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On the stability of the equilibrium of conservative systems with an infinite number of degrees of freedom
ON TBE STABILITY OF TBE EQUILIBRIUM OF CONSERVATIVESYSTEMS WITH AN INFINITE NUMBER OF DEGREES OF FREEDOM (OB USTOICUIVOSTI BAVNOVEBIIA EONSEUVATIVNYKU SISTBM S BEZKONECliNYllCBISLOM STBPENEI SVOBODY) PW
Yo1.26, No.2, 1962, pp. 356-358 A. Y. SLOBODISIN (Yoscon) (Received
Saptcrbrr
18,
1961)
BY means of examples involving some simple systems with an infinite number of degrees of freedom, an analysis is made of the possibility of the existence of a direct method of proof of stability with the aid of
Lia9unov'sfunctionalof the incrementof the total energy of a system. The direct method of proof is used to establish the stability of equilibrium
defined
in the linear
theory
of elasticity.
1. The basis of the direct proof of the classical theorem on the stability of the equilibrium of a conservative system with a rinite number of degrees of freedom 111, is the following property, of a COBtinuous function of a finite number of variables: the lower boundary, of the difference between the values of a function on a sphere of sufficiently small radius with center at a point of a strict relative minimum, and the minimum value, is positive. This property is the essential point in the proof of the general theorem on stability [2,31. In somewhat stronger form, it plays a role in the theory of the direct method of Liapunov c41 (definition (4.,1)). In order to avoid the introduction of a new terminology, we will call this property the definite positiveness of the increment of the function. When one goes over from discrete to continuous systems, which corresponds to the shift from functions to functionals, then there arise the following questions: (1) For what metrics of the function space is the property of the definite positiveness of the increment of the functional preserved? (2) To what extent does the energy integral give a complete picture of the stability as compared to that given by the results that follow froa the direct solution of Cauchy’s problem?
513
A.M. Slobodkin
514
2, We will consider the problem on the stability of the equilibrium of a uniform free string stretched between two fixed points. The plane motion is described by the equations =
Wt
TIN,
(OGz
u (0,
t&o),
t)
=
u
0,
(1,
t)
=o
(2.1)
Here p is the linear density, and T, is the tension of the string. The zero initial conditions correspond to the equilibrium (2.2)
u (I, t) ZZ0 The total
energy 1
1 H
=
$
pq
dx
+
;
’
s
0 remains constant
Tlu,” dx
(2.%j
s 0
and is coctinuous
along the motion,
with respect
to the
metric
. s
Because of the inequalities 2
1
c
uxadx>
UPdx,
0”
the
total
energy
1
s
f
u,” dx >
0
is positive
(2.5)
0
definite
with respect
to the metrics
pZ,
and P? Pa=suP,luI This implies [4,51 the stability spect to the metrics pl, p2 and pl, As is
known [41,
the
Solution
of
the
(ogz
equilibrium
(2.6)
(2.2)
with re-
is given
bs the
ps*
of Cauchy’s
problem
formula u (2, f) = +
11
(x -
ot) +
UIJ
(a=j/-)
(x + at) + ; ;];:,,,,,]
(2.7)
where u. and v. are functions obtained by an odd, period 22) to the entire real axis of the initial tions and velocities of the points of the string.
periodic extension (of values of the devia-
From (2.7) one can easily with respect to the metrics
of equilibrium
obtain the p4 and p3
P4=auP,/~I+sup*Iu~I
stability
(o
(2.2)
t2.8)
The
and also
stability
of
the stability
the
of
equilibriur
(2.2)
of
with respect
The total energy H is not positive pI or with respect to the “statistical”
conservative
515
systems
to PI.
definite metric
with respect
to the metric
(O
P,*=suP,laI+suP,Iu,I
This example shows that the energy integral gives picture of the equilibrium stability of a system.
only a partial
3. If the potential energy of the system is a functional of a function of one variable. then the fulfillment of the hypotheses of Osgood’s II71 theorem (which are somewhat stronger than those that are sufficient for the existence a strong minimum of the potential energy at the position of equilibrium) will guarantee the definite positiveness, with respect to the metric pJ, of the increment of the total energy of the system, and, hence, the equilibrium stability with respect to the metrics PI, pJ. It was in this manner that Kneser proved the equilibrium stability of a string fastened at both ends in a gravitational field [8]. Born [9] used analogous arguments in his study of the equilibrium stability of rods. But for systems, whose than one variable, and on one cannot expect [lo. ll] ment for a metric of type
potential energy depends on functions of more their derivatives of order not greater than one, the positive definiteness of the energy increp8.
Let us consider, for example. the equilibrium stability of a membrane, clamped along the periphery, under the action of a constant transverse load. The increment of the potential energy of the membrane has the form (3.1)
if one takes account of the membrane’s deviation from the position equilibrium. Here, T2 is the stress in the membrane. Hadamard [IO] has shown that this relative to the metric
functional
Ps = sup,,, 1u 1. lowever,
an analog
of the first
is not positive
of
definite
(2, Y) ED
inequality
of
(3.2)
(2.5)
is valid
Lll]: (3.3)
516
A.M.
Slobodkin
here C1 is a fixed positive constant of this, we can draw certain conclusions with respect to the metrics pgs p7:
independent of u. On the basis on the equilibrium stability
Ps = sup,*v I u I + SUP,,I/ I % I + sup,&! I ug I + sup,,,
I UtI,
6%Y)E D
(3.4)
P? = { \\ (ua + u,” + nva+ “t’) dz:dy)“’
(3.5)
4. We consider next the question on the equilibrium stability in a field of the mass forces (constant in time) of an elastic body in the linear theory of elasticity. Suppose that on the part a’ of the surface of a body there is given a displacement u”, independent of time n = Ilo(z) on the part
a’:
the tension
(wE 0’)
(4.1)
(ZEc”)
(4.2)
is tno t, = tn” (z)
while
on the part a”‘,
the contact
condition
‘L, = uno(I),
is
t, = 0
The case when one of the parts from consideration.
(XE c”‘)
au and a”’
(4.3)
is absent,
is not excluded
The increment of the potential energy of a body in the investigation of its displacement from the position of equilibrium, has the form V=f
s
Cijklui, j ‘k.1
do
0
u=o
t,=o
(XEQ’),
u, = 0,
(ZE5”);
s
u.1.1. u.1.j dm),Cz
\ ;,
CO s
‘ijkl
ui,j
uk.l
do,,&
s 0
UiBj
where Cq and Cg are fixed positive implies the equilibrium stability PS = suPx
Ui
5j do
(4.4)
(ZG G”)
is the tensor of the elastic constants. The repeated summation from 1 to 3. The inequality [I31
Here, c.. dices indii:,“e’
0
t, = 0
in-
(4.5)
uiutda
(4.6)
(Kern’ s inequality)
constants that do not depend on u, with respect to the metrics ps, p9:
I u I + suPx v"isjui;j
+
suPz I u( I
(XEW)
(4.7)
The stability
PO=
Is
of
tua+
equilibrium
the
%.j
of
conservative
systcas
517
f4.6)
“i,j + ur8)do
a
1.
Lejeune-Dirichlet, P. G., 9b ustoichivosti ravnovesiia (On the stability of equilibrium). (Russian translation) Lagrange Journal. Rnalitichtskaia ~ekhanika (Anaiytic lechanics). Vol. 1, Appendix 11, GTTI, 1950.
2.
Liapunov,
A. 1. , ~bshchaia Problem
(General
dvitheniia
GTTI, 1950.
of Motion).
3.
Chetaev, 1955.
4.
Movchan. A. A., of processes
5.
Movchan, A. A., Ob ustoichivosti dvizhenfia sploshnykh tel. Teorema Lagranzha i ee obrashcbenie (On the stability of motion of solid bodies. Theorem of Lagrange and its inversion). Inzh. sb. Vol 29, 1960.
6.
Sobolev, eatical
N.Q.,
ob ustoichivosti
zadacha
on Stability
Ustoichivost’
(Stability
dvizheniia
Ustoichfvoat’ with respect
of Motion).
GTTX,
protsessov po dvum metrikam (Stability to two metrics). PMMVol. 24. X0.6, 1960.
natenaticheskoi S. L. , Uravneniia Physics). GTTI, 1947.
(Equations
fiziki
of Mathc-
7.
ischisLavrent’ ev. M.A. and Liusternik, L.A., Osnovy variatsionnogo leniia (Foundations of Variational Calculus). Vol. 1, Part II. ONTI, 1935.
8.
des Gleichgewichts Kneser, A., Die StabilitXt FGdeu. Journal fur die reine und angeoandt Heft 3, 1903.
Mathenatik
Born, M., ~ntersuchungen
der
9.
in Ebene
iiber
Stabilitiit
und Aaum. Dissertation.
10. Hadamard, I, , Sur quelgues Annafes
die
scientifiques
de
questions l’icole
h’a;ngender scbwerer Bd. 125.
Gidttingen, de calcul normale
efastischen
Linit
1906. des variations. tome 24, 1907.
superieurc
delle variazioni 11. Fubini, G., 11 teoreea di Osgood nel calcolo integrali multipli. R.C. Accad. Lincei Vol. 19, 1910.
degli
A.M. Slobodkin
518
12. Courant, of
R. and Hilbert,
Mathematical
Physics).
D.,
Metody
(Russiar
natenaticheskoi
translation).
fiziki
Vol.
(Methods
2, Chap. 6.
GTTI, 1945. funktsionala (The 13. Mikhlin, S.G., Problema minimuma kvadratichnogo Problem of the minimum of a Quadratic Functional). GTTI, 1952.
Trans
lated
by H.
P. T.
Yo1.26, No.2, 1962, pp. 356-358 A. Y. SLOBODISIN (Yoscon) (Received
Saptcrbrr
18,
1961)
BY means of examples involving some simple systems with an infinite number of degrees of freedom, an analysis is made of the possibility of the existence of a direct method of proof of stability with the aid of
Lia9unov'sfunctionalof the incrementof the total energy of a system. The direct method of proof is used to establish the stability of equilibrium
defined
in the linear
theory
of elasticity.
1. The basis of the direct proof of the classical theorem on the stability of the equilibrium of a conservative system with a rinite number of degrees of freedom 111, is the following property, of a COBtinuous function of a finite number of variables: the lower boundary, of the difference between the values of a function on a sphere of sufficiently small radius with center at a point of a strict relative minimum, and the minimum value, is positive. This property is the essential point in the proof of the general theorem on stability [2,31. In somewhat stronger form, it plays a role in the theory of the direct method of Liapunov c41 (definition (4.,1)). In order to avoid the introduction of a new terminology, we will call this property the definite positiveness of the increment of the function. When one goes over from discrete to continuous systems, which corresponds to the shift from functions to functionals, then there arise the following questions: (1) For what metrics of the function space is the property of the definite positiveness of the increment of the functional preserved? (2) To what extent does the energy integral give a complete picture of the stability as compared to that given by the results that follow froa the direct solution of Cauchy’s problem?
513
A.M. Slobodkin
514
2, We will consider the problem on the stability of the equilibrium of a uniform free string stretched between two fixed points. The plane motion is described by the equations =
Wt
TIN,
(OGz
u (0,
t&o),
t)
=
u
0,
(1,
t)
=o
(2.1)
Here p is the linear density, and T, is the tension of the string. The zero initial conditions correspond to the equilibrium (2.2)
u (I, t) ZZ0 The total
energy 1
1 H
=
$
pq
dx
+
;
’
s
0 remains constant
Tlu,” dx
(2.%j
s 0
and is coctinuous
along the motion,
with respect
to the
metric
. s
Because of the inequalities 2
1
c
uxadx>
UPdx,
0”
the
total
energy
1
s
f
u,” dx >
0
is positive
(2.5)
0
definite
with respect
to the metrics
pZ,
and P? Pa=suP,luI This implies [4,51 the stability spect to the metrics pl, p2 and pl, As is
known [41,
the
Solution
of
the
(ogz
equilibrium
(2.6)
(2.2)
with re-
is given
bs the
ps*
of Cauchy’s
problem
formula u (2, f) = +
11
(x -
ot) +
UIJ
(a=j/-)
(x + at) + ; ;];:,,,,,]
(2.7)
where u. and v. are functions obtained by an odd, period 22) to the entire real axis of the initial tions and velocities of the points of the string.
periodic extension (of values of the devia-
From (2.7) one can easily with respect to the metrics
of equilibrium
obtain the p4 and p3
P4=auP,/~I+sup*Iu~I
stability
(o
(2.2)
t2.8)
The
and also
stability
of
the stability
the
of
equilibriur
(2.2)
of
with respect
The total energy H is not positive pI or with respect to the “statistical”
conservative
515
systems
to PI.
definite metric
with respect
to the metric
(O
P,*=suP,laI+suP,Iu,I
This example shows that the energy integral gives picture of the equilibrium stability of a system.
only a partial
3. If the potential energy of the system is a functional of a function of one variable. then the fulfillment of the hypotheses of Osgood’s II71 theorem (which are somewhat stronger than those that are sufficient for the existence a strong minimum of the potential energy at the position of equilibrium) will guarantee the definite positiveness, with respect to the metric pJ, of the increment of the total energy of the system, and, hence, the equilibrium stability with respect to the metrics PI, pJ. It was in this manner that Kneser proved the equilibrium stability of a string fastened at both ends in a gravitational field [8]. Born [9] used analogous arguments in his study of the equilibrium stability of rods. But for systems, whose than one variable, and on one cannot expect [lo. ll] ment for a metric of type
potential energy depends on functions of more their derivatives of order not greater than one, the positive definiteness of the energy increp8.
Let us consider, for example. the equilibrium stability of a membrane, clamped along the periphery, under the action of a constant transverse load. The increment of the potential energy of the membrane has the form (3.1)
if one takes account of the membrane’s deviation from the position equilibrium. Here, T2 is the stress in the membrane. Hadamard [IO] has shown that this relative to the metric
functional
Ps = sup,,, 1u 1. lowever,
an analog
of the first
is not positive
of
definite
(2, Y) ED
inequality
of
(3.2)
(2.5)
is valid
Lll]: (3.3)
516
A.M.
Slobodkin
here C1 is a fixed positive constant of this, we can draw certain conclusions with respect to the metrics pgs p7:
independent of u. On the basis on the equilibrium stability
Ps = sup,*v I u I + SUP,,I/ I % I + sup,&! I ug I + sup,,,
I UtI,
6%Y)E D
(3.4)
P? = { \\ (ua + u,” + nva+ “t’) dz:dy)“’
(3.5)
4. We consider next the question on the equilibrium stability in a field of the mass forces (constant in time) of an elastic body in the linear theory of elasticity. Suppose that on the part a’ of the surface of a body there is given a displacement u”, independent of time n = Ilo(z) on the part
a’:
the tension
(wE 0’)
(4.1)
(ZEc”)
(4.2)
is tno t, = tn” (z)
while
on the part a”‘,
the contact
condition
‘L, = uno(I),
is
t, = 0
The case when one of the parts from consideration.
(XE c”‘)
au and a”’
(4.3)
is absent,
is not excluded
The increment of the potential energy of a body in the investigation of its displacement from the position of equilibrium, has the form V=f
s
Cijklui, j ‘k.1
do
0
u=o
t,=o
(XEQ’),
u, = 0,
(ZE5”);
s
u.1.1. u.1.j dm),Cz
\ ;,
CO s
‘ijkl
ui,j
uk.l
do,,&
s 0
UiBj
where Cq and Cg are fixed positive implies the equilibrium stability PS = suPx
Ui
5j do
(4.4)
(ZG G”)
is the tensor of the elastic constants. The repeated summation from 1 to 3. The inequality [I31
Here, c.. dices indii:,“e’
0
t, = 0
in-
(4.5)
uiutda
(4.6)
(Kern’ s inequality)
constants that do not depend on u, with respect to the metrics ps, p9:
I u I + suPx v"isjui;j
+
suPz I u( I
(XEW)
(4.7)
The stability
PO=
Is
of
tua+
equilibrium
the
%.j
of
conservative
systcas
517
f4.6)
“i,j + ur8)do
a
1.
Lejeune-Dirichlet, P. G., 9b ustoichivosti ravnovesiia (On the stability of equilibrium). (Russian translation) Lagrange Journal. Rnalitichtskaia ~ekhanika (Anaiytic lechanics). Vol. 1, Appendix 11, GTTI, 1950.
2.
Liapunov,
A. 1. , ~bshchaia Problem
(General
dvitheniia
GTTI, 1950.
of Motion).
3.
Chetaev, 1955.
4.
Movchan. A. A., of processes
5.
Movchan, A. A., Ob ustoichivosti dvizhenfia sploshnykh tel. Teorema Lagranzha i ee obrashcbenie (On the stability of motion of solid bodies. Theorem of Lagrange and its inversion). Inzh. sb. Vol 29, 1960.
6.
Sobolev, eatical
N.Q.,
ob ustoichivosti
zadacha
on Stability
Ustoichivost’
(Stability
dvizheniia
Ustoichfvoat’ with respect
of Motion).
GTTX,
protsessov po dvum metrikam (Stability to two metrics). PMMVol. 24. X0.6, 1960.
natenaticheskoi S. L. , Uravneniia Physics). GTTI, 1947.
(Equations
fiziki
of Mathc-
7.
ischisLavrent’ ev. M.A. and Liusternik, L.A., Osnovy variatsionnogo leniia (Foundations of Variational Calculus). Vol. 1, Part II. ONTI, 1935.
8.
des Gleichgewichts Kneser, A., Die StabilitXt FGdeu. Journal fur die reine und angeoandt Heft 3, 1903.
Mathenatik
Born, M., ~ntersuchungen
der
9.
in Ebene
iiber
Stabilitiit
und Aaum. Dissertation.
10. Hadamard, I, , Sur quelgues Annafes
die
scientifiques
de
questions l’icole
h’a;ngender scbwerer Bd. 125.
Gidttingen, de calcul normale
efastischen
Linit
1906. des variations. tome 24, 1907.
superieurc
delle variazioni 11. Fubini, G., 11 teoreea di Osgood nel calcolo integrali multipli. R.C. Accad. Lincei Vol. 19, 1910.
degli
A.M. Slobodkin
518
12. Courant, of
R. and Hilbert,
Mathematical
Physics).
D.,
Metody
(Russiar
natenaticheskoi
translation).
fiziki
Vol.
(Methods
2, Chap. 6.
GTTI, 1945. funktsionala (The 13. Mikhlin, S.G., Problema minimuma kvadratichnogo Problem of the minimum of a Quadratic Functional). GTTI, 1952.
Trans
lated
by H.
P. T.