
Q, where
(t) is the angular velocity in the moving reference frame and corresponds to the
skew-symmetric matrix (t) = B
−1
(t)
˙
B(t).
Here one concludes that it is natural to consider the angular velocity as a skew-
symmetric matrix. The element ω
12
corresponds to the rotation in the plane determined
by the first two axes Ox and Oy, and similarly for the other elements. In the three-
dimensional case we have a natural correspondence given above, and one can consider
the angular velocity as a vector. But, in higher-dimensional cases, generally speaking,
such a correspondence does not exist. We will see later how in dimension four, using
isomorphism between so(4) and so(3) × so(3) two vectors in the three-dimensional
space are joined to an 4 × 4 skew-symmetric matrix.
The moment of inertia with respect to the axis u, defined with the unit vector u
through a fixed point O is :
I (u) =
B
d
2
dm =
B
u ×
Q, u ×
Qdm =
B
Q × (u ×
Q), u
dm =I u, u,
where d is the distance between corresponding point and axis u, I is inertia operator
with respect to the point O defined with
I u =
B
Q × (u ×
Q)dm,
and integrations goes over the body B. The diagonal elements I
1
, I
2
, I
3
are called
the principal moments of inertia, with respect to the principal axes of inertia.The
ellipsoid I , =1isthe inertia ellipsoid of the body at the point O.Inthe
principal coordinates its equation is:
I
1
2
1
+ I
2
2
2
+ I
3
2
3
= 1.
123