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Destabilization paradox due to breaking the
Hamiltonian and reversible symmetry
Oleg N. Kirillov
To cite this version:
Oleg N. Kirillov. Destabilization paradox due to breaking the Hamiltonian and reversible
symmetry. International Journal of Non-Linear Mechanics, Elsevier, 2007, 42 (1), pp.71.
�10.1016/j.ijnonlinmec.2006.09.003�. �hal-00501733�
www.elsevier.com/locate/nlm
Author’s Accepted Manuscript
Destabilization paradox due to breaking the
Hamiltonian and reversible symmetry
Oleg N. Kirillov
PII: S0020-7462(07)00025-X
DOI: doi:10.1016/j.ijnonlinmec.2006.09.003
Reference: NLM 1310
To appear in: International Journal of Non-
Linear Mechanics
Received date: 30 June 2006
Revised date: 5 September 2006
Accepted date: 29 September 2006
Cite this article as: Oleg N. Kirillov, Destabilization paradox due to breaking the Hamil-
tonian and reversible symmetry, International Journal of Non-Linear Mechanics (2007),
doi:10.1016/j.ijnonlinmec.2006.09.003
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Accepted manuscript
Destabilization paradox
due to breaking the Hamiltonian and reversible symmetry
∗
Oleg N. Kirillov
†
Abstract
Stability of a linear autonomous non-conservative system in the presence of potential, gyroscopic, dis-
sipative, and non-conservative positional forces is studied. The cases when the non-conservative system is
close to a gyroscopic system or to a circulatory one, are examined. It is known that marginal stability of
gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action
of small non-conservative positional and velocity-dependent forces. The present paper shows that in both
cases the boundary of the asymptotic stability d omain of the perturbed system possesses singularities such
as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. In case of two
degrees of freedom, approximations of the stability boundary near the singularities are found in terms of
the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified
Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems
with stationary and rotating damping.
Keywords: non-conservative system, dissipation-induced instabilities, destabilization paradox
Submitted to: Int. J. of Nonlin. Mechs.
1 Intro duction
Consider an autonomous non-conservative system described by a linear differential equation of second order
¨
x +(ΩG + δD)
˙
x +(K + νN)x =0, (1)
where dot denotes time differentiation, x ∈ R
m
,andrealmatrixK = K
T
corresponds to potential forces.
Real matrices D = D
T
, G = −G
T
,andN = −N
T
are related to dissipative (damping), gyroscopic, and non-
conservative positional (circulatory) forces with magnitudes controlled by scaling factors δ,Ω,andν respectively.
∗
Dedicated to the memory of Karl Popp.
†
Dynamics group, Department of Mechanical Engineering, Technical University of Darmstadt, Hochsch ulstr. 1, 64289 Darm-
stadt, Germany (e-mail: kirillov@dyn.tu-darmstadt.de, Tel: +49 6151 16 6828, Fax: +49 6151 16 4125) and Institute of Mechanics,
Moscow State Lomonosov University, Michurinskii pr. 1, 119192 Moscow, Russia (e-mail: kirillov@imec.msu.ru).
1
Accepted manuscript
General non-conservative system (1) has two important limiting cases corresponding to circulatory and
gyroscopic systems. A circulatory system is obtained from (1) by neglecting velocity-dependent forces
¨
x +(K + νN)x =0, (2)
while a gyroscopic one has no damping and non-conservative positional forces
¨
x +ΩG
˙
x + Kx =0. (3)
Circulatory and gyroscopic systems (2) and (3) possess fundamental symmetries that are easily seen after
transformation of equation (1) to the Cauchy form
˙
y = Ay with
A =
⎛
⎝
−
1
2
ΩGI
1
2
δΩDG +
1
4
Ω
2
G
2
− K − νN − δD −
1
2
ΩG
⎞
⎠
, y =
⎛
⎝
x
˙
x +
1
2
ΩGx
⎞
⎠
, (4)
where I is the identity matrix.
Indeed, in the absence of damping and gyroscopic forces (δ =Ω=0)thematrixA changes as RAR = −A
due to a coordinate transformation with the matrix
R = R
−1
=
⎛
⎝
I 0
0 −I
⎞
⎠
. (5)
This means that the matrix A has a reversible symmetry, and equation (2) describes a reversible dynamical
system [1, 2]. Due to this property,
det(A − λI)=det(R(A − λI)R)=det(A + λI), (6)
and the eigenvalues of circulatory system (2) appear in pairs (−λ, λ). Consequently, the equilibrium of a
circulatory system is either unstable or all its eigenvalues lie on the imaginary axis of the complex plane
implying marginal stability if they are semi-simple.
Without damping and non-conservative positional forces (δ = ν =0)thematrixA possesses the Hamiltonian
symmetry JAJ = A
T
,whereJ is a unit symplectic matrix [3]
J = −J
−1
=
⎛
⎝
0 I
−I 0
⎞
⎠
. (7)
As a consequence,
det(A − λI)=det(J(A − λI)J)=det(A
T
+ λI)=det(A + λI), (8)
which implies that if λ is an eigenvalue of A then so is −λ, similarly to the reversible case. Therefore, gyroscopic
system (3) can be only marginally stable with its spectrum belonging to the imaginary axis of the complex plane.
In the presence of all the four forces the Hamiltonian and reversible symmetries are broken and the marginal
stability is generally destroyed. Instead, system (1) can be asymptotically stable if its characteristic polynomial
P (λ)=det(Iλ
2
+(ΩG + δD)λ + K + νN), (9)
2
Accepted manuscript
satisfies the criterion of Routh and Hurwitz. The most interesting for many applications is the situation
when system (1) is close either to circulatory system (2) with δ, Ω ν (imperfect reversible system) or to
gyroscopic system (3) with δ, ν Ω (imperfect Hamiltonian system). Furthermore, the effect of small damping
and gyroscopic forces on the stability of circulatory systems as well as the effect of small damping and non-
conservative positional forces on the stability of gyroscopic systems are regarded as paradoxical,sincethe
stability properties are extremely sensitive to the choice of the perturbation, and the balance of forces resulting
in the asymptotic stability is not evident [4–46]. This characterization sounds even more justified if to take into
account the connection of the destabilization paradox with the physical paradoxes such as “tippe top inversion”
and “rising egg phenomenon” [34, 35, 42, 45].
Historically, the destabilization paradox appeared first in a study of a gyroscopic system with dissipation by
Thomson and Tait, who found that the dissipative perturbation destroys the gyroscopic stabilization so that the
system is neither marginally nor asymptotically stable [4]. The terminology dissipation-induced instabilities has
its roots in that classical work [20, 45]. A similar effect of non-conservative positional forces on the stability of
gyroscopic systems has been established almost a hundred years later by Lakhadanov and Karapetyan [11, 12].
These ideas have been extensively developed, e.g., in the works [10, 18–21, 23, 34, 35, 42–46].
A more sophisticated form of the destabilization paradox has been discovered by Ziegler on the example of
a double pendulum loaded by a follower force with the damping non-uniformly distributed among the natural
modes [7]. Without dissipation, the Ziegler pendulum is a circulatory system and it is marginally stable for the
loads non-exceeding some critical value. Small dissipation makes the pendulum either unstable or asymptotically
stable with the critical load, which can be significantly lower than that of the undamped system. This is caused
by the singular nature of the new critical load, which is a non-differentiable at the origin function of the damping
parameters, having no limit when the damping coefficients uniformly tend to zero [8, 9, 26]. Numerous other
aspects of the destabilization paradox by small velocity-dependent forces in circulatory systems, including more
general settings and non-linear effects, have been investigated, e.g., in [8, 13–15, 17, 24–32, 36–41].
The destabilization paradox in Ziegler’s form has been revealed recently by Crandall in his study of a gyro-
scopic pendulum with stationary and rotating damping [22]. Contrary to the Ziegler pendulum, the undamped
gyropendulum is a gyroscopic system that is marginally stable when its spin exceeds a critical value. Stationary
damping corresponding to dissipative velocity-dependent force destroys the gyroscopic stabilization in accor-
dance with the theorem of Thomson and Tait [4]. However, the Crandall gyropendulum with stationary and
rotating damping, where the latter is related to non-conservative positional force, can be asymptotically stable
for the rotation rates exceeding considerably the critical spin of the undamped system.
The growing number of other physical and mechanical examples demonstrating the destabilization paradox
due to an interplay of non-conservative effects, requires a unified treatment of this phenomenon taking into
account all types of forces presented in equation (1), as reported recently by Krechetnikov and Marsden [45].
The goal of the present paper is to find and analyze the domain of asymptotic stability of system (1) in the
space of the parameters δ,Ω,andν with special attention to imperfect reversible and Hamiltonian cases.
Below we show that the boundary of the asymptotic stability domain of a circulatory system perturbed by
3