Kapitsa pendulum and principle of whirligig
A comparison of the mechanism of stabilization of unstable states of physical systems in a rapidly oscillating external field (on example the Kapitsa’s pendulum) with a whirligig type stabilization mechanism is presented in this work. It is shown that the whirligig mechanism is more efficient than t...
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irk-123456789-1122262017-01-19T03:02:56Z Kapitsa pendulum and principle of whirligig Buts, V.A. Нелинейные процессы в плазменных средах A comparison of the mechanism of stabilization of unstable states of physical systems in a rapidly oscillating external field (on example the Kapitsa’s pendulum) with a whirligig type stabilization mechanism is presented in this work. It is shown that the whirligig mechanism is more efficient than the stabilization in rapidly changing fields for stabilizing unstable states if or when it can be actually used. Дано порівняння механізму стабілізації нестійких станів фізичних систем у зовнішньому швидкоосцилюючому полі (на прикладі маятника Капиці) з механізмом стабілізації дзиґи. Показано, що в тих випадках, коли для стабілізації може бути використаний принцип дзиґи, він є більш ефективним, ніж механізм стабілізації в швидкозмінних полях. Дано сравнение механизма стабилизации неустойчивых состояний физических систем во внешнем быстроосциллирующем поле (на примере маятника Капицы) с механизмом стабилизации юлы. Показано, что в тех случаях, когда для стабилизации неустойчивых состояний может быть использован принцип юлы, он оказывается более эффективным, чем механизм стабилизации в быстроменяющихся полях. 2015 Article Kapitsa pendulum and principle of whirligig / V.A. Buts // Вопросы атомной науки и техники. — 2015. — № 4. — С. 237-240. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 05.45.Pq ; 05.45.-a; 05.45.Mt; 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/112226 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Buts, V.A. Kapitsa pendulum and principle of whirligig Вопросы атомной науки и техники |
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A comparison of the mechanism of stabilization of unstable states of physical systems in a rapidly oscillating external field (on example the Kapitsa’s pendulum) with a whirligig type stabilization mechanism is presented in this work. It is shown that the whirligig mechanism is more efficient than the stabilization in rapidly changing fields for stabilizing unstable states if or when it can be actually used. |
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Buts, V.A. |
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Buts, V.A. |
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Kapitsa pendulum and principle of whirligig |
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Kapitsa pendulum and principle of whirligig |
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Kapitsa pendulum and principle of whirligig |
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Kapitsa pendulum and principle of whirligig |
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Kapitsa pendulum and principle of whirligig |
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kapitsa pendulum and principle of whirligig |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы в плазменных средах |
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Kapitsa pendulum and principle of whirligig / V.A. Buts // Вопросы атомной науки и техники. — 2015. — № 4. — С. 237-240. — Бібліогр.: 11 назв. — англ. |
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Вопросы атомной науки и техники |
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AT butsva kapitsapendulumandprincipleofwhirligig |
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ISSN 1562-6016. ВАНТ. 2015. №4(98) 237
KAPITSA PENDULUM AND PRINCIPLE OF WHIRLIGIG
V.A. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
A comparison of the mechanism of stabilization of unstable states of physical systems in a rapidly oscillating ex-
ternal field (on example the Kapitsa’s pendulum) with a whirligig type stabilization mechanism is presented in this
work. It is shown that the whirligig mechanism is more efficient than the stabilization in rapidly changing fields for
stabilizing unstable states if or when it can be actually used.
PACS: 05.45.Pq ; 05.45.-a; 05.45.Mt; 52.35.Mw
INTRODUCTION
At present, the mechanism of stabilization of unsta-
ble states in a variety of physical systems in rapidly
changing external fields has been well studied and wide-
ly used. Kapitsa works [1, 2] initiated a whole field of
investigations – the vibrational mechanics. Numerous
publications dedicated to the description of various
physical effects observed in this case. Sufficient exam-
ple is the publications [1 - 6]. The most known example
of stabilization of unstable states with the rapid change
in the system’s parameters is the stabilization of the
upper unstable location of a mathematical pendulum by
rapidly changing the position of its suspension point
known as the Kapitsa pendulum. The same mechanism
is the basis of high-frequency radio field pressure forces
(Gaponov-Miller forces).
Works [7 - 10] suggest a mechanism to stabilize a
wide range of unstable physical systems. This mecha-
nism allows to stabilize both classical and quantum sys-
tems. In particular, the possibility of stabilization of a
radiation flow in plasma and a flow of charged particles
in plasma has been suggested. The possibility to sup-
press the local instability has been demonstrated using
the example of the Lorenz system. For quantum systems
this mechanism is reminiscent of the quantum Zeno
effect. The main feature of this mechanism is the intro-
duction to a system under stabilization an additional
degree of freedom. The interaction of this degree of
freedom with one of the degrees of freedom of an un-
stable system under certain conditions, allows you stabi-
lize the system. These conditions are conveniently de-
scribed using a visual image of a children's toy – whirli-
gig. Let us imagine that we have a real whirligig which
does not rotate. Its vertical position is unstable. The
whirligig will fall down during a characteristic time T .
If we now rotate the whirligig, and the period of its rota-
tion is significantly smaller than T , the vertical position
of the whirligig is stable. This simple visual image con-
tains two basic characteristics required to implement the
stabilization mechanisms. Indeed, time of the whirligig
fall (when it is not running) is convenient to associate
with the lifetime of the unstable state of the physical
system under investigation. To make this state stable it
is necessary to introduce a mechanism similar to the
rotation of the whirligig. Such a mechanism can be the
introduction of an additional degree of freedom, which
effectively interacts with one of the unstable physical
system’s own degree of freedom.
At first glance, whirligig mechanism looks equiva-
lent to the mechanism of stabilization in rapidly oscillat-
ing external fields. Indeed, for the whirligig-type stabi-
lization one needs to introduce rapidly changing pro-
cesses (the analogue to whirligig rotation). However, as
one can see below, in all cases where the principle of
whirligig can be used, it is more efficient than the stabi-
lization using the rapidly changing external fields. This
work proves this fact. Here in the second section we
transform the Kapitsa equations to the form which al-
lows the comparison of the whirligig stabilization
mechanism with the stabilization mechanism using ex-
ternal rapidly changing fields.
1. GENERAL CONSIDERATIONS
At first, let us briefly express some general consid-
erations that allow us to understand the whirligig mech-
anism of stabilization. In most cases, the unstable states
of dynamic systems are locally characterized by singu-
lar points of "saddle" type. The unstable nodes and fo-
cuses occur much less frequently. Here is the example
of transforming an unstable point of "saddle" type into
an elliptical point (to the point of "center" type). Phase
portraits of the neighborhood of the saddle point are
shown in Figs. 1,a,b,c.
Fig. 1,a. Phase portrait
of a singular point
of the "saddle" type
Fig. 1,b. Phase portrait
of a singular point
of the "saddle" type
Fig. 1,c. Singular point
of the "saddle" type
on the phase plane
Fig. 2. Phase portrait
of a singular point
of "center" type
The equations in the phase plane, which describe the
dynamics of phase trajectories in the neighborhood of
the saddle point, have the form:
0 1x xγ= ⋅ 1 0x xγ= . (1)
mailto:vbuts@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2015. №4(98) 238
In most real cases, each of the dependent variables
in equations (1) is some characteristic a degree of free-
dom of the system under investigation. For example,
this may be complex amplitudes of interacting nonlinear
waves. Therefore, in this paper we assume that each of
these first-order equations describes one of the degrees
of freedom of the system under investigation. Suppose
now that we want to transform a neighborhood of a sin-
gular saddle point into the neighborhood, which corre-
sponds to a stable singular point, for example, elliptical
point (Fig. 2).
To achieve this, one can do as follows. We introduce
into the system an additional degree of freedom con-
nected to the one of the degrees of freedom of the un-
stable system. The simplest model that describes the
dynamics of the system in the vicinity of the saddle
point after such modification differs from the equation
(1) only by additional equation:
0 1 2x x xγ δ= ⋅ + ⋅ ; 1 0x xγ= ; 2 0x xδ= − ⋅ . (2)
This new degree of freedom is connected to one of
the degrees of freedom of the original system by the
coupling coefficientδ . The system of equations (2) is
equivalent to the equation of the linear pendulum:
( )2 2
0 0 0x xδ γ+ − = . (3)
From equation (3) is immediately obvious that as
soon as the coefficient that describes the connection
between the degrees of freedom is greater than the in-
stability increment (δ γ> ), an unstable saddle point
turns into an elliptical point. The phase space, shown on
Fig. 1,c, became the phase space which is shown on
Fig. 2. This simple algorithm of transforming the unsta-
ble saddle point into an elliptical point well characteriz-
es the whirligig principle. Indeed, if we do not introduce
an additional degree of freedom, our system is unstable
(whirligig falls). Moreover, the characteristic instability
time ( ~ 1/T γ ) can be associated with the whirligig fall
time. The inclusion of an additional degree of freedom
which stabilizes our system is similar to the presence of
the whirligig rotation. Moreover, the whirligig stabiliza-
tion has not only qualitative but also quantitative analo-
gy. Indeed, for the vertical position of a whirligig to be
stable, it is necessary that the rotation period is shorter
than the time of falling. In our model (see equation (3)),
the system becomes stable if the coupling coefficient
becomes greater than the instability increment ( δ γ> ).
Moreover, if the increment of the instability is zero,
then the system of equations (2) describes oscillations
with frequency 2 / rotTδ π= . Thus, there are qualitative
and quantitative similarities between the considered
mechanism and a mechanism of stabilizing a whirligig’s
vertical position.
Let us make the following remark. We are accus-
tomed to the fact that the increase in the number of de-
grees of freedom of a dynamical system leads to tougher
conditions for the realization of its steady state. Indeed
assume our physical system is described by the follow-
ing system of equations:
(Z, t)n nZ F=
. (4)
The nature of the stability of this system at the se-
lected point of the phase space is described by a linear
system of equations that describes the dynamics of
small deviations 0Z Z x= +
:
ˆ .x Ax=
(5)
In many cases, the coefficients of the matrix can be
considered constant. Then, to determine the type of the
singular point of the system (5), we must find the roots
of the characteristic equation:
1 2
0 1 2 1... 0.n n n
n nα λ α λ α λ α λ α− −
−+ + + + + = (6)
Rause-Gurvitz criterion states that the higher order
of the equations makes it more difficult to satisfy the
conditions of stability for this system. In the example
above we have increased the number of degrees of free-
dom, but have achieved the opposite result. It may look
like an exceptional case. However, it is not true. [7 - 11]
contain numerous examples showing how the introduc-
tion of an additional degree of freedom to complex sys-
tems such as those that describe the stabilization of ra-
diation fluxes in the plasma can also lead to a stabiliza-
tion of unstable states. This result echoes the result pro-
vided in Haken’s book [12] about the role of the order
parameters. Thus, the introduction of an additional de-
gree of freedom is an analogue of the Haken’s parame-
ter of order.
2. DYNAMICS OF KAPITSA PENDULUM
Let’s briefly provide the main results that character-
ize the dynamics of the Kapitsa pendulum. Such pendu-
lum is described by a mathematical pendulum with pe-
riodically changed parameters:
( )( )2 1 cos sin 0x t xε ω+Ω + ⋅ = . (7)
For small values of the angle ( 1x ϕ≡ << ), this equa-
tion is the Mathieu equation. The solution of equation
(7) is studied in all areas of the parameters. We are in-
terested in the area of non-resonant parameters. We as-
sume that the frequency of parametric excitation is large
(ω >> Ω ), and the parameter ε has no constrains. The
stability of the upper position of a mathematical pendu-
lum can be achieved in this parameters area. It is well
known (e.g., [3]) that the dynamics of such nonlinear
oscillator is determined by the effective potential:
2( ) cos( ) sin ( )U x x xα= − + ⋅ , (8)
where −
2 2
2
a
l
ωα = ⋅ Ω
; l − length of the pendulum;
l a>> − oscillation amplitude of the point of suspen-
sion; /l gΩ = − the frequency of the linear oscilla-
tion of the pendulum; ω − the frequency of the oscilla-
tion of the point of suspension.
Stable position corresponds to the minimum of the
potential (8). Fig. 3 shows the potential in the case when
the conditions of the top position stability of the pendu-
lum are not satisfied. One can see that the stable posi-
tion is the lowest position of the pendulum. However, if
the oscillation frequency of the suspension point or its
oscillation amplitude increases, the upper position
( x ϕ π≡ = ) may become stable too. This case is shown
on Fig. 4. One can see that the upper position of the
pendulum ( 3.14x ϕ≡ = ) gets local minima. The loca-
ISSN 1562-6016. ВАНТ. 2015. №4(98) 239
tion of the pendulum is stable at these points. These
results are well known before (see, e.g., [1 - 5]).
Fig. 3. The effective potential at 0.5α =
Fig. 4. The effective potential at 1.2α =
The mathematical model of the Kapitsa pendulum
inconvenient for comparison with other stabilization
mechanisms in the form presented above. Therefore, for
convenience of comparison, we somewhat simplified
the model (7). Namely, we are interested in the dynam-
ics in the vicinity of the unstable position of the pendu-
lum only, i.e. in the vicinity of the point x π= . The
dynamics of Kapitsa pendulum in this region of angles
can be described by the system of equations:
( )2 2
0 0 0 1 0cosx x t x x xε ω−Ω = Ω ⋅ = ⋅ , (9)
2
1 1 0x xω+ = , ( ) ( )2
1 10 ; 0 0x xε= Ω = .
The first equation of the system (9) describes the
dynamics of the Kapitsa pendulum in the vicinity of an
unstable equilibrium point. However, we have added to
this equation the equation of a linear oscillator, which
oscillations together with initial conditions describe the
vibrations of the suspension point of the pendulum.
Such modification is useful for comparison of the
known mechanism of the Kapitsa pendulum stabiliza-
tion with the mechanism, which uses the principle of
whirligig. The system of equations (9) describes the
movement in constant (in time) potential and in the ex-
ternal rapidly oscillating field. It is convenient to rewrite
it as:
( )cos ( , )Nx x x f xε ω τ τ− = ⋅ ≡ , (10)
here 2( ) / 2U x x= − , tτ = Ω⋅ ;
/x dx dτ≡ ( )cos ( , )t x f x tε ω ⋅ ≡ ;
/ 1Nω ω= Ω >> .
Then, using standard averaging procedure (see, for
example, [3]), we find the following expression for the
effective potential, which determines the dynamics of
the pendulum in the vicinity of its upper unstable posi-
tion:
2
2
2
1( )
2 2eff
N
xU x f
ω
= − + ;
2 2 2 2 2
2 2( ) 1
2 4 2 2eff
N N
x x xU x ε ε
ω ω
⋅
= − + = − −
. (11)
The steady state corresponds to the minimum of this
potential. The stability condition has the form:
2 22 1.Nε ω> >> (12)
When expressed in familiar variables it transforms to
well-known expression: 2 ;l
a
ω > ⋅Ω⋅ /l gΩ = . By
definition / 1Nω ω= Ω >> is a large parameter.
Thus, we obtained known Kapitsa result on stabili-
zation of the upper unstable position of the pendulum.
The stabilization was achieved by applying external
influence to the parameters of the pendulum. At the
same time, if we consider the system (9), we can say
that stabilization mechanism, in some sense is a nonlin-
ear mechanism. Indeed, if we have kept the reverse in-
fluence of the pendulum dynamics on the oscillator,
which causes fluctuations in the point of suspension,
then it would be a non-linear system. However, we ig-
nored this bond.
As a result, the first equation (9) remains linear
equation with parameters subjected to external high-
frequency disturbances. Note also that equation (10)
shows the parameters under which the external high-
frequency perturbations can change the character of the
unstable saddle point. Indeed, if we drop the right side
of the system (10), the remaining part of this equation
describes the dynamics in a neighborhood of a singular
unstable point of "saddle" type. The presence of external
high-frequency perturbations (right-hand side of equa-
tion (10)) with the parameters defined by the inequality
(12) transforms an unstable singular point of "saddle"
type to an elliptic point. Below we show that there is
another opportunity to change the nature of an unstable
point turning it into a stable singular point. To achieve
this let’s rewrite equation (10) without the right side:
0 1x x= Ω⋅ 1 0x x= Ω⋅ . (13)
These equations describe the unstable state of "sad-
dle" type. Let’s modify this system of equations assum-
ing that the variable of the first equation in (13) is line-
arly connected to some other (additional) variable 2x :
0 1 2 ,x x xδ= Ω⋅ + ⋅
1 0x x= Ω⋅ , 2 0x xδ= − ⋅ , (14)
where δ − is the coefficient of the connection.
The system (14) is equivalent to the equation of the
linear pendulum:
( )2 2
0 0 0.x xδ+ −Ω =
(15)
We have already dealt with such equation (see equa-
tion (2) and (3)). From (15) follows that, in order to
stabilize the unstable upper position of the pendulum, it
is necessary to satisfy the conditionδ > Ω only.
This stabilization condition is much weaker than the
stabilization as the result of rapidly changing fields (12).
CONCLUSIONS
Is was shown above, that the usage of the whirligig
principle allows to stabilize an unstable state in much
simple and easier way than it can be achieved using
external rapidly oscillating fields. However, it should be
ISSN 1562-6016. ВАНТ. 2015. №4(98) 240
understood that the whirligig principle is not always
possible to realize. For example, one can hardly imagine
a mathematical Kapitza pendulum, oscillation of which
somehow connected with some other degree of freedom
and the frequency of these oscillations is large than the
fundamental frequency of the pendulum. However, in
many other cases (see, for example, [7-11]) it would be
easy to achieve. Let us formulate the main conclusions
of this work:
1. The principle of whirligig stabilization of unstable
states cannot be reduced to the stabilization mecha-
nism, based on usage an external rapidly oscillating
fields.
2. In all cases where the whirligig principle can be
used for stabilization of unstable states this mecha-
nism is more effective than other stabilization
mechanisms.
REFERENCES
1. P.L. Kapitza. Dynamic stability of the pendulum
with vibrating suspension point // Soviet Physics
JETP. 1951, v. 21, p. 588-597.
2. Collected papers of P.L. Kapitza / Ed. D. Ter Haar,
London: “Pergamon”, 1965, v. 2, p. 714-726.
3. L.D. Landau, E.M. Lifschitz. Mechanics. New York:
“Pergamon”, 1976.
4. V.P. Krainov. Selected mathematical methods in
theoretical physics. Publisher MIPT. 1996 (in Rus-
sian).
5. E.I. Butikov. Extraordinary oscillations of an ordi-
nary forced pendulum // Eur. J. Phys. 2008, v. 29,
p. 215-233.
6. E.I. Butikov. On the dynamic stabilization of an
inverted pendulum // Am. J. Phys. 2001, v. 69,
p. 755-768.
7. V.A. Buts. Stabilization of classic and quantum sys-
tems // Problems of Atomic Science and Technology.
2012, № 6, p. 146-148.
8. V.A. Buts. The mechanism of suppression of quan-
tum transitions (quantum whirligig) // Problems of
Atomic Science and Technology. 2010, № 4, p. 259-
263.
9. V.A. Buts. Stabilization of instable states // MIKON.
2014, 20th International Conference on Microwaves,
Radar and Wireless Communications, June 16-18,
Gdansk, Poland, 2014, p. 681-685.
10. V.A. Buts. Mechanisms to enhance the frequency
and degree of coherence of the radiation // Problems
of Theoretical Physics. Proceedings. Series “Prob-
lems of Theoretical and Mathematical Physics”.
Proceedings. Kharkiv, 2014, Issue 1, p. 82-247 (in
Russian).
11. H. Haken. Synergetics. Springer-Verlag. 1978, p. 408.
Article received 05.06.2015
МАЯТНИК КАПИЦЫ И ПРИНЦИП ЮЛЫ
В.А. Буц
Дано сравнение механизма стабилизации неустойчивых состояний физических систем во внешнем быст-
роосциллирующем поле (на примере маятника Капицы) с механизмом стабилизации юлы. Показано, что в
тех случаях, когда для стабилизации неустойчивых состояний может быть использован принцип юлы, он
оказывается более эффективным, чем механизм стабилизации в быстроменяющихся полях.
МАЯТНИК КАПИЦІ ТА ПРИНЦИП ДЗИҐИ
В.О. Буц
Дано порівняння механізму стабілізації нестійких станів фізичних систем у зовнішньому швидкоосци-
люючому полі (на прикладі маятника Капиці) з механізмом стабілізації дзиґи. Показано, що в тих випадках,
коли для стабілізації може бути використаний принцип дзиґи, він є більш ефективним, ніж механізм стабілі-
зації в швидкозмінних полях.
|