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Qualitative analysis of periodic oscillations in classical autonomous Hamiltonian systems
D
a32%7462/93 16.00 + .oa 1993 Pnfjamoa Press Ltd
QUALITATIVE ANALYSIS OF PERIODIC QSCILLATIONS IN CLASSICAL AUTONOMOUS HAMILTONIAN SYSTEMS A. A.
ZEVIN
Transmag Research fnstitute, Ukraine Academy of Sciences, 32000.5 Dniepro~trovsk, Piesargevskogo 5, Ukraine (Received
12 June 1992)
Abstract-The paper deals with periodic oscillations of an autonomous Hamiltonian system which are qualitatively the same as the corresponding normal-mode oscillations of the linear&d system. The conditions that guarantee the existence of a continuous branch of such solutions coinciding with a Lyapunov one-parameter family in the n~ghbourho~ of the ~uilibrium point and reaching the boundary of a given region of the configuration space are obtained. Bilateral estimates of the oscillation periods are derived. Under a certain condition of concavity or convexity of the potential function, the Lyapunov family of periodic solutions has a unique continuation in the parameter to the boundary of the region; the corresponding period is a monotonic function of the parameter. As an example, a system of coupled oscillators is treated.
1. INTRODUCTION
Periodic oscillations of autonomous Hamiltonian systems have been studied in many papers (see surveys [l-3]). The majority of the results are pure existence theorems that guarantee the existence of periodic oscillations with a prescribed total energy or period. The theory of normal-mode vibrations originating from the works of Rosenberg gives some info~ation also on the behaviour of periodic solutions of second-order Hamiltonian systems (see, for example, [4-83). So, it has been found that a system with a homogeneous potential function has similar normal-mode vibrations (the corresponding trajectories in the configuration space are straight lines). For a common class of potential functions, the existence of such solutions that simultaneously reach the boundary surface and, provided the potential function is symmetric, simultaneously pass the origin has been proved. This paper is concerned with more detailed characterizations of periodic solutions. According to the Lyapunov theorem [9], in the neighbourhood of the equilibrium point, there exist one-parameter families of periodic solutions whose periods are close to that of the linearized system. Clearly, these solutions are qualitatively the same as the normal vibrations of the linearized system. In particular, in a system with symmetric potential function all coordinates maintain their signs in the course of half a period or even vary monotonically between the extreme values; a non-linear mode preserves the nodal properties (has the same number of zeros) of the corresponding linear mode. Below we give non-local criteria for the existence of such solutions. It turns out that if the potential function satisfies some conditions in a given region of the configuration space then there exists a continuous branch of such solutions coinciding with the corresponding Lyapunov family in the vicinity of the equilibrium point and reaching the boundary of the region. The periods of the solutions satisfy bilateral estimates depending on the region. Under a certain additional condition of concavity or convexity (that is verified via the Hessian of the potential function), the Lyapunov family is uniquely prolongable in the parameter to the boundary of the region and there is a monotone relationship between the period and the parameter.
2. THE SYSTEM
Let us consider a classical conservative Hamiltonian system having a finite number n of degrees of freedom. Denote by x = (xl,. . . , x,) the vector of generalized coordinates, by Contributed by P. Hagedorn. 281
282
A. A. ZEWN
V(x) the potential energy, by K = (1/2)(Mi, ir) the kinetic energy where M is a positivedefinite symmetric inertia matrix, and (a, b) the inner product of vectors a and b. V(x) is supposed to be twice differentiable and symmetric The function [V(x) = V(-x)]. The corresponding differential equation is
MR + f(x) = 0,
f(x) = v,(x).
(1)
Denote by or (wp I III;+,) and xp = (xfI,. . . ,xz) quencies and modes of the linearized system
(i = 1,. . . ,k < n) the natural fre-
Mj2 + A(O)x = 0,
(2)
where A(x) =
Xj(t,
+
O(t,
S),
where x;(t) is the corresponding solution of (2) [9]. By (3), x7(t) = x3 sin o)‘f; so, supposing that x; # 0 for all i, we obtain for small s SgtlXji(t,
S)
=
on (0, q/2), i = 1,. . . , n.
SgllX$
To simplify the next account, let us change the positive directions of those coordinates for which x$ < 0. Then, keeping the above designations, we have xp > 0; therefore, xj(t, s) satisfies x(t) > 0 on (0, T/2). (4) We suppose that x3 is the only sign-constant mode (if M is a diagonal matrix, this supposition certainly holds due to the known property of orthogonality: (Mxy, x,“) = 0 for k #j). Let R be a given region in the configuration space containing the origin (for example, n = (x: V(x) 5 h} or R = (X: IXil I A:, i = 1,. . . ,n}), and cX~its boundary. The aim of this paper is to obtain conditions that guarantee the existence in R of a continuous branch of periodic solutions satisfying (3) and (4). It means that for every open region 9 E R (0 E 8) there exists a solution x(t) of the kind (3) and (4) such that x(t) E 3 for all t, and x(t,) E 88 for some t,.
3. EQUIVALENT
INTEGRAL
EQUATION
In this section we reduce the problem to the equivalent integral equation. N = InitIy,tXl, P = IPikI;e=, be symmetric matrices such that for x 2 0, XER Nx I f(x) I Px.
Let (5)
By the integral mean-value theorem, f(x) can be written in the form f(x) = C(x)x, 1
C(x) = Icik(x)lik=lv
C(x) =
A(ux 0
(6)
Periodic oscillations in classical autonomous Hamittonian systems
283
So one can take nik =
fXlillCi~(XX
Pik =
for
lllZtXCik(X)
0, X E R
X 2
(7)
Having made in (1) the change of the variable t + rT, we can write it as follows: Mx” + T’Nx = -T’+(x), t)(x) = f(x) - Nx,
’=
$
Taking into account (3) and (4), we shall seek one-periodic solutions x(r) =
-x(-r)
=
-x(7
+
x(7)
of (8) such that
w
(9)
and x(7)
>
0
on (0, l/4].
(IO)
BY (9)s x(0) = x’( l/4) = 0.
(11)
Conversely, equality (11) for any solution x(r) implies identity (9). So, a solution x(r) of the desired type coincides on (0, l/4 3 with a positive solution of the boundary problem given by equations (8) and (11). Denote by Ai and xi = (xii,. . . ,xi,) the eigenvalues and eigenvectors of M-r N [i = 1,. . . ,n; &I Ri+1, (MXi, x{) = 13. Suppose that for Li > 0 the values T;11’2/2~ are not integers; then the linear boundary problem Mx” f T’Nx = 0,
x(0) = x’(1/4) = 0
(12)
has only the trivial solution. So, there exists a Green matrix T(t, a, T) = ITit(r, u, T)If.,= 1 by the use of which the solution x(7) of (8) and (11) can be written as foflows: x = G( T)x,
s l/4
[G(T)x](T)
= -T*
UT, u, W4x(u)l
du.
(13)
as E +O,
(14)
0
By definition,
T(7,
u,
T) satisfies (12) for t # t and
lim(Y(r
+ E, T, T) - T’(r - E,r, T)) = M-l
where M-l is the inverse matrix. It is easy to check that rik
=
“:I
~“kX~i~“(7)~“(~)
for
7 I
u
rik
=
“$I
xvkxvipv(~)qv(7)
for
7 2
u,
(W
where P&f
= -
P*(s) = -
sin(o,sT) w T
Y
sinh(w,sT) o T
Y
1
4*(s) = 9
44s) =
cos[ 0, T( l/4 - s)]
cosw,T/4
’
cosh[o,T( l/4 -s)] cosh(w, T/4)
P”(S) =’ -s,
q”(s) = 1
’
co,=&
ifA,>O.
w, = &j
if I., < 0.
if R, = 0.
In fact, expression (15) satisfies (12) and lim(T;k(r + &,t, T) - &(r
- &,t, T)) = i x,kx,i V=l
as E 4 0.
(16)
Since MXTX = I (X = [XT, . . . , XT], T denotes transpose, I is the identity matrix) then X’X = M-‘. By a direct calculation, one gets that an element rik of the matrix X*X is equal to the right-hand side of equality (16). Thus, the fact that (14) is satisfied proves the correctness of (15).
~r~l~u~u? ue stu~i3J
Suo~ln[Os
aA!g!sod jo *f# las aql ‘illewqla
s! s uo$a~
sy ‘([z’&
k%.IU.IaJ
‘or]) 'f3x uoyl~os au0 peal II2sey *f) = 3 qI!M ~fl)Uo?l~nba ‘(*J!‘_Jf;3_&~~0s 3oj ‘0sf(7 ‘(*A )J)A # (7 ‘( _J.)d)L uaql _A = & w 7 3 x suognlos ou aJt? a~aql j! ‘snqL ‘(C&E uia~oaql ‘or]) 0 = (7 ‘(.J;)J)A ‘a~oja~aql :o > tf sagdur! 7 3 x JeaU *J 3~110s JOJ Uaql‘+_l;cl SE [t/1 ‘o)uo oa+-(A %'L)J aixJ!s
ql!M(IZ) Alyznba
+J
'I = IIOqII ‘)II3Oq ‘Oqd -I" x(J)*3
(IZ)
= X
uoyenba ICITE ~=oaq~ ‘013) I = (7 ‘(_,~)d)~ '~~=~l~(~~)JOUO~~~~oS~S~XUa~l~3XawOSJOJ~ 4p=baaql‘os ~0 anfehuaSl!a
aql ~ap!suos sn 137
u9q1 7 3 x II~ ~03 I > y JI 2 =L =yjI'] 5’(sa!ldLu!xy=x(__t)*f)
pue
!o r7 x JOJ x(-.1)+5 5 xf_~I*f) ‘69 01 ana %[~Iz WaJoaql ‘I I])(_.A)).~ w%lal ayl s! I = y ‘awaq !ro$eJado an!l!sod-011 e s! (_d),g ‘(0~) 68 =loi~a~ rllguapf ayl s! 01 aJaqM ‘CtJ/I ‘03 Uo
(05)
OI4
5 i~)Cx(_..z)+!?,f
5 Oila
leql q3ns 0 < gi pue 0 < 13 siueisuo3
ale aJay
‘0s ‘0 f (z)x
‘OZ tl)x~u~JoJo = (O)LX(_J)+91‘CfdI‘o)uoo e (l)CX(_J)+91 uaq1‘0 = (4 ‘n‘0k.I JOJ sp[oq (81) a3u!s .(z) +x uo~l~un~ aA$sod aql 01 %u!puodsaJ.toci pus -45 =I SnonuyIo:, AlalaldtuoD aqi JO anlI?AUafa!a III? s! 1 = y ‘(61) JO Ma!n. UI
( _J) +g Jowado
0 &I)
‘“P(fl)W
- d)(_.Jc
“2 ‘1)&l
V/Ii
‘x(-L)+!I uojwnba ‘7 uo
pray loi3aA aql jo uoye3o.i aql(7
,(_.L)-
aA!l!sod
s! (d)*f)
= (1)CX(_L)+!Il
= x
aql saysyas x(&E)*9
‘(d)d)C,
-
(J;I;O)U!S
‘3 u! s JO hepunoq
a.y%auuou h?
la?
/(lalaldmoD s! (~)*r,
hpuodsaJlo3
aql (~)*g
ql!M %U!p!DU!o:, UO!]DUnJ snonu!luo~ aql uo sa!IaJ uraJoaql
.auo aA! lowado
put? ‘0 z x
I? 2q (X)*J 127
aq$ JO JooJd
aq& $IoAd
qI!M 0 %U!l%UUO3 q3U"'CjSnOnU!lUOS ESWOJ
Jo 8 las aql uaq_L *w/u2
‘n ‘1)~
3~1 se u x = 7
JO au03 r! aq y
SULIOJSU~J~ (J )*g -a-! ‘(+J '_J]~3~;
‘(81) pue (5) Cg ‘3 u! snonuyo:, PUE u U! (X)J
fo:oi] Splay JOWA Jo UO!lBlOJ JO hOaql
Sanfett ~u~puodsaJ~o~ aqL*Q ahye%au-uou
jo sJowhwa%!a
‘U‘ ’ ’ ’ ‘1 = J/‘! ‘[b/I
ix = (I) +x uo!~~un~ aye,
x = xLL).5f
icq alouaa
‘0 %up~yuo:, J tq ws uado ue s ‘suoyunJ
-l!sod e olu! (1)x uo!lc~un~ [v//r ‘0) uo aA@au-uou Joj
aql ‘SnOnUyIO3 s! (.,! 'n'2)~ aDu!S ‘($1) Jowado IlE ‘oJ (5) %U!@!lI?S
-(,J '_J]3d &I) jo suoyqos
(8f)says!]esput? s$s!xa (l -dr_H (81)
= _J
a2aqM ‘( +_z ‘_J: J 3J; 301
ti awes .toj +f < :y put? ‘0 < ;x pua sanlcAuafI!a aql aq (2x
asoddns
‘1 uraroay~
= 2x pue ;y 137
. . “2~)
‘0) 3 n ‘1 JOJ 0 > (J; ‘n ‘I)J!J
‘0 > (L ‘11‘O)Q ‘aJoja3aql
‘puv 1+J: -J
J
(11) * uoyyos
any&au-uou
au10s .ro~fg 3 (*1)x pue ‘[*/I ‘013 r JOJ f 3 (1)x ieqi qctns (EI)JO auo lseaI it? sis!xa aIaql0
pug a~ ‘{[@/I ‘013 q3ueJq
1 foj $3 (1)~ :(1)x)
snonupuo2
a,t!le8au-uou
&~!u!eluos
0 3 6 uo!%ar uado Ithaca 30~ $ayl
= 7 %u!soddns ‘wojaJaq_t
‘se I@M 0 %uyauuo:,
IS SWJOJ 8 ‘suoy~puo~ amos Japun ‘layi Moqs 1p2qs aM *(fI)jo
suoynlos
Jo las I? s! 8 ‘([b//r ‘013 1 JOJ u 3 (2)x :( 2)~) = s asoddns aM uo MOU rn0.I~ *[or] hldwa-uou
haAaJ0
hpunoq
~IJOJ 01 p+ suopDunj
aql qI!M ~30 uo!lDaualu!
s! 8 $2~ v ‘0 &~!u!eluo~
s! 0 %.~!uyuos
U3ClNn
s z> 7 ias undo
aqlJ! se ~I!M 0 %u!uauuoD qwwq
3 u! las uado 1.112s pue ‘(( I)“x‘
snonu!iuoD
~0133~ [V/I ‘01 uo NOI.LV2i3ClISN03
JO
acwds
SNOIlfllOS
q3eueg
3HL 30
e
snonu!iuol,
‘t!
. . . ‘( L)!x) = (2)x 3
aiouap
23N31SIXx
ial
‘P
Periodic oscillations in classical autonomous Hamiltonian systems
285
long branch (i.e. intersection of B, with the boundary of any region in E containing 0 is non-empty). Since G = G, in f2, the set B of non-negative solutions of (13) coincides with B, in S. Thus, f3 forms a continuous branch connecting 0 with 8s. As seen from the proof, the period of any such solution TE CT-, T+ ). The theorem is proved. Remark. For T < T+ the rth item in (15) diminishes on [0, l/4). Hence, r’(r, u, T) < 0
and x‘(7) >
0
on [O, l/4)
(22)
if TEI:T’,T+),(T’> T-). Thus, the corresponding solution x(t) with a period Tf CT’, T+ ) changes monotonically between the extreme values xi( - 1/4T) = -Ai and xi( 1/4T) = Ar, like the solution x?(t) of the linearized system. Inequality (22) holds for all positive solutions (i.e. T’ = T-) if eik(x)
20
= y
forx20,xoR;i,k=
l,...,
n.
In fact, by differentiating (8), we obtain that v(r) = x’(r) satisfies the equation Mv” + T=Nv = -T’Y(x)v, y(X) =
(24)
l$ik(x)l%&=l*
Taking into account that v’(0) = v( l/4) = 0, the corresponding written as follows: v = G,( T)v, W G,(T)v = -T* I
integral equation can be
1-(T/4 - T, T/4 - u, T)Y(x~~))v(u)d~.
(25)
0
By (18) and (23). the operator G,(T) for TE [T-, T+) is positive. So, one can analogously prove that the set of positive solutions of (25) forms in E a continuous branch connecting 0 with as, where S = (v(r):
*v(u)du ER I
for
7E
[O, l/4]).
0
Clearly, the solutions of (13) X(T) =
v(u)du
satisfy (22) and form a continuous branch connecting 0 with aS. The Lyapunov family xj(rT, s) E B for small s. By supposition, xj’ is the only signconstant mode of the linearized system; so, the branch B near 0 consists of solutions xj(7T, s) only. Suppose now that, in addition to the conditions of theorem 1, C(x)>A(x)
forx>O,xoiZ,
(26)
i.e. (C(x)y, y) > (A(x)y, y) for all y # 0. Theorem 2. The Lyapunov family xj (t* s) is uniquely prolongable in s to the boundary of the region Q i.e. Xj(t, s) E R for s E (0, s * ), and Xj(t, s*) E dn for some r, s *. The solutions Xj(t, s) satisfy (3) and (4), the period q(s) increasing monotonically.
Prooj Let us show that q(s) > TT for small s. Denote by J,(s) (A, < &+t) the positive eigenvalues of the problem Mx” + I.C(r, s)x = 0, x(0) = x’( l/4) = 0,
C(T, s) = C(Xj(ZT, s)).
(27)
A. A. ZEVIN
286
By (6) and (8), T;(s) is one of the eigenvalues, xj ( ST, s) being the associated eige~unction. In view of (26), for x > 0, k > 0, kx E R iK( kx)
T=;[;
(28)
j;A(ux)du]=+(kx)-C(kx)]cO.
Hence, C(0) = A(0) > C(x), and C(r, s) < A(0). If C(7, s) > 0, the eigenvalues J.,(C) increase with a decrease in C [ 121. As is obvious from the proof of this assertion, the condition C > 0 can be substituted for 114
E=
(C(sfx(zXx(r))dt
s0
> 0,
where x(7) is any eigenfunction of (27). BY (271, l/A E=f (Mx’(7), x’(7))d7 - f(Mx’(r), *I 0
(2%
~(r))lA’~.
(30)
Since x(0) = x’( l/4) = 0, the term outside the integral equals 0; so, E > 0 (by supposition, M >O).
The eigenvalues 1.,(O) are formed from the quantities (2nr/o~)z, i = 1,. . . ,n, 1,2,. . . . Let Tj’* = J+,(O);this eigenvalue is simple because c&‘/w: are not integers. So, T;(s) = Ai for small s; therefore, Tjfs) increases in s (C(r, s) < C(7,O) = A(O)). So, for small T - T;, there exists a solution x(7, T) of the boundary problem given by equations (8) and (1 t) such that x(7, q(s)) = xj(7T, s). According to the perturbation theory, x(7, T) has a unique continuation in the parameter T if the linearized problem r =
My” + T’A(r,
Y(0) = Y’(l/4) = 0,
T)y = 0,
A(7, T) = A(x(7,T))
(31)
has only the trivial solution. Let us consider self-conjugate boundary value problem My” + T*Ny + i.T*R(r)y y(0) = y’( l/4) = 0,
The corresponding
R(7) =
Irikf~)llk=
19
= 0, R(s) = RT(7) > 0.
(32)
integral equation is “Y = GdT)y, I/4
CGR(T)YI(~) = -T*
I
r(7, ~9T)R(u)y(u)dn,
Y= l/i_.
(33)
0
Since X(T, T) satisfies My” -t C(r, T)y = 0,
C(T, T) = C(x(7, T)),
(34)
problems (32) and (33) at R(r) = C(7, T) - N have eigenvalues J = 1 and v = 1. The operator GR( T) is 710-positive and completely continuous for TE [T”, T' ); so, the eigenvalues v = 1 and R = 1 associated with the positive eigenfunction x(t, Tf (Xj(r, s) > 0 on (0, T/4) for small s) are, respectively, the largest and the smallest eigenvalue of problems (33) and (32) at R(7) = C(7) - N. Let R(t) = A(r, T) - N. By (261, A(7, T) < C(7, 7’); SO, A,(A) > i.:(C) [12]. Hence, Ai > 1. Therefore, problem (31) has only the trivial solution that ensures the uniqueness of the continuation of a positive solution x(5, T) in T. The inequality x(7, T) > 0 cannot break under continuation. Indeed, otherwise x(7, Tl) >Oon(O, 1/43,x,(ri, T,) = Oorx;(O, T,) = Oforsome k, Ti,sl ~(0, 1/4].But,by (lf!), [Gx](r) > 0 on (0, l/4], and [ Gx J’(0) > 0 for any x(7) 2 0, x( 7) + 0. Thus, x(7, T) > 0 and ~‘(0, T) > 0; so, x(s, T) is continuable in T. Since x3 is the only sign-constant mode of (2), x(7, T) # 0 for T > T/O. Hence, x(r,,T*)~dQ for some 7i, T* c Ti+ [r(r,u,T)+a;, as T+Tjc]. Let T(s) be any
Periodic oscilfations in classical autonomous Hamiltonian systems
287
monotonic fun~ioR coinciding with q(s) for smali s, Z’(s*) = T*. Evidently, the function x(r/~~s), T(s)) is the desired one-parameter family xi(r, s). The theorem is proved. Remark. Provided C(x)
forx>O,xEQ
(35)
one can analogously show that in problem (32) rl I < 1 at R = A - N; so, xj(t, s) is unique, and q(s) decreases monotonically until A,(s) > 1. Let 0 < .4(x) < A + for x c !& x > 0; denote by w: the eigenfrequencies of the system M% + A+x = 0. It has been shown in ref. [13] that under (35) and regardless of (5), the inequality AZ > I is true if ~~~~~/~~,~~/~~I,
3w,O>o:,
i#j,m=l,3,5
,...
(36)
Note that for Hamiltonian systems inequality (26) or (35) (where A(x) = H,,(x), H(x) is a Hamiltonian), together with conditions analogous to (36), ensures the uniqueness of x,(r, s) and monotonicity of q(s) regardless of the behaviour of Xj( t, s) 113). Van Groesen found the monotonicity conditions for the minimal period Z’(E) of system (1) (E is the total energy) [14]. In particular, T(E) increases if (,4(x)x, x) < (C(x)x, x); evidently, this condition is weaker than (26).
5. DISCUSSION
AND EXAMPLES
Elements (7) of N and P depend on region ft. If Q retracts to the equilib~um point, N,P+A(O), TJ&T;-+T, O. So, there always is a sufficiently small R for which theorem 1 holds. It is necessary to emphasize that, in view of the inequality x7 > 0, the analysis of another branch of solutions requires the change of directions of some coordinates; as a result, in the general case the matrices N and P will be different for different families. Inequality (5) and, therefore, matrices N and P are invariant with respect to the mentioned change of coordinates if
v(X) = t
Y(Xi) + (CO%X)9
l$(Xi)
= I$(-Xi).
(37)
i=l
Note that, in particular, the potential energy of n non-linear oscillators connected to each other by linear strings is of the form (37). Let nixi
S.&(xi)
d U-d
-
=
dx
I;
for Xi > 0, XiE a.
PiXi
(38)
i
Then one can put for all branches N=COfdiag(nl
,...,
n,),
P=Cc+diag(p,
,...,
p,).
(39)
If &(-Xi)z~JO)X: or ,$(xf)S&(O)Xi (i = 1,. . . , tt, _& = d_#dx) then, respectively, N = A(O), Wi = W: or P = A(O), O: = 0,.9 In the first case the rigidities of a non-linear system exceed that of the linearized one; here condition (17) certainly holds (k = j). If .fIXXi)/Xi
>_&.z(xih
i =
1,.
. * t f3
(401
then the inequality C(x) > A(x) holds. Let in (1) M = diag(mi,. . . ,mJ,
fi’(x)TOforxZO,i=
l,...,
n,f(x)#Oforx#O.
(41)
for t 2 U.
(42)
Then one can take N = 0; the Green matrix is I = diag(F,(s, Ti = -T/Rli
for T S u,
u),. . . ,T;((z, u)), Ii = -u/m2
A. A. ZEVIN
288
Here the operator G(T) is positive for all T.By (41), for x(5) EL some component G( T)x+x! as T-tax So,equality (21) implies p c 0; therefore, y(F( T),L) = 0 for large T. On the other hand, for small T the equality G( T)x = lxyields I < 1 (G(T)x+ 0 as T + 0); so,here y(F( T),L)= 1. Thus, there is a solution on L,sounder condition (41), the set of positive solutions forms a continuous infinitely long branch. Note that this result was obtained using different arguments by Rabinovitz [IS] and (for a less general class of f(x)) by Duffin [16]. The existence of an infinitely long continuous branch of non-negative on (0, T/2) periodic solutions was proved by Krasnoselskii [l 11. Some properties of the solutions under condition (26) or (35) were established in ref. [17], where non-autonomous systems were also treated. Since T’(r, u) I 0, the solution x(t) increases monotonically on [0, l/4] (that in a common case does not rule out the identity Xi(T) z 0 for some i). Condition (41) can be satisfied, in particular, for a system of oscillators coupled successively by elastic constraints and for a string or a beam carrying lumped masses [16, 173. To comply with (41), positive xi values should correspond to opposite displacements of adjacent masses. Thus, the solution in question describes such oscillations that adjacent masses always move in opposite directions. For small amplitudes they belong to the Lyapunov family associated with the highest natural frequency (positive eigenfunction x$’of non-negative, by (41), matrix M-’ A(0) corresponds to the largest eigenvalue [18]). To illustrate an application of the results obtained, let us consider a system of two non-linear oscillators connected by a linear string:
mlf, +/l(xl)+c(xI -x,)=0,
m,i2 +fz(x2)+c(xI-xl)= 0.
(43)
Let fliXi Ij(Xi) 5 PiXi for Xi E R, Xi > 0 (i= 1,2). In the first mode of the linearized system x7 > 0, i.e. both masses move in the same direction. Let us obtain conditions that guarantee the existence of a continuous branch of such solutions in R. Here
Suppose N > 0; then in (15) w:, o: are the eigenvalues of the matrix M- ’ N, x1,x2 are the corresponding eigenvectors. The condition T(T, u, T) < 0 on (0, l/4) can be considerably simplified. Suppose 02/w1 < 3; then cos(o,T/4)> 0,cos(wzT/4)< 0 for TE(~~c/~I~,~x/w~). So,one can easily check that rl 1(~, u, T) < 0 on (0, l/4) if r\ ,(O, u, T) I 0 and r( l/4, u, T) I 0.These inequalities hold if, respectively, r;,(O,
l/4, T) 5 0 and
ar,
l(
l/4, u, T)
au
IO, u=o
which implies x:~ cos(o, T/4) + x:~ cos(o, T/4) I 0.
(44)
Analogous considerations give the conditions ensuring the inequalities rz2(r, u, T) < 0 and rlZ(Tru, T)= rzl(T, U, T)
+ X:~COS(O~T/4) I 0. + 02x21x22tan(wzT/4)
2 0.
(45)
Let w: be the first eigenfrequency of the system Mfi+ Px = 0,T; = 2x/w:. If inequalities (44) and (45) hold for TE [T;, T:) (T: = 2x/wl) then, by theorem 1, for any region 9 E R containing 0, there is a solution x(t) E 9 satisfying (3) and (4) and lying on 89 for some t. If dJ(xi)/dxi 2 ni (i = 1,2) then, in accordance with the remark to theorem 1, x1(t),x2(t) vary monotonically between the extreme values Xi(- T/4) = - Ai and Xi(T/4) = Ai. If w2/w1 I 2, in both the items in the second inequality (45) are positive; so, it holds trivially. Evidently, the rest of the inequalities hold for T E [T;, T:)provided they are true for T= T;. Let m, = m2 = m and n, = n2 = n; then xl1 = xr2, xzl = -x22 and (44) and
Periodic oscillations
in classical auronomous
Hamiltonian
systems
289
(45) imply
1 w: 5 +
+ 02) =
(46)
This condition can be expressed by means of the elements of the matrix P. In particular, so, (46) yields let Pr = p2 = p. Then of = fi; :+1
+J_I?
(47)
where c/n 5 1.5 in view of the condition used, wJol 5 2. The values p, n depend on the region R. If nx 0 (i = 1,2) then, under condition (47), solutions x,(t) form a continuous infinitely long branch. It means that there are oscillations in question with any prescribed total energy. Let us now turn to the solutions x,(t) corresponding to the Lyapunov family xz(t, s). First of all, it is necessary to change the direction of the axis xi or x2 to comply with the condition x’: > 0. Let n, + c > O,nz + c > 0, then condition (41) is satisfied. Therefore, solutions x2(t) form a continuous infinitely long branch; under such oscillations, the masses m, , m2 always move in opposite directions. As a specific example, let us consider oscillations of coupled pendulums. Here mi = ,UiIf andi = /.Iiglisinxi, where pi is the mass and li the length of the ith pendulum, g is the acceleration due to gravity, and x1, x2 are angle coordinates. Let Q = {x: lxil I Ai < 7~); evidently, one can take pi = pig1i and hi = pig/i( sin Ai)/Ai. Since (sin x)/x > cos x for x < x* = 4.49. . . , the inequality A < C is true. So, if w2/o, I 2 and inequalities (44) and (45) hold for T = 2n/w: then, in accordance with theorem 2, the Lyapunov family x,(t, s) is uniquely continuable in s to &2; the period T,(s) increases monotonically. For solutions x,(t), one can take N = 0. Since here the operator G(T) is positive for all T, theorem 2 is also applicable; so, the family x2(& s) can be continued to dQ, and the period T2(s) increases. Unlike the previous case, here the pendulums rotate in opposite directions. If C + &CJli(SiIlX*)/X* > 0 (i = 1,2) then (41) holds; so, solutions x,(t) form a continuous infinitely long branch. To analyse the oscillations of the pendulums about the upper equilibrium point, it is necessary to putj(x) = -piglisinxi. Clearly, here solutions X,(t) do not exist. Let c > pig/i (i = 1,2); then (41) holds. So, solutions x2(t) form a continuous infinitely long branch. Here the pendulums rotating in opposite directions meet at the upper equilibrium point.
REFERENCES
1. N. Desolneux-Moilis, 2. 3. 4. 5. 6. 7. 8. 9. IO. Il. 12.
13. 14. 15.
Orbites periodiques des systems Hamiltonians autonomes. Lecture Notes Math. 842, 156-173 (1981). P. H. Rabinowitz. Periodic solutions of Hamiltonian systems: a survey. SIAM J. Morh. Anal. 13(3), 343-352 (1982). E. Zehnder, Periodic solutions of Hamiltonian equations. Lecture Notes Moth. 1031, 172-213 (1983). R. M. Rosenberg, The normal modes of nonlinear n-degree-of-freedom systems. J. appl. Mrch. 29.7-14 (1962). R. M. Rosenberg, On nonlinear vibrations of systems with many degrees of freedom. Ada. appl. Mech. 9, 156-242 ( 1966). C. H. Pak and R. M. Rosenberg, On the existence of normal mode vibrations in nonlinear systems. Q. appl. Math. 26. 403416 (1968). A. Weinstein, Normal modes for nonlinear Hamiltonian systems. Inuenrions Math. 20. 47-57 (1973). E. W. C. van Groesen, On normal modes in classical Hamiltonian systems. hr. J. Non-Lineor Mech. l&55-70 (1983). A. Lyapunov, Probleme generale de la stabilite du mouvement. Ann. Fus. Sci. Toulouse. 2, 203474 (1907). M. A. Krasnoselskii and P. P. Zabreiko, Geomerricol Methods of Nonlinear Analysis. Springer, Berlin (1984). M. A. Krasnoselskii. Posiriue Solutions o/Operator Equorions. Noordhotf, Groningen. The Netherlands (1968). M. G. Krein. Principles of the theory of I-zones of stability of the canonical system of linear differential eouations with neriodic coefficients in: In memory_ of _ A. A. Andronou. Izd. AN SSSR, Moscow (1955). (in Russian). . A. A. Zevin. Anisochronicitv conditions of autonomous Hamiltonian systems. Me&. Solids (Mekh. Tcer. Tekr) 23(I), 33-39 (1988). E. van Groesen. Duality between period and energy of certain periodic Hamiltonian motions. J. London Marh. Sot. 34(2). 435-448 (1986). P. H. Rabinowitr A note on a theorem of R. Duffin. Arch. Rarional Mech. Anal. 93(l). 91-102 (1986).
290
A. A. ZEWN
16. R. J. Duffin. Vibration of a beaded string analyzed topologically. Arch. Rational Mech. Anal. 56(J), 287-293 (1974). 17. A. A. Zevin. Existence, stability and some properties of the class of periodic motions in nonlinear mechanical systems. Much. Solids (Mekh. Tuer. Bela) B(4). 42-51 (I985). 18. R. Bellman, J~t~oducfio~ ru Marrix Analysis. M~raw-Hill, New York (1960).
a32%7462/93 16.00 + .oa 1993 Pnfjamoa Press Ltd
QUALITATIVE ANALYSIS OF PERIODIC QSCILLATIONS IN CLASSICAL AUTONOMOUS HAMILTONIAN SYSTEMS A. A.
ZEVIN
Transmag Research fnstitute, Ukraine Academy of Sciences, 32000.5 Dniepro~trovsk, Piesargevskogo 5, Ukraine (Received
12 June 1992)
Abstract-The paper deals with periodic oscillations of an autonomous Hamiltonian system which are qualitatively the same as the corresponding normal-mode oscillations of the linear&d system. The conditions that guarantee the existence of a continuous branch of such solutions coinciding with a Lyapunov one-parameter family in the n~ghbourho~ of the ~uilibrium point and reaching the boundary of a given region of the configuration space are obtained. Bilateral estimates of the oscillation periods are derived. Under a certain condition of concavity or convexity of the potential function, the Lyapunov family of periodic solutions has a unique continuation in the parameter to the boundary of the region; the corresponding period is a monotonic function of the parameter. As an example, a system of coupled oscillators is treated.
1. INTRODUCTION
Periodic oscillations of autonomous Hamiltonian systems have been studied in many papers (see surveys [l-3]). The majority of the results are pure existence theorems that guarantee the existence of periodic oscillations with a prescribed total energy or period. The theory of normal-mode vibrations originating from the works of Rosenberg gives some info~ation also on the behaviour of periodic solutions of second-order Hamiltonian systems (see, for example, [4-83). So, it has been found that a system with a homogeneous potential function has similar normal-mode vibrations (the corresponding trajectories in the configuration space are straight lines). For a common class of potential functions, the existence of such solutions that simultaneously reach the boundary surface and, provided the potential function is symmetric, simultaneously pass the origin has been proved. This paper is concerned with more detailed characterizations of periodic solutions. According to the Lyapunov theorem [9], in the neighbourhood of the equilibrium point, there exist one-parameter families of periodic solutions whose periods are close to that of the linearized system. Clearly, these solutions are qualitatively the same as the normal vibrations of the linearized system. In particular, in a system with symmetric potential function all coordinates maintain their signs in the course of half a period or even vary monotonically between the extreme values; a non-linear mode preserves the nodal properties (has the same number of zeros) of the corresponding linear mode. Below we give non-local criteria for the existence of such solutions. It turns out that if the potential function satisfies some conditions in a given region of the configuration space then there exists a continuous branch of such solutions coinciding with the corresponding Lyapunov family in the vicinity of the equilibrium point and reaching the boundary of the region. The periods of the solutions satisfy bilateral estimates depending on the region. Under a certain additional condition of concavity or convexity (that is verified via the Hessian of the potential function), the Lyapunov family is uniquely prolongable in the parameter to the boundary of the region and there is a monotone relationship between the period and the parameter.
2. THE SYSTEM
Let us consider a classical conservative Hamiltonian system having a finite number n of degrees of freedom. Denote by x = (xl,. . . , x,) the vector of generalized coordinates, by Contributed by P. Hagedorn. 281
282
A. A. ZEWN
V(x) the potential energy, by K = (1/2)(Mi, ir) the kinetic energy where M is a positivedefinite symmetric inertia matrix, and (a, b) the inner product of vectors a and b. V(x) is supposed to be twice differentiable and symmetric The function [V(x) = V(-x)]. The corresponding differential equation is
MR + f(x) = 0,
f(x) = v,(x).
(1)
Denote by or (wp I III;+,) and xp = (xfI,. . . ,xz) quencies and modes of the linearized system
(i = 1,. . . ,k < n) the natural fre-
Mj2 + A(O)x = 0,
(2)
where A(x) =
Xj(t,
+
O(t,
S),
where x;(t) is the corresponding solution of (2) [9]. By (3), x7(t) = x3 sin o)‘f; so, supposing that x; # 0 for all i, we obtain for small s SgtlXji(t,
S)
=
on (0, q/2), i = 1,. . . , n.
SgllX$
To simplify the next account, let us change the positive directions of those coordinates for which x$ < 0. Then, keeping the above designations, we have xp > 0; therefore, xj(t, s) satisfies x(t) > 0 on (0, T/2). (4) We suppose that x3 is the only sign-constant mode (if M is a diagonal matrix, this supposition certainly holds due to the known property of orthogonality: (Mxy, x,“) = 0 for k #j). Let R be a given region in the configuration space containing the origin (for example, n = (x: V(x) 5 h} or R = (X: IXil I A:, i = 1,. . . ,n}), and cX~its boundary. The aim of this paper is to obtain conditions that guarantee the existence in R of a continuous branch of periodic solutions satisfying (3) and (4). It means that for every open region 9 E R (0 E 8) there exists a solution x(t) of the kind (3) and (4) such that x(t) E 3 for all t, and x(t,) E 88 for some t,.
3. EQUIVALENT
INTEGRAL
EQUATION
In this section we reduce the problem to the equivalent integral equation. N = InitIy,tXl, P = IPikI;e=, be symmetric matrices such that for x 2 0, XER Nx I f(x) I Px.
Let (5)
By the integral mean-value theorem, f(x) can be written in the form f(x) = C(x)x, 1
C(x) = Icik(x)lik=lv
C(x) =
A(ux 0
(6)
Periodic oscillations in classical autonomous Hamittonian systems
283
So one can take nik =
fXlillCi~(XX
Pik =
for
lllZtXCik(X)
0, X E R
X 2
(7)
Having made in (1) the change of the variable t + rT, we can write it as follows: Mx” + T’Nx = -T’+(x), t)(x) = f(x) - Nx,
’=
$
Taking into account (3) and (4), we shall seek one-periodic solutions x(r) =
-x(-r)
=
-x(7
+
x(7)
of (8) such that
w
(9)
and x(7)
>
0
on (0, l/4].
(IO)
BY (9)s x(0) = x’( l/4) = 0.
(11)
Conversely, equality (11) for any solution x(r) implies identity (9). So, a solution x(r) of the desired type coincides on (0, l/4 3 with a positive solution of the boundary problem given by equations (8) and (11). Denote by Ai and xi = (xii,. . . ,xi,) the eigenvalues and eigenvectors of M-r N [i = 1,. . . ,n; &I Ri+1, (MXi, x{) = 13. Suppose that for Li > 0 the values T;11’2/2~ are not integers; then the linear boundary problem Mx” f T’Nx = 0,
x(0) = x’(1/4) = 0
(12)
has only the trivial solution. So, there exists a Green matrix T(t, a, T) = ITit(r, u, T)If.,= 1 by the use of which the solution x(7) of (8) and (11) can be written as foflows: x = G( T)x,
s l/4
[G(T)x](T)
= -T*
UT, u, W4x(u)l
du.
(13)
as E +O,
(14)
0
By definition,
T(7,
u,
T) satisfies (12) for t # t and
lim(Y(r
+ E, T, T) - T’(r - E,r, T)) = M-l
where M-l is the inverse matrix. It is easy to check that rik
=
“:I
~“kX~i~“(7)~“(~)
for
7 I
u
rik
=
“$I
xvkxvipv(~)qv(7)
for
7 2
u,
(W
where P&f
= -
P*(s) = -
sin(o,sT) w T
Y
sinh(w,sT) o T
Y
1
4*(s) = 9
44s) =
cos[ 0, T( l/4 - s)]
cosw,T/4
’
cosh[o,T( l/4 -s)] cosh(w, T/4)
P”(S) =’ -s,
q”(s) = 1
’
co,=&
ifA,>O.
w, = &j
if I., < 0.
if R, = 0.
In fact, expression (15) satisfies (12) and lim(T;k(r + &,t, T) - &(r
- &,t, T)) = i x,kx,i V=l
as E 4 0.
(16)
Since MXTX = I (X = [XT, . . . , XT], T denotes transpose, I is the identity matrix) then X’X = M-‘. By a direct calculation, one gets that an element rik of the matrix X*X is equal to the right-hand side of equality (16). Thus, the fact that (14) is satisfied proves the correctness of (15).
~r~l~u~u? ue stu~i3J
Suo~ln[Os
aA!g!sod jo *f# las aql ‘illewqla
s! s uo$a~
sy ‘([z’&
k%.IU.IaJ
‘or]) 'f3x uoyl~os au0 peal II2sey *f) = 3 qI!M ~fl)Uo?l~nba ‘(*J!‘_Jf;3_&~~0s 3oj ‘0sf(7 ‘(*A )J)A # (7 ‘( _J.)d)L uaql _A = & w 7 3 x suognlos ou aJt? a~aql j! ‘snqL ‘(C&E uia~oaql ‘or]) 0 = (7 ‘(.J;)J)A ‘a~oja~aql :o > tf sagdur! 7 3 x JeaU *J 3~110s JOJ Uaql‘+_l;cl SE [t/1 ‘o)uo oa+-(A %'L)J aixJ!s
ql!M(IZ) Alyznba
+J
'I = IIOqII ‘)II3Oq ‘Oqd -I" x(J)*3
(IZ)
= X
uoyenba ICITE ~=oaq~ ‘013) I = (7 ‘(_,~)d)~ '~~=~l~(~~)JOUO~~~~oS~S~XUa~l~3XawOSJOJ~ 4p=baaql‘os ~0 anfehuaSl!a
aql ~ap!suos sn 137
u9q1 7 3 x II~ ~03 I > y JI 2 =L =yjI'] 5’(sa!ldLu!xy=x(__t)*f)
pue
!o r7 x JOJ x(-.1)+5 5 xf_~I*f) ‘69 01 ana %[~Iz WaJoaql ‘I I])(_.A)).~ w%lal ayl s! I = y ‘awaq !ro$eJado an!l!sod-011 e s! (_d),g ‘(0~) 68 =loi~a~ rllguapf ayl s! 01 aJaqM ‘CtJ/I ‘03 Uo
(05)
OI4
5 i~)Cx(_..z)+!?,f
5 Oila
leql q3ns 0 < gi pue 0 < 13 siueisuo3
ale aJay
‘0s ‘0 f (z)x
‘OZ tl)x~u~JoJo = (O)LX(_J)+91‘CfdI‘o)uoo e (l)CX(_J)+91 uaq1‘0 = (4 ‘n‘0k.I JOJ sp[oq (81) a3u!s .(z) +x uo~l~un~ aA$sod aql 01 %u!puodsaJ.toci pus -45 =I SnonuyIo:, AlalaldtuoD aqi JO anlI?AUafa!a III? s! 1 = y ‘(61) JO Ma!n. UI
( _J) +g Jowado
0 &I)
‘“P(fl)W
- d)(_.Jc
“2 ‘1)&l
V/Ii
‘x(-L)+!I uojwnba ‘7 uo
pray loi3aA aql jo uoye3o.i aql(7
,(_.L)-
aA!l!sod
s! (d)*f)
= (1)CX(_L)+!Il
= x
aql saysyas x(&E)*9
‘(d)d)C,
-
(J;I;O)U!S
‘3 u! s JO hepunoq
a.y%auuou h?
la?
/(lalaldmoD s! (~)*r,
hpuodsaJlo3
aql (~)*g
ql!M %U!p!DU!o:, UO!]DUnJ snonu!luo~ aql uo sa!IaJ uraJoaql
.auo aA! lowado
put? ‘0 z x
I? 2q (X)*J 127
aq$ JO JooJd
aq& $IoAd
qI!M 0 %U!l%UUO3 q3U"'CjSnOnU!lUOS ESWOJ
Jo 8 las aql uaq_L *w/u2
‘n ‘1)~
3~1 se u x = 7
JO au03 r! aq y
SULIOJSU~J~ (J )*g -a-! ‘(+J '_J]~3~;
‘(81) pue (5) Cg ‘3 u! snonuyo:, PUE u U! (X)J
fo:oi] Splay JOWA Jo UO!lBlOJ JO hOaql
Sanfett ~u~puodsaJ~o~ aqL*Q ahye%au-uou
jo sJowhwa%!a
‘U‘ ’ ’ ’ ‘1 = J/‘! ‘[b/I
ix = (I) +x uo!~~un~ aye,
x = xLL).5f
icq alouaa
‘0 %up~yuo:, J tq ws uado ue s ‘suoyunJ
-l!sod e olu! (1)x uo!lc~un~ [v//r ‘0) uo aA@au-uou Joj
aql ‘SnOnUyIO3 s! (.,! 'n'2)~ aDu!S ‘($1) Jowado IlE ‘oJ (5) %U!@!lI?S
-(,J '_J]3d &I) jo suoyqos
(8f)says!]esput? s$s!xa (l -dr_H (81)
= _J
a2aqM ‘( +_z ‘_J: J 3J; 301
ti awes .toj +f < :y put? ‘0 < ;x pua sanlcAuafI!a aql aq (2x
asoddns
‘1 uraroay~
= 2x pue ;y 137
. . “2~)
‘0) 3 n ‘1 JOJ 0 > (J; ‘n ‘I)J!J
‘0 > (L ‘11‘O)Q ‘aJoja3aql
‘puv 1+J: -J
J
(11) * uoyyos
any&au-uou
au10s .ro~fg 3 (*1)x pue ‘[*/I ‘013 r JOJ f 3 (1)x ieqi qctns (EI)JO auo lseaI it? sis!xa aIaql0
pug a~ ‘{[@/I ‘013 q3ueJq
1 foj $3 (1)~ :(1)x)
snonupuo2
a,t!le8au-uou
&~!u!eluos
0 3 6 uo!%ar uado Ithaca 30~ $ayl
= 7 %u!soddns ‘wojaJaq_t
‘se I@M 0 %uyauuo:,
IS SWJOJ 8 ‘suoy~puo~ amos Japun ‘layi Moqs 1p2qs aM *(fI)jo
suoynlos
Jo las I? s! 8 ‘([b//r ‘013 1 JOJ u 3 (2)x :( 2)~) = s asoddns aM uo MOU rn0.I~ *[or] hldwa-uou
haAaJ0
hpunoq
~IJOJ 01 p+ suopDunj
aql qI!M ~30 uo!lDaualu!
s! 8 $2~ v ‘0 &~!u!eluo~
s! 0 %.~!uyuos
U3ClNn
s z> 7 ias undo
aqlJ! se ~I!M 0 %u!uauuoD qwwq
3 u! las uado 1.112s pue ‘(( I)“x‘
snonu!iuoD
~0133~ [V/I ‘01 uo NOI.LV2i3ClISN03
JO
acwds
SNOIlfllOS
q3eueg
3HL 30
e
snonu!iuol,
‘t!
. . . ‘( L)!x) = (2)x 3
aiouap
23N31SIXx
ial
‘P
Periodic oscillations in classical autonomous Hamiltonian systems
285
long branch (i.e. intersection of B, with the boundary of any region in E containing 0 is non-empty). Since G = G, in f2, the set B of non-negative solutions of (13) coincides with B, in S. Thus, f3 forms a continuous branch connecting 0 with 8s. As seen from the proof, the period of any such solution TE CT-, T+ ). The theorem is proved. Remark. For T < T+ the rth item in (15) diminishes on [0, l/4). Hence, r’(r, u, T) < 0
and x‘(7) >
0
on [O, l/4)
(22)
if TEI:T’,T+),(T’> T-). Thus, the corresponding solution x(t) with a period Tf CT’, T+ ) changes monotonically between the extreme values xi( - 1/4T) = -Ai and xi( 1/4T) = Ar, like the solution x?(t) of the linearized system. Inequality (22) holds for all positive solutions (i.e. T’ = T-) if eik(x)
20
= y
forx20,xoR;i,k=
l,...,
n.
In fact, by differentiating (8), we obtain that v(r) = x’(r) satisfies the equation Mv” + T=Nv = -T’Y(x)v, y(X) =
(24)
l$ik(x)l%&=l*
Taking into account that v’(0) = v( l/4) = 0, the corresponding written as follows: v = G,( T)v, W G,(T)v = -T* I
integral equation can be
1-(T/4 - T, T/4 - u, T)Y(x~~))v(u)d~.
(25)
0
By (18) and (23). the operator G,(T) for TE [T-, T+) is positive. So, one can analogously prove that the set of positive solutions of (25) forms in E a continuous branch connecting 0 with as, where S = (v(r):
*v(u)du ER I
for
7E
[O, l/4]).
0
Clearly, the solutions of (13) X(T) =
v(u)du
satisfy (22) and form a continuous branch connecting 0 with aS. The Lyapunov family xj(rT, s) E B for small s. By supposition, xj’ is the only signconstant mode of the linearized system; so, the branch B near 0 consists of solutions xj(7T, s) only. Suppose now that, in addition to the conditions of theorem 1, C(x)>A(x)
forx>O,xoiZ,
(26)
i.e. (C(x)y, y) > (A(x)y, y) for all y # 0. Theorem 2. The Lyapunov family xj (t* s) is uniquely prolongable in s to the boundary of the region Q i.e. Xj(t, s) E R for s E (0, s * ), and Xj(t, s*) E dn for some r, s *. The solutions Xj(t, s) satisfy (3) and (4), the period q(s) increasing monotonically.
Prooj Let us show that q(s) > TT for small s. Denote by J,(s) (A, < &+t) the positive eigenvalues of the problem Mx” + I.C(r, s)x = 0, x(0) = x’( l/4) = 0,
C(T, s) = C(Xj(ZT, s)).
(27)
A. A. ZEVIN
286
By (6) and (8), T;(s) is one of the eigenvalues, xj ( ST, s) being the associated eige~unction. In view of (26), for x > 0, k > 0, kx E R iK( kx)
T=;[;
(28)
j;A(ux)du]=+(kx)-C(kx)]cO.
Hence, C(0) = A(0) > C(x), and C(r, s) < A(0). If C(7, s) > 0, the eigenvalues J.,(C) increase with a decrease in C [ 121. As is obvious from the proof of this assertion, the condition C > 0 can be substituted for 114
E=
(C(sfx(zXx(r))dt
s0
> 0,
where x(7) is any eigenfunction of (27). BY (271, l/A E=f (Mx’(7), x’(7))d7 - f(Mx’(r), *I 0
(2%
~(r))lA’~.
(30)
Since x(0) = x’( l/4) = 0, the term outside the integral equals 0; so, E > 0 (by supposition, M >O).
The eigenvalues 1.,(O) are formed from the quantities (2nr/o~)z, i = 1,. . . ,n, 1,2,. . . . Let Tj’* = J+,(O);this eigenvalue is simple because c&‘/w: are not integers. So, T;(s) = Ai for small s; therefore, Tjfs) increases in s (C(r, s) < C(7,O) = A(O)). So, for small T - T;, there exists a solution x(7, T) of the boundary problem given by equations (8) and (1 t) such that x(7, q(s)) = xj(7T, s). According to the perturbation theory, x(7, T) has a unique continuation in the parameter T if the linearized problem r =
My” + T’A(r,
Y(0) = Y’(l/4) = 0,
T)y = 0,
A(7, T) = A(x(7,T))
(31)
has only the trivial solution. Let us consider self-conjugate boundary value problem My” + T*Ny + i.T*R(r)y y(0) = y’( l/4) = 0,
The corresponding
R(7) =
Irikf~)llk=
19
= 0, R(s) = RT(7) > 0.
(32)
integral equation is “Y = GdT)y, I/4
CGR(T)YI(~) = -T*
I
r(7, ~9T)R(u)y(u)dn,
Y= l/i_.
(33)
0
Since X(T, T) satisfies My” -t C(r, T)y = 0,
C(T, T) = C(x(7, T)),
(34)
problems (32) and (33) at R(r) = C(7, T) - N have eigenvalues J = 1 and v = 1. The operator GR( T) is 710-positive and completely continuous for TE [T”, T' ); so, the eigenvalues v = 1 and R = 1 associated with the positive eigenfunction x(t, Tf (Xj(r, s) > 0 on (0, T/4) for small s) are, respectively, the largest and the smallest eigenvalue of problems (33) and (32) at R(7) = C(7) - N. Let R(t) = A(r, T) - N. By (261, A(7, T) < C(7, 7’); SO, A,(A) > i.:(C) [12]. Hence, Ai > 1. Therefore, problem (31) has only the trivial solution that ensures the uniqueness of the continuation of a positive solution x(5, T) in T. The inequality x(7, T) > 0 cannot break under continuation. Indeed, otherwise x(7, Tl) >Oon(O, 1/43,x,(ri, T,) = Oorx;(O, T,) = Oforsome k, Ti,sl ~(0, 1/4].But,by (lf!), [Gx](r) > 0 on (0, l/4], and [ Gx J’(0) > 0 for any x(7) 2 0, x( 7) + 0. Thus, x(7, T) > 0 and ~‘(0, T) > 0; so, x(s, T) is continuable in T. Since x3 is the only sign-constant mode of (2), x(7, T) # 0 for T > T/O. Hence, x(r,,T*)~dQ for some 7i, T* c Ti+ [r(r,u,T)+a;, as T+Tjc]. Let T(s) be any
Periodic oscilfations in classical autonomous Hamiltonian systems
287
monotonic fun~ioR coinciding with q(s) for smali s, Z’(s*) = T*. Evidently, the function x(r/~~s), T(s)) is the desired one-parameter family xi(r, s). The theorem is proved. Remark. Provided C(x)
forx>O,xEQ
(35)
one can analogously show that in problem (32) rl I < 1 at R = A - N; so, xj(t, s) is unique, and q(s) decreases monotonically until A,(s) > 1. Let 0 < .4(x) < A + for x c !& x > 0; denote by w: the eigenfrequencies of the system M% + A+x = 0. It has been shown in ref. [13] that under (35) and regardless of (5), the inequality AZ > I is true if ~~~~~/~~,~~/~~I,
3w,O>o:,
i#j,m=l,3,5
,...
(36)
Note that for Hamiltonian systems inequality (26) or (35) (where A(x) = H,,(x), H(x) is a Hamiltonian), together with conditions analogous to (36), ensures the uniqueness of x,(r, s) and monotonicity of q(s) regardless of the behaviour of Xj( t, s) 113). Van Groesen found the monotonicity conditions for the minimal period Z’(E) of system (1) (E is the total energy) [14]. In particular, T(E) increases if (,4(x)x, x) < (C(x)x, x); evidently, this condition is weaker than (26).
5. DISCUSSION
AND EXAMPLES
Elements (7) of N and P depend on region ft. If Q retracts to the equilib~um point, N,P+A(O), TJ&T;-+T, O. So, there always is a sufficiently small R for which theorem 1 holds. It is necessary to emphasize that, in view of the inequality x7 > 0, the analysis of another branch of solutions requires the change of directions of some coordinates; as a result, in the general case the matrices N and P will be different for different families. Inequality (5) and, therefore, matrices N and P are invariant with respect to the mentioned change of coordinates if
v(X) = t
Y(Xi) + (CO%X)9
l$(Xi)
= I$(-Xi).
(37)
i=l
Note that, in particular, the potential energy of n non-linear oscillators connected to each other by linear strings is of the form (37). Let nixi
S.&(xi)
d U-d
-
=
dx
I;
for Xi > 0, XiE a.
PiXi
(38)
i
Then one can put for all branches N=COfdiag(nl
,...,
n,),
P=Cc+diag(p,
,...,
p,).
(39)
If &(-Xi)z~JO)X: or ,$(xf)S&(O)Xi (i = 1,. . . , tt, _& = d_#dx) then, respectively, N = A(O), Wi = W: or P = A(O), O: = 0,.9 In the first case the rigidities of a non-linear system exceed that of the linearized one; here condition (17) certainly holds (k = j). If .fIXXi)/Xi
>_&.z(xih
i =
1,.
. * t f3
(401
then the inequality C(x) > A(x) holds. Let in (1) M = diag(mi,. . . ,mJ,
fi’(x)TOforxZO,i=
l,...,
n,f(x)#Oforx#O.
(41)
for t 2 U.
(42)
Then one can take N = 0; the Green matrix is I = diag(F,(s, Ti = -T/Rli
for T S u,
u),. . . ,T;((z, u)), Ii = -u/m2
A. A. ZEVIN
288
Here the operator G(T) is positive for all T.By (41), for x(5) EL some component G( T)x+x! as T-tax So,equality (21) implies p c 0; therefore, y(F( T),L) = 0 for large T. On the other hand, for small T the equality G( T)x = lxyields I < 1 (G(T)x+ 0 as T + 0); so,here y(F( T),L)= 1. Thus, there is a solution on L,sounder condition (41), the set of positive solutions forms a continuous infinitely long branch. Note that this result was obtained using different arguments by Rabinovitz [IS] and (for a less general class of f(x)) by Duffin [16]. The existence of an infinitely long continuous branch of non-negative on (0, T/2) periodic solutions was proved by Krasnoselskii [l 11. Some properties of the solutions under condition (26) or (35) were established in ref. [17], where non-autonomous systems were also treated. Since T’(r, u) I 0, the solution x(t) increases monotonically on [0, l/4] (that in a common case does not rule out the identity Xi(T) z 0 for some i). Condition (41) can be satisfied, in particular, for a system of oscillators coupled successively by elastic constraints and for a string or a beam carrying lumped masses [16, 173. To comply with (41), positive xi values should correspond to opposite displacements of adjacent masses. Thus, the solution in question describes such oscillations that adjacent masses always move in opposite directions. For small amplitudes they belong to the Lyapunov family associated with the highest natural frequency (positive eigenfunction x$’of non-negative, by (41), matrix M-’ A(0) corresponds to the largest eigenvalue [18]). To illustrate an application of the results obtained, let us consider a system of two non-linear oscillators connected by a linear string:
mlf, +/l(xl)+c(xI -x,)=0,
m,i2 +fz(x2)+c(xI-xl)= 0.
(43)
Let fliXi Ij(Xi) 5 PiXi for Xi E R, Xi > 0 (i= 1,2). In the first mode of the linearized system x7 > 0, i.e. both masses move in the same direction. Let us obtain conditions that guarantee the existence of a continuous branch of such solutions in R. Here
Suppose N > 0; then in (15) w:, o: are the eigenvalues of the matrix M- ’ N, x1,x2 are the corresponding eigenvectors. The condition T(T, u, T) < 0 on (0, l/4) can be considerably simplified. Suppose 02/w1 < 3; then cos(o,T/4)> 0,cos(wzT/4)< 0 for TE(~~c/~I~,~x/w~). So,one can easily check that rl 1(~, u, T) < 0 on (0, l/4) if r\ ,(O, u, T) I 0 and r( l/4, u, T) I 0.These inequalities hold if, respectively, r;,(O,
l/4, T) 5 0 and
ar,
l(
l/4, u, T)
au
IO, u=o
which implies x:~ cos(o, T/4) + x:~ cos(o, T/4) I 0.
(44)
Analogous considerations give the conditions ensuring the inequalities rz2(r, u, T) < 0 and rlZ(Tru, T)= rzl(T, U, T)
+ X:~COS(O~T/4) I 0. + 02x21x22tan(wzT/4)
2 0.
(45)
Let w: be the first eigenfrequency of the system Mfi+ Px = 0,T; = 2x/w:. If inequalities (44) and (45) hold for TE [T;, T:) (T: = 2x/wl) then, by theorem 1, for any region 9 E R containing 0, there is a solution x(t) E 9 satisfying (3) and (4) and lying on 89 for some t. If dJ(xi)/dxi 2 ni (i = 1,2) then, in accordance with the remark to theorem 1, x1(t),x2(t) vary monotonically between the extreme values Xi(- T/4) = - Ai and Xi(T/4) = Ai. If w2/w1 I 2, in both the items in the second inequality (45) are positive; so, it holds trivially. Evidently, the rest of the inequalities hold for T E [T;, T:)provided they are true for T= T;. Let m, = m2 = m and n, = n2 = n; then xl1 = xr2, xzl = -x22 and (44) and
Periodic oscillations
in classical auronomous
Hamiltonian
systems
289
(45) imply
1 w: 5 +
+ 02) =
(46)
This condition can be expressed by means of the elements of the matrix P. In particular, so, (46) yields let Pr = p2 = p. Then of = fi; :+1
+J_I?
(47)
where c/n 5 1.5 in view of the condition used, wJol 5 2. The values p, n depend on the region R. If nx
REFERENCES
1. N. Desolneux-Moilis, 2. 3. 4. 5. 6. 7. 8. 9. IO. Il. 12.
13. 14. 15.
Orbites periodiques des systems Hamiltonians autonomes. Lecture Notes Math. 842, 156-173 (1981). P. H. Rabinowitz. Periodic solutions of Hamiltonian systems: a survey. SIAM J. Morh. Anal. 13(3), 343-352 (1982). E. Zehnder, Periodic solutions of Hamiltonian equations. Lecture Notes Moth. 1031, 172-213 (1983). R. M. Rosenberg, The normal modes of nonlinear n-degree-of-freedom systems. J. appl. Mrch. 29.7-14 (1962). R. M. Rosenberg, On nonlinear vibrations of systems with many degrees of freedom. Ada. appl. Mech. 9, 156-242 ( 1966). C. H. Pak and R. M. Rosenberg, On the existence of normal mode vibrations in nonlinear systems. Q. appl. Math. 26. 403416 (1968). A. Weinstein, Normal modes for nonlinear Hamiltonian systems. Inuenrions Math. 20. 47-57 (1973). E. W. C. van Groesen, On normal modes in classical Hamiltonian systems. hr. J. Non-Lineor Mech. l&55-70 (1983). A. Lyapunov, Probleme generale de la stabilite du mouvement. Ann. Fus. Sci. Toulouse. 2, 203474 (1907). M. A. Krasnoselskii and P. P. Zabreiko, Geomerricol Methods of Nonlinear Analysis. Springer, Berlin (1984). M. A. Krasnoselskii. Posiriue Solutions o/Operator Equorions. Noordhotf, Groningen. The Netherlands (1968). M. G. Krein. Principles of the theory of I-zones of stability of the canonical system of linear differential eouations with neriodic coefficients in: In memory_ of _ A. A. Andronou. Izd. AN SSSR, Moscow (1955). (in Russian). . A. A. Zevin. Anisochronicitv conditions of autonomous Hamiltonian systems. Me&. Solids (Mekh. Tcer. Tekr) 23(I), 33-39 (1988). E. van Groesen. Duality between period and energy of certain periodic Hamiltonian motions. J. London Marh. Sot. 34(2). 435-448 (1986). P. H. Rabinowitr A note on a theorem of R. Duffin. Arch. Rarional Mech. Anal. 93(l). 91-102 (1986).
290
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16. R. J. Duffin. Vibration of a beaded string analyzed topologically. Arch. Rational Mech. Anal. 56(J), 287-293 (1974). 17. A. A. Zevin. Existence, stability and some properties of the class of periodic motions in nonlinear mechanical systems. Much. Solids (Mekh. Tuer. Bela) B(4). 42-51 (I985). 18. R. Bellman, J~t~oducfio~ ru Marrix Analysis. M~raw-Hill, New York (1960).