Parametric Resonance in the Oscillations of a Charged Pendulum Inside a Uniformly Charged Circular Ring Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 513-526. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220703

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 37N05, 70H14, 70J40, 70J25

Parametric Resonance in the Oscillations of a Charged Pendulum Inside a Uniformly Charged Circular Ring

H. E. Cabral, A. C. Carvalho

We study the mechanical system consisting of the following variant of the planar pendulum. The suspension point oscillates harmonically in the vertical direction, with small amplitude e, about the center of a circumference which is located in the plane of oscillations of the pendulum. The circumference has a uniform distribution of electric charges with total charge Q and the bob of the pendulum, with mass m, carries an electric charge q. We study the motion of the pendulum as a function of three parameters: e, the ratio of charges /J> = q and a parameter a related to the frequency of oscillations of the suspension point and the length of the pendulum. As the speed of oscillations of the mass m are small magnetic effects are disregarded and the motion is subjected only to the gravity force and the electrostatic force. The electrostatic potential is determined in terms of the Jacobi elliptic functions. We study the parametric resonance of the linearized equations about the stable equilibrium finding the boundary surfaces of stability domains using the Deprit-Hori method.

Keywords: planar charged pendulum, Hamiltonian systems, parametric resonance, Deprit-Hori method, Jacobi elliptic integrals

1. Introduction

We consider the dynamics of a planar pendulum of length l with a suspension point oscillating harmonically in the vertical around the center of a circumference of radius R > l situated in a vertical plane. The bob of the pendulum has a mass m and carries an electric charge q which can be positive or negative. The circumference has a uniform distribution of positive

Received November 04, 2021 Accepted July 12, 2022

Hildeberto E. Cabral hild@dmat.ufpe.br

Department of mathematics, Universidade Federal de Pernambuco CEP 50670-901 Recife-PE, Brazil

Adecarlos C. Carvalho adecarlos.carvalho@ufma.br

Department of mathematics, Universidade Federal do Maranhäo Av. dos Portugueses, 1966, 65080-805, Säo Luis-MA, Brazil

electric charge a with total charge Q = 2nRa. For reasons of symmetry, at any position of the suspension point the electrostatic force on the bob of the pendulum is directed along the line passing through the bob and the center of the circumference, directed towards the center if the charge q is positive and away from the center if q is negative.

The dynamics is described by a one-degree of freedom Hamiltonian which is 2n-periodic in time t and depends on three parameters, the ratio i of the charges, a parameter a related to the frequency of the harmonic oscillation of the suspension point and a small parameter e defined in terms of the amplitude of this harmonic oscillation, see Eq. (2.8).

The electrostatic potential and its derivative involve the Jacobi complete elliptic integrals of first and second kinds, see Eqs. (2.4) and (2.7).

The Hamiltonian has two equilibria corresponding to the vertical lowest and highest positions of the bob of the pendulum: a stable and an unstable equilibrium, respectively. For the stable equilibrium we study the stability of the system in the parameter space (a, e), constructing the regions of parametric resonance.

2. Formulation of the problem

We consider a pendulum of length l with a suspension point oscillating vertically in a harmonic way. The bob of the pendulum has a mass m and carries an electric charge q which can be positive or negative. The point of suspension O2 moves along the vertical through the center O1 of a circumference of radius R situated in a vertical plane and the point O1 is at a distance d> R from the origin O, at the ground level, of an inertial Cartesian coordinate system. The circumference has a uniform distribution of positive electric charge a with total charge Q = 2nRa. See Fig. 1.

Fig. 1. Figure of the problem

The Lagrangian function is of the form L = T — U with U = Ug + Uc, where T is the kinetic energy, Ug is the potential energy due to gravity and Uc is the electrostatic potential energy. We assume that the speed of the bob of the pendulum is small so that we can ignore the magnetic influence in the motion of the pendulum.

It is easy to see, for reasons of symmetry, that at any position of the suspension point the electrostatic force on the bob of the pendulum is directed along the line passing through the bob

and the center of the circumference, directed towards the center if the charge q is positive and away from the center if q is negative. This force will be found as the gradient of the electrostatic potential which we now consider.

If 0 is the angle that the pendulum makes with the vertical the coordinates of the pendulum are x = l sin 0 and y = d — p — l cos 0, where p is the distance from the suspension point O2 to the center 01 of the circumference. The kinetic energy is given by T = + y2), hence

T = — 2

(lè cos e)2 + (p - lè sin <

and the potential energy Ug is given by Ug = mg(d — p — l cos 0).

For a position of the pendulum given by the angle 0, denote by b the distance from the bob of the pendulum to the center of the circumference and by r the distance of the bob to a point on the circumference. We assume p and l are such that b is always less than R. By the cosine law we have r2 = b2 + R2 — 2bR cos 3 and b2 = p2 + l2 + 2pl cos 0.

Then the electrostatic potential energy Uc(0) is given by

f d3

Uc = 2 K,0qcrR

J Jb2 + R2 - 2bRcos/3' o v

where k0 is the Coulomb constant.

Using the change of variables /3 = tt — 7 in the integral and the identity cos 7 = 1 — 2 sin2 we obtain

n/2

u = 4K0qab f __

b+R J /1 _ 4br - 2 •

0 v1 (6+i?)2 sm 1

Since b < R are positive numbers we have 0 < < 1 because 0 < (^/b — VR^' = b + R —

- 2 Vb /R. Therefore, the potential Uc is given by

where

n/2

K(Jb) = f dn

J \/l — k2 sin2 7 0 v 1

is the complete elliptic integral of the first kind of modulus k. Then the Lagrangian function of the problem is given by

L m ~ ~2

(p — 10 sin 0)2 + (l 0 cos 0)2 — mg(d — p — l cos 0) — Uc, which determines the equation of motion of the pendulum

dU

ml29 — m lp sin 9 + ml a sin 9 + —f = 0.

d0

Let p = a cos ut define the harmonic motion of the suspension point where the amplitude a is considered to be small. Rescaling time t as t = ut and denoting by primes the derivatives

2

with respect to r, we have 9 = uj29" and p = oj2p". Setting a = jjfe and t = f, the equation of

motion becomes

9'' + a sin 9 + e cos t sin 9 +

1 dUc

ml2u2 d9

We write b in the form b = If, where f is given by the expression

0.

/ = f(9, t, e) = \j(e cos r + cos 9)2 + sin2 9. Now we take R = 2l. Then the modulus of the elliptic integral is given by

V8f

k(f ) =

2 + f

and the electrostatic potential assumes the form

UM r, e) = f^K(fc).

(2.2)

(2.3)

(2.4)

Since Q = 2ttRa, denoting by n the ratio of charges n = we write

4k0qa _ k0Q2

rnPuj2 ^rnl3u)2ir

and notice that the expression multiplying i is a dimensionless constant, since the dimensionality of k0 is Nm2 C"2 and N is kgms"2. We will take Q so that this expression is equal to 1. Then with k given by (2.3), taking

2

W = (2.5)

2 + f

the equation of motion becomes

9" + a sin 9 + e cos t sin 9 + ß

dW d9 '

I t, e) = 0.

In the Appendix A we will prove that

dW

(9, t, e) = — e cos t sin 9

f 2(2 — f)

E(k) —

f 2(2 + f)

K(k)

(2.6)

(2.7)

where the elliptic modulus k = k(f) is given by (2.3). Here

n/2

E(k) = J \J 1 - k2 sin2 7 dq

is the complete elliptic integral of the second type.

When i = 0, Eq. (2.6) describes the motion of a pendulum with a suspension point oscillating harmonically in the vertical direction, see [3].

Setting x = 9 and y = 9', Eq. (2.6) can be written in the canonical form

X = y, y' = —a sin x — e cos t sin x — ß

dW dx '

1

1

0

with the Hamiltonian function given by

y2

H(x, y, t, a, /j,, e) = ——acosx — ecosrcosx + ¡j,W(x, r, t). (2.8)

The points P1 = (0, 0) and P2 = (n, 0) are equilibria of this Hamiltonian system, for all values of the parameters. When e = 0, we have the dynamics of a simple pendulum, which is described by an autonomous one-degree-of-freedom system. For both equilibria we have Hyy = 1 and Hxy = 0. For Hxx we obtain

Hxx = a for P1 and Hxx = —a for P2.

So, as a > 0, the equilibrium P1 is always stable and P2 is always unstable, independently of the value of j.

3. Stability of periodic linear Hamiltonian systems

In the articles [1, 2, 5-7] we have used the expression parametric stability in the sense that the Hamiltonian depends on parameters, and studied the regions in the parameter space corresponding to the unstable systems of the parametric family (regions of parametric resonance) and the regions that correspond to the stable systems of the parametric family of Hamiltonians. In these articles we have not made a formal definition of parametric stability.

The expression parametrically stable linear Hamiltonian systems has already been used in the literature in the meaning of strongly stable linear Hamiltonian systems, see [10, 13]. So we take this opportunity not only to make precise our definition, but also to introduce a nonconflicting expression for this kind of stability.

To this end we first recall the definition of strongly stable systems introduced in the research works of Krein, Gel'fand and Lidskii around the year 1950, see [8, 11].

We fix a norm |||| in the linear space H of real continuous t-periodic Hamiltonian matrices A(t) of order 2n. Consider a continuous t-periodic real linear Hamiltonian system

x = A(t)x, x e R2n, A(t + t) = A(t). (3.1)

Definition 1. The linear system (3.1) is said to be strongly stable if it is stable and there exists an e > 0 such that any t-periodic linear Hamiltonian system x = A(t)x with ||A — A|| < e is stable.

Now suppose that the t-periodic linear Hamiltonian system (3.1) depends on a parameter j in a set P C Rfc, that, is A = A(t, j).

Definition 2. We say that the system x = A(t, j0)x, j0 e P, is ¡-stable if it is stable and there exists an e > 0 such that the system x = A(t, j)x is stable for any j e Be(j0).

We do not call such systems parametrically stable in order to avoid conflict with the use of this expression as a synonym for strongly stable. The prefix ¡ means parameter not necessarily the parameter ¡ in the definition.

If x = A(t, j0)x is strongly stable, then it is clear that it is ¡-stable, but the converse is, of course, not necessarily true.

The following theorem classifies the strongly stable systems, see [14]:

Theorem 1 (Krein — Gel'fand — Lidskii). The real continuous periodic linear Hamilto-nian system x = A(t)x is strongly stable if and only if all of its multipliers are in the unit circle S1 and all of them are definite.

Let us explain the meaning of definite multiplier. Let X(t) = Q(t)etB be the Floquet decomposition of the matrizant of the system (3.1). The eigenvalues of the monodromy matrix X(2t) = e2TB with B a real matrix are the multipliers of the linear system. For a stable system they are all on the unit circumference S1 of the complex plane.

Let (x, y) = x1 y1 +... + x2ny2n be the scalar product in R2n and J = [ °j j] the standard symplectic matrix of order 2n. The multiplier p is said to be of the first kind if the symplectic product {r, s} = (r, Js) is positive for any eigenvector v = r + is of X(2t) belonging to p. It is of the second kind if {r, s} < 0, for any eigenvector v of p. A multiplier is said to be definite if it is either of the first or second kind.

The problem of parametric resonance. Consider a real t-periodic linear Hamiltonian

system

x = A(t, f, e)x (3.2)

depending on a small real parameter e and an additional parameter f belonging to a set P c Rk.

Suppose that the unperturbed system x = A(t, f, 0)x is stable for f G P.

The question is: if f* G P, is the perturbed system (3.2) stable for (f, e) near (f*, 0)?

If the system x = A(t, f*, 0)x is strongly stable, then, as we have mentioned above, it is f-stable, so x = A(t, f, e)x is stable for all (f, e) in a small neighborhood of (f*, 0). So the question is posed only if the system x = A(t, f*, 0)x is not strongly stable.

The unperturbed system x = A(t, f*, 0)x is stable if and only if the multipliers p1(f*), ..., p2n(f*) are all on the unit circumference and the monodromy matrix is diagonalizable. If the multipliers are all simple, that is, if they have multiplicity one, then, by continuity of the eigenvalues, for f near f* and e small, the system (3.2) has only simple multipliers. Moreover, all of them are on the unit circumference. Indeed, by the Lyapunov - Poincare theorem they come in quadruples p, p_1, p, p_1, since the Hamiltonian system is real. So if a multiplier is off the unit circumference the number of multipliers exceeds 2n, which is not possible. Consequently, if the multipliers are all simple, the system (3.2) is stable for (f, e) in a small neighborhood of (f*, 0).

Therefore, we assume that there is at least one multiple multiplier p** = p- (p*) among the multipliers p1(f*), ..., p2n(f*).

Let \k = iuk, uk ^ 0 be a characteristic exponent of the linear system x = A(t, f, 0)x and let pk = e2rXk be the corresponding multiplier. We have taken the period 2t in order to work with a real Hamiltonian matrix B in the Floquet decomposition X(t) = Q(t)etB of the matrizant X(t) of the system x = A(t, f, 0)x. As p-1 = e~2rXk is also a multiplier of this equation we see that, if pk is a multiple multiplier, then either pk = pl or pk = p-1, for some l. Therefore, the existence of a multiple multiplier means that. Lok ± oj1 = ^N for some integer N. By rescaling time we can take the period of the system to be t = n, so this relation can be written as

Uk ± ui = N. (3.3)

We will call KGL resonance1 a resonance of the type (3.3). One such resonance is called a basic resonance if 2uk = N where N is a natural number. If uk + ul = N or uk — ul = N with 2uk or 2ul an integer, we get a pair of basic resonances. A KGL resonance uk+ul = N or uk — ul = N

1 This acronym KGL stands for Krein-Gel'fand-Lidskii.

with 2wk or 2wl not an integer is called a combined resonance. A resonance is simple if there is only one of its type among the resonance relations (3.3), otherwise we say that it is a multiple resonance.

Definition 3. A value /* £ P for which the system x = A(t, /*, 0)x has a multiple multiplier is said to be a value of parametric resonance.

We assume that the values of parametric resonance are isolated in the set P.

Assume that /* is a value of parametric resonance. By the Krein-Gel'fand-Lidskii theorem 1 the unperturbed system x = A(t, 0)x may still be strongly stable, in which case it is /-stable so for (/, e) near (/*, 0) the perturbed system x = A(t, e)x is stable.

Now assume that the unperturbed system, x = A(t, 0)x, is not strongly stable. Then in any neighborhood of it there are stable and unstable systems. Therefore, in the family (3.2) we may have parameters which give stable systems and parameters which give unstable systems. So maybe the parameters can be separated by continuous surfaces which bound, in the parameter space, the regions of stable systems of the family and the regions of unstable systems.

The goal is then to look for such continuous surfaces in the parameter space (/, e). The instability regions in the space of parameters are called regions of parametric resonance.

The problem of determining the boundary surfaces of regions of stability and instability for the Hamiltonian (3.2) is the problem of parametric resonance.

After these considerations on stability of linear Hamiltonian systems let us go back to our one-degree-of-freedom Hamiltonian (2.8). We observe that for one-degree-of-freedom systems the only possible KGL resonances are 2w = N, where w is the frequency of the unperturbed system and N is a natural number.

Setting £ = x and n = y, the linearized Hamiltonian is given by

1

1

H = tïï + ô [« + e cos T + ¿tF(T> e)] Ç >

(3.4)

where F is defined by

d2W

F(T>= T'e) = ~£COST

1 E (^L)

/2(2-/) \2 + f)

1 k(^L)

/2(2 + /) \2 + f)

x=0

which, since f (0, t, e) = 1 + e cos t, can be written in the form

F (t, £) = -

£ COS t

(1 + £ COS T)2

1

1 — £ COS t

E(A) -

1

3 + £ COS t

K(A)

(3.5)

with

A = A(t, £) =

/8(1 + £ COS t )

3 + e cos t )2

Applying the symplectic linear change of variables £, n ^ x, y defined by the formulas

£ = a-l/4 x, n = y to the Hamiltonian (3.4) and expanding it into a power series of e, we obtain

a/«, 2 2n x2 cos(r) H(x, y, t, e) = ^-(x2 + y2) +

+ E

x2 dmF

m>\

2 At de"

(t, 0)

mls/a'

(3.6)

m

£

Using the formulas

dK E(fc)-(l-fc2)K(fc) 9E E(fc)-K(fc)

k( 1-Jfc2) and k :

we compute expressions for 0), obtaining (see the Appendix B)

dm F j 8 \

— (r, 0) = -Pm(r, 0)E U- + Qm(r, 0)K

8 9/'

m > 0,

with

Pa

e cos t

(1 + e cos t)2(1 — ecos t): and for m ^ 1, with Ae =

Qo

e cos t

(1 + e cos t )2 (3 + e cos t )

p =^i,4p

-Гг™ " ~г л r m— 1

Ae n япН (1 - d^m-l I Ae p _ Kn

de A

Below we list the coefficients Pm(t, 0), Qm(T, 0) up to order 6:

m 0 1 2 3 4 5 6

P * m 0 cos T — 1 cos2 T 47 3 ^ cos T -^cos4T 132I50 cos5T 8826J6 cos6 T

Qm 0 1 cos T 4 2 — 1 cos T 1 q ч cos T -fWr ¿pcos5T 4029„00 cos6 T

The Hamiltonian (3.6) for e = 0 has the frequency a = л/а. So the KGL resonances 2a(a0) = = N determine the sequence of values of parametric resonance

N 2

«o = — N = 1,2,3,....

After applying the Deprit-Hori method [2, 7, 9, 12] to normalize (3.6) in order to get an autonomous quadratic Hamiltonian K(x, y, а, e) which will be used to construct the boundary surfaces of the regions of parametric resonance, the frequency а(а, e) of the Hamiltonian K depends on the three parameters. Since a (a, p, 0) = л/а, a resonance relation 2a(a0) = N for the unperturbed system in (3.6) persists along the entire straight line (а0, 0) of the parameter plane e = 0. So if we enter the half-space e > 0 from a point on this line, the system (3.6) may be stable or unstable for e > 0 small. Therefore, we want to construct surfaces in the half-space e > 0 emanating from this line such that, if the point (а, e) lies on one of the sides defined by the surface, the system is stable and, if it lies on the other side, the system is unstable.

We will look for these surfaces as graphs of functions defined on the plane (0, e), expressing such functions as power series of e with coefficients depending on that is,

a = a0 + a1e + a2e2 + a3 e3 + a4e4 + O(e5),

(3.7)

where a - = a: •(//), j ^ 1 are functions of p and 0:0 = "^.

Inserting (3.7) into the Hamiltonian (3.4) and making the symplectic change of variables

С = a,

-1/4

X,

ao/ Y,

n

followed by the rotation

X = X cos (a/S^t) + Y sin (a/o^t) , Y = —X sin (a/S^t) + Y cos (a/S^t) ,

we obtain the new Hamiltonian

ft = e

dF

a1 + cos(r) + /x—— (r, 0) de

N

+ £

m>2

e

m!

d mF

iV'

(3.8)

where a0 = ^ and 5 = A" cos (^r) + F sin (4n). The above rotation is important to eliminate the term H0 of the Hamiltonian which reduces substantially the computations in the Deprit-Hori method, see [5].

4. Boundary surfaces of the regions of parametric resonance

In this section we apply the Deprit-Hori method [2, 7, 9, 12] to the Hamiltonian (3.8) in order to construct the autonomous Hamiltonian K to be used in the construction of the boundary surfaces of the regions of parametric resonances.

The method allows us to transform via a symplectic change of variables X, Y ^ p, P a Hamiltonian function of the form

H(X, Y, v, e) = Y, y, (4-1)

m=0

into an autonomous Hamiltonian of the form

K(p, P) = k02P2 + klipP + k20P2, (4.2)

where kj = ^ kjem with k(m depending on a1, ..., am.

m=1

Applying the Deprit-Hori method to the Hamiltonian (3.8), we obtain a Hamiltonian of the form (4.2) with the term k11 identically zero, which implies that the characteristic equation is of the form A2 + 4k20k02 = 0. So, the stability region is determined by the inequality k20k02 > 0, whose boundary is given by the equation k20k02 = 0, that is,

k20 = 0 or k02 = 0.

Setting equal to zero the coefficients of all the powers of e in the expressions k20 and k02, we find the coefficients aj(/) in (3.7) and, therefore, the boundary surfaces in the space of the

parameters (a, ¡jl, e). These surfaces emanate from the straight line a = ^ defined by the equation 2a(a) = N, N ^ 1 in the plane e = 0.

Applying the method for the resonance 2a = 1, that is, when N = 1, we obtain a Hamiltonian of the form (4.2). From the equations k20 = 0 and k02 = 0 we obtain, respectively, the surfaces

1

1 '(][t -f Lti2fc -f U^gfc -f Lti^t

a+ = - + cine + a12t2 + a13t3 + au e4 + 0( e5),

4

a~ = - — ane + a12t2 — a13t3 + au e4 + 0( e5),

with

an = ^ + ^(K-3E)p,

a™ = -I + 4(18K - 30E)p - ^(K2 - 6KE + 9E2)p2,

8 72 1 1

72 1

«13 = + 7^t(9K - 23E)p - —(11K2 - 58KE + 75E2)p2 - —(K - 3E)3p

r>2\ ,.2

1

\3,.3

32 32 11

288

864

a14 = -— + —— (44K - 83E)p -

1

(165K2 - 850KE + 1097E2)p2+

+

384 1

144

1728

2592

(11E - 5K)(K - 3E)2p3 -

1

(K - 3E)4p4

31104

where K = K

'§ ) and E

E

These two surfaces determine the region of instability (region of parametric resonance) associated to the resonance 2a = 1. In Fig. 2 on the left we have the surfaces bounding the region of parametric resonance, while on the right we have the planar section formed by the intersection of this region with the plane p =

M

0.4 0.20.0

Fig. 2. Surfaces for N = 1; the planar section formed by the intersection with ¿t = ^

For the resonance N = 2 the equations k20 = 0 and k02 = 0 define, respectively, the surfaces

a+ = 1 + a+2 e2 + a+4 e4 + O(e5),

a

22

2

24

4

1 + a22 e2 + a24 e4 + O(e5)

with

+ 5 198E - 84K

ai, =---

22 12

a

22

v+

«24 =

1 12 763 3456

108 18E — 12K

p +

108

-P

45E2 - 30EK + 5K2 108

9E2 - 6EK + K2

p2

p

615 924E - 293 760K 1947E2 - 1246EK + 201K2

279 936

+

(3015E - 1126K)(3E — K) 23 328

108

" 5184

2 3 763(3E - K)4

p2+

p

279 936

p4,

48 276E — 26 784K

a24 =

3456

279 936

(57E — 26K)(3E — K)2 23 328

V3 +

987E2 - 734EK + 137K2

5184

"V

5(3E - K)4 279 936

V4.

When N = 3 the equations k20 = 0 and k02 = 0 give, respectively,

a

+ _

+ a32e2 + a33e3 + a34e4 + O(e5 ),

with

a = 1 + «32t2 - a33e3 + «34t4 + 0{e5),

a32

1 16

114E - 54K 9E2 - 6EK + K2 2 ■fJi + — "

144

144

-V

1 507E - 213K

«33 = +-—-At

a34 =

32 13

864

574 020E — 263 628K

(25E - 11K)(3E — K) 2 ¡-l +

(3E - K)3

288

864

V

5120 414 720

(135E - 61K)(3E - K)2

+

8167E2 - 6378EK + 1247K2

23 040

V2

4

34 560

13(3E-Kr h 414 720 h

Continuing the process for N = 4, 5, 6, ..., we obtain a decomposition of the space of parameters (a, ¡, e) by alternating regions of stability and instability. In Fig. 3, the figure on left shows this decomposition for N = 1, 2, 3 in the plane ^ = jq- The figure on right is an amplification of the case N = 3.

2.0

1.5

a

1.0

0.5

0.0

stable

unstable •

stable

unstable

2.250

0.00 0.05 0.10 0.15 0.20 0.25 0.30

e

0.00 0.05 0.10 0.15 0.20 0.25 0.30 e

Fig. 3. Decomposition for N = 1, 2, 3 (¿t = amplification of the case N = 3

5

5. Conclusion

We have studied the ¡ -stability of the linearized system around the stable equilibrium.

For each KGL resonance which is located on the lines aN = e = 0, N = 1, 2, 3, ... of

the (a, ¡, 0) plane of the space of parameters (a, ¡, e) we have found, by applying the Deprit-Hori method, the boundary surfaces of the regions of parametric resonance, emanating from each of these lines.

We observe that these regions get narrower as N increases, so the regions of stability become widened.

Appendix A. Computation of the derivative of W

From the theory of elliptic functions (see [4, 12]) we have

K'(k)

E(k) K(k)

k(1 - k2) k '

and since (see the expression (2.3) of k = k(f)) d 1 1 df

dQ 2 + f (2 + f )2 dQ

and

dk = 72/(2 - /) df dQ /(2 + /)2 66'

we obtain from (2.5)

dW ~dd

= 2 = 2 = 2 = 2 = 2

1

(2 + f )2 1

(2 + f )2 1

'(2+7) 1

(2+7F 1

K(k) +

K(k) +

dQ

/(2 + /)3 /2/(2 -f) ( E(fc) K(k)

dQ

/(2 + /)3 \k(l-fc2) k

K(k) +

/(2 + /)3 \2VV(2 -/)2 1 2- f

E(k) -

2f (2 - f) 1

K(k)

2f (2 + f )2

c2l dQ'

2y/2/

9/

dQ

L2f (2 - f) 2f (2 + f)-

This, together with = — (t cost sin 0)j obtained from (2.2), proves the formula (2.7).

Appendix B. Computation of the derivative of ^

In the Hamiltonian (3.6) we need to compute the derivatives r, 0), m ^ 0, of the function F(t, e) given in (3.5).

Proposition 1. For the function F in (3.5), we have

dmF ( [8\ ( [8\

— (r, 0) = -Pm(r, 0)bU-\ + Qm(r, 0)K ( Y g j > m > °>

with Pm, Qm for m = 0 given by

P.

e cos t

(1 + e cos t)2(1 — e cos t)

, Qo =

e cos t

(1 + e cos t )2(3 + e cos t )

(B.1)

and for m ^ 1,

= dPm_! A n n =9Qm-i A

m "T A m~1 yl(l — A2) C^ A A m_1'

where Ae denotes

Proof. From (3.5) we have F(t, e) = -P0E(A) + Q0K(A) with P0 and Q0 given in (B.1). This gives *qem (t, 0) for m — 0. Now, using the formulas

dK.,, E(k) - (1 - k2)K(k) , dE.,, E(k) - K(k)

-z-r(k) =-T,-rrrr--and -z-r(k) =-:-,

dk y ' k(l - k2) dky ' k

we get

dF (dPo^ , D 4 dE..\ dQo, ^ , dK

B(A) + P0A—(A) + -^K(A) + Q0A£ — (A) =

de \de y ' 0 £ dky 'J de v 7 £ dk

dpo^ D „ E(A) - K(A) , dQ0^ , ^ 4 E(A) - (1 - A2)K(A)

]-B(A) - PQA£ — + -^K(A) +

d^ 0 £ A dE £ A(1 - A2)

f + " 1(1^°) + Ow + -

Setting A = ^ + ^P0 ~ A{^Aj)Qo and Q1 = ^ + - we have

Inductively we obtain the other cases. □

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1] Cruz Araujo, G. and Cabral, H.E., Parametric Stability in a P + 2-Body Problem, J. Dynam. Differential Equations, 2018, vol. 30, no. 2, pp. 719-742.

[2] Cruz Araujo, G. and Cabral, H. E., Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point, Regul. Chaotic Dyn, 2021, vol. 26, no. 1, pp. 39-60.

[3] Bardin, B.S. and Markeyev, A. P., The Stability of the Equilibrium of a Pendulum for Vertical Oscillations of the Point of Suspension, J. Appl. Math. Mech., 1995, vol. 59, no. 6, pp. 879-886; see also: Prikl. Mat. Mekh, 1995, vol. 59, no. 6, pp. 922-929.

[4] Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., rev., Grundlehren Math. Wiss., vol. 67, Heidelberg: Springer, 1971.

[5] Cabral, H.E. and Carvalho, A. C., Parametric Stability of a Charged Pendulum with Oscillating Suspension Point, J. Differential Equations, 2021, vol. 284, pp. 23-38.

[6] de Menezes Neto, J.L. and Cabral, H.E., Parametric Stability of a Pendulum with Variable Length in an Elliptic Orbit, Regul. Chaotic Dyn., 2020, vol. 25, no. 4, pp. 323-329.

[7] Brandäo Dias, L. and Cabral, H. E., Parametric Stability in a Sitnikov-Like Restricted P-Body Problem, J. Dynam. Differential Equations, 2018, vol. 30, no. 1, pp. 81-92.

[8] Gel'fand, I. M. and Lidskii, V. B., On the Structure of Stability Regions of Linear Canonical Systems of Differential Equations with Periodic Coefficients, Uspekhi Mat. Nauk, 1955, vol. 10, no. 1, pp. 3-40 (Russian).

[9] Kamel, A. A., Expansion Formulae in Canonical Transformations Depending on a Small Parameter, Celestial Mech, 1969, vol. 1, no. 2, pp. 190-199.

[10] Meyer, K.R. and Offin, D.C., Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 3rd ed., Appl. Math. Sci., vol. 90, Cham: Springer, 2017.

[11] Krein, M. G., Generalization of Certain Investigations of A. M. Liapunov on Linear Differential Equations with Periodic Coefficients, Dokl. Akad. Nauk SSSR (N.S.), 1950, vol. 73, no. 3, pp. 445-448 (Russian).

[12] Markeev, A. P., Linear Hamiltonian Systems and Some Problems of Stability of the Satellite Center of Mass, Izhevsk: R&C Dynamics, Institute of Computer Science, 2009 (Russian).

[13] Moser, J., New Aspects in the Theory of Stability of Hamiltonian Systems, Comm. Pure Appl. Math., 1958, vol. 11, no. 1, pp. 81-114.

[14] Yakubovich, V. A. and Starzhinskii, V.M., Linear Differential Equations with Periodic Coefficients: In 2 Vols., New York: Wiley, 1975.