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Act~ lVIechaniea 50, 163-- 175 (1984) ACTA M E C H A N I C A �9 by Springer-Verlag 1984

An Extremum Variational 1Jriaciple

for Classical tIamiltonian Systems*

By

Dj. S. Djukic, Novi Sad, Yugoslavia, and T. M. Atanackovic, Berlin

(Received .February 1, 1983; revised July 18, 1983)

A variational principle for a mechanical systems with n-degrees of freedom, which is subject to conservative generalized forces, is formulated. I~ecessary conditions for its extremum are given in detail. Two possibilities for ~0onstructing the functional are established. The first one is applicable if the canonical equations have certain simple algebraic properties, the second one is applicable in the general case. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting Hamiltonian description. Finally, the theory is used for obtaining approximate solution of nonlinear mechanical proble m with two-degrees of freedom.

1. Introduction

I t is well known that variational description of mechanical systems with n-degrees of freedom have been subject of many investigations. At the present t ime here are two main problems in this area. The first problem is faced if we want to include in variational description classical nonconservative mechanics with purely noneonservative generalized forces. The second problem is concerning extremali ty of the variational principles.

An extremum principle is one which establishes the equivalence between an equation and the fact tha t some functional attains an extremum value, either a maximum or a minimum. In the Classical mechanics there are few treatments of the extrema] properties .which are different, in details, than those usually given in variational calculus (see for example [1]). One proof of the extremality, [2, p. 650], is valid for a small t ime interval of motion. Another, which is due to Bobylev, [3, p. 504] and [2, p. 653], needs our knowledge of the Cauchy integral of the differential equations of motion. Morse [4] has developed a method for dealing with necessary conditions of extremum of the classical action integrM. He transformed the sufficient condition ~ problem into a boundary value problem. The method is applied [5] to harmonic and anharmonie oscillator in one dimension, and proof is given tha t the action integral is minimum for certain t ime subinterval during the motion.

Dual principles (called complementary when they are extremal) or reciprocal,

* Research supported by the U.S. NSI~, Grant No.Yor 82/062.

I64= Dj. S. Djukic ~nd T. 3/L Atanackovic:

form another class of variational principles, which have received much attention recently, mainly due to the work of Noble and Sewell [6] and [7]. However, the first notion concerning dual principles can be found in [8]. There it is shown that by the so called Friedrieks transformations one can construct complementary Lagrange formalism in momentum space. Further development of the dual principles can be found in [1], [9]--[13]. The idea of dual principles is applied in [14] to harmonic oscillator and the Kepler problem. When a dual principle is available, t hep rob lem is formulated, variationally, in two different but i~ter- related ways. In one a solution is characterized by a maximum principle and in the other by a minimum principle. The maximum and minimum values of the respective funetionals are the same. The importance of dual variational principle in application results frora the fact that the difference between the values of both functionals for two different trial functions can be used as a measure of the accuracy of approximate solution (see for example [15]--[19]). In many cases the significance of dual variational principles is enhanced because the functional involved have by themselves some relevant physical meaning (see [20]). Recently, dual variational principles have served as basis for the development of finite- element techniques for different problems.

Usually variational principles are defined for two-point boundary problems. To derive variational principle for initial value problems it is necessary to introduce the initial conditions into governing equations. A system of integro-differential equations, which contains the initial conditions implicitly and for which variational principles can be derived, are obtained in [21], [22], [23].

In general if the dual variational principles are to become complementary or extremum principles they must satisfy restrictive conditions on their structure and the corresponding two-point boundary conditions. This necessary conditions reduce the number of the boundary value problems, which can be incorporated in the existing theory, and for which error estimate of approximate solution can be given.

In our previous paper [24] we formulated, for a mechanical system of one degree of freedom, a variational principle as difference between primal and dual functionals, calculated on the same trial function. This principle has several important properties: 1) The value of the functional on the exact solution is equal to zero; 2) I t is an extremum principle in the ease when primal and dual principles are extremal, but also in many eases when they are not; 3) its sta- t ionarity and extremal properties do not depend on the boundary conditions ; 4) The principle is applicable to the problems with initial conditions without any adaption.

Our intention is to extend the results of [24:] to general mechanical systems with n degrees of freedom which possess ttamiltonian structure. In the first part we will assume, as it is common in the theory of complementary principles, that we are able to find an explicit solution to a specific algebraic problem. In the second part we will analyze the case when that explicit solution does not exist. After that, influence of an approximate solution to the algebraic problem on the value of the functional will be established. Finally, the functional will be used for finding approximate solution of a nonlinear mechanical problem with two degrees of freedom.

An Extremum Variational Principle for Hamiltonian Systems 165

2. An Extremum Variational Principle for the ffamiltonian Systems

Let us consider a holonomic mechanical system with n-degrees of freedom where the q~ are the generalized coordinates, the p~ are the generalized momenta and t is the time. In this paper the summation convention is adopted and small greek indices imply a range of values from 1 to n. The system is described by a Lagrangian function L(t, q~, ~ ) or by the corresponding Hamiltonian H(t, q", io~) = p , q " - - L Where q:--= dq"/dt and p, ~ ~L/~q~. The governing equations of motion of the mechanical system are the Hamilton canonical equations

~=0__HH L ~ - ~ b ~ @ 0 H = 0 , for a<~t<_b, (1.1,2) Op, ' ~q=

which are subject to arbitrary initial or boundary conditions. The exact solution of the differential equations (1) and the corresponding initial or boundary conditions will define a path in the phase space q:, p,. Variations round the exact path are given by

Q~ = q" q- 8q ~, P~ = p~ q- @,. (2)

The variations are arbitrary, subject only to the limitation that 8q= and dp~ are sufficiently smooth and small. Let us consider the following functional with Q~ and P~ as variables

b /

J(Q, P) = f [P,@ -- H(t, Q, P)] dr, (3) a

where a and b are the initial and the terminal time, respectively. Integration by parts leads to

b

J(Q, P) = (PoQ.): - f [P,Q" q- H(t, Q, P)] dr. (4) a

I-Iere Q= and P , belong to the class C= of functions which have continuous derivatives up to second order for a _< t ~ b and H is assumed to possess continuous second-order derivatives with respect to all its arguments.

In our analysis we shall use two different forms of the functionals (3) and (4).

Form 1: Let us consider (1.1) along the variations Q: and P~ as algebrMe equations and suppose that they could be solved for P, , i.e.

DL P , = v~(t, Q, Q) = --~ (5) OQ=

and that the v, are differentiable functions of all arguments. Now, substituting P: according to (5) into (3) we obtain the following first functional with the Q's as variables

b

E(Q) = f {vo(a, Q, (~) @ - H[t, v(t, Q, Q), Q]} dr. (6) a

I t is obvious from the procedure of constructing the functional that for the cMculation of the first variation (dE) and the second variation (8rE) we can use the functional (3). The needed variations of the generalized momenta are obtained

166

from (1.1) :

where

C~ -- ~H

Dj. 8. Djukic and T. M. Atanaekovic:

(7)

B"~ = o~--H B~ = B~:, B~/~v = (V, det (B"~) # 0, (8) ap~ @p'

and ~y~ are the Kroneeker deltas. Now, the first and second variation of the functional E(Q) become

b

~ E - - - - I p ~ q ' V + f [ ( ~ - - ~ ) ~ P o - - ( ~ , + ~ ) ~ ] d ' , (~) r

and b

a

where ~p, is given by (7) and

A,~- - ~2ft A ~ = A~,, A~A~ ~ = (~ , det (A,~) # 0. (11) ~q~ ~q~ '

We must underline the fact that the functional (6) is the usuM I-Iamilton's action integral for the Eqs. (1) where under the integral sign we have Lagrange's function, i.e.

b

E(Q) = f L(t, Q, Q) dr. (6.1) a

Also, from (5) we can deduce

~2L 0~L ~p~ -- ~ " q- ~q~. (7.1)

If we use (7.1) instead of (7) and follow the previous procedure, or if we use (6.1) to calculate the first and second variations of E, then in the corresponding final results we will have terms eMculated by Langrangian and Hamiltonian function or solely by Lagrangian function. In construction of the second form of the functional (4) and calculation of the appropriate first and second variations use of the canonical variables and Hamiltonian function is essential. To exceed this possible difference in the given Form 1 of the functional (3) and future Form 2 of the functional (4) we did not used (6.1) and (7.1) for our analysis.

Form 2: Let us consider (1.2) along the variations Q~ and P~ as algebraic equations and suppose that they could be solved for the Q~, i.e.

Q~ = ~(t , P, P), (12)

and that the ~b ~ are differentiable functions of all argmnents. The second form of the functional we will obtain in two steps. First we substitute (12) into the inte- grand of (4)

b

G(P,.Q) : (P,Q~)a ~ - f {P,r P, P) + H[t, P, ga(t, P, P)]} dt. (13) a


Then we substitute (5) into this result and obtain the following second funqtional with the Q's as variables

b

G(Q) ---- [v~(t, Q, (2) Q~]a b - a f {~,(t, Q, (2) q~"(t, v(t, Q, Q), ~(t, Q, Q)) (14)

§ H[t, v(t, Q, 0), •(t, v(t, Q, Q), i;(t, Q, Q))]} dt.

Remark. 1: The procedure used for constructing the functional (14) is slightly different from that proposed in [1] for calculating dual variational principles: According to [1] the generalized coordinates must be eliminated, by (12), in the term (P,Q~)b a of the functional (13). In construction of the functional (14) the partial integration in (3), which yields (4), is essential, becouse the starting form (4) must be independent of first derivatives Q~'s. Another passage from (3) to the functional (14) could be based on involutory transformations [1, p. 5].

Remark 2: If we have an exact solution q" of the Eqs. (1), then from the technique used for deriving G, it is obvious that E(q) ~ G(q).

For the calculation of the first variation (~G) and the second variation ($2G) we can use the functional (4) along the steps which lead to (14): First, from fig and 5~G we eliminate the variations ($q" under the integral sign using the relations

aq~ = - - Z ' ( a ~ + OJap~) (15)

obtained from the Eqs. (1.2). Second, we use the Eqs. (7) to eliminate 6p~ from the result of the previous step. By (15) 3G and 3~G become

(~G = [ap~LyX~ ~ + p, aq~]a b

b (16) -- f {(~p~ (~p~ --(~") § ap. [--LyA~C~ -+ - ~ (A~L~)]I dt,

a

b 'Y a

where (~p: and Ly are given by (7) and (1.2). Now, we define a functional I(Q) as a difference between the functional E(Q),

Eq. (6), and the functional G(Q), Eq. (14),

I(Q) d~ E(Q) -- G(Q) = -[v,(t, Q, (2)Q~]b b

+ f {v.(t, Q, (2) Q~- H[t, v(t, Q, Q), Q] ~ (18)

§ ~(t,~Q, Q) ~[t, v(t, Q, Q), ~(t, Q, Q)]

§ H[t, v(t, Q, Q), ~(t, v(t, Q, Q), r Q, Q))]} dt.

By means of I(Q) we shall constitute our new variational principle.

168 Dj. S. Djukic and T. M. At~nackovic:

I f q: and Q~ are two close curves, which are connected by the Eq. (2), due to the smoothness of I(Q), the following expansion is valid

I(Q) = I(q) + ~I + O~I + (O(6q)~). (19)

Combining (7), (9), (10), (16) and (17) we get the first and second variations of the functional (18) as

t b

(t

b

1 f Xr dt (21) d2I - - 2

a

where

(22)

Here, all coefficients of dq ~ are calculated along the curve q=, p~ = v~(t, q, (t). Let if= be the solution of Eqs. (1) plus appropriate boundary or initial conditions.

Then we have the following theorems :

Theorem 1: The functional I(Q) is s tat ionary at q= and the stat ionary value is ~(q) = 0.

Proo/: From (1) and (20) we conclude that M(q) = O. From (18) and Remark 2, following Eq. (14), it is obvious that I(q) must be zero.

Theorem 2: A necessary condition for the functional I(Q) to have a local maximum at Q~ = q~ is that following quadratic form with respect to F~

X~.F~ (23)

is positive definite function for every t s [a, b]. Similarly, for a minimum at Q~ = q" the condition is tha t (23) is a negative definite function.

Proo/: The Theorem 2 is obvious from Eq. (21).

Remark 3: I f the quadratic form (23) is not definite we can not say anything about the character of the stat ionari ty of I(Q).

Remark 4: The usuM necessary condition (see details for a mechanical system with one degree of freedom in [1, p. 24]) for the functionM (6) to have a local extremum is that quadratic forms A,~qZ 5q" and B~(~pfip~ are definite with the oposite signs, or, if one of the systems A~, B ~ is equal to zero then the remained form must be definite.

An Extremum Variational Principle for I-Iamiltonian Systems 169

Remark 5: From (1) and (20) we conclude tha t (3I(q) = 0 for arbi trary values of (3!/~ and (3~ at t - - a and t = b. Also, (3V does not depends on this quantities. Hence, the Theorems 1 and 2 are valid for any initial or boundary conditions which are imposed to the Eqs. (1). Usualy, the boundary conditione have a great influence on the necessary conditions for extremum.

!

Remark 6: From the first variation (3I it is obvious tha t the solution of Eqs. (1) is not the only stationarity point of I(Q). Other points of stationarity could be obtained by equating (3I to zero but without use of the Eqs. (1). I t would furnish differential equations of the fourth order (zero values of the brackets by (3q" under the integral sign in (20)) and corresponding boundary conditions. For example, if we consider (3~ and (3q~ at t ~ a and t = b as arbi trary quantities, then the corresponding boundary conditions are zero values of the coefficients by (3~" and (3q" in (20) at t = a and t = b. We must underline the fact tha t the value of I at these points of s tat ionari ty is not equal to zero and the second variation (21) does not have any useful meaning.

Remark 7: As an illustration for Theorem 2 we concentrate our at tention on an undamped mechanical system with n degrees of freedom. The system is vibrating around its stable equilibrium position. Hence, in the linear case the coefficients A,~ = @H/~q ~ ~q~ are constants, H denotes the potential energy. In that case A,r is a positive definite quadratic form and so is A~r where A,~A~ = (3,q. Therefore, the principle (18) is a maximum principle in that c a s e .

3. The Variational Principle as a Constrained Variational Problem

In the procedure for constructing Our variational principle (18) we needed the solution (5) of Eq. (1.1) and solution (12) of Eq. (1.2). The possibility for an explicit construction of ' the second solution (12) is very restricted. Here, we will accomodate our theory for such cases by the usual Lagrange multiplier rule. We must remember tha t in (13) and in the functional (18), everywhere where we have eliminated the generalized coordinates by the solution (12), we have ~b :. Now, for the generalized coordinates at this places we will use notation M% So, we make a fundamental difference between coordinates at this places and coordinates at other places where they are not solution of (1.2). Therefore P, , /5, and M ~ are connected by the set of Eqs. (1.2)

OH ( t , M , P ) = O , for a < _ t < b , (24)

which are equality constraints, to be taken into account, when extremize the following functional

I(Q, M) ~_ E(Q) -- G(Q, M) = _(p.Q~) b

a

+ f [P~@ -- H(t, P, Q) + P~M ~ -f- H(t, M, P)] dt, b

(25)

here, the P . ' s must be eliminated by (5).

170 Dj, S. Djukic and T, M. Atanackovie:

The variational problem is equivalent to extremizing the modified functional

b

i(Q, M, ~) = I((2, M) -- f 2~(t) (L.)~ dr, (26)

where Lagrange's multipliers 32 are unknown functions of time. Taking the first variation of (26), using (7), the stai~ionarity condition dI = 0 for independent variations dm" and dq~ yields the, following differential equations

d ~ d 0~ - - L , - - ~ { , ~ [ ~ ~p~~ § ((x,o)~

- N = + [ ( / ' ) = (L , ) : ]

- ( X ~ ) = (0r (L~)=~} = 0,

,~ : (X'"),~ ( L : ) J , for t ~ [a, b] (28)

and the following boundary condition

~p. m

where

(29)

OH (t, m, p) ---- O, p. = v~(t, q, (t) for t ~ [a, b], (30) (L~)=~ -~ ~oo + ~

B,= = ~,o(t, q, p ) , OH _ OH (t, q, p ) , ~,+ = O,~(t, ~, p ) , (3~) Op~ Op~

(A~)" = (X~=)~(t'm'P)' (~P,)m -- (~0-~=),, ( t ,m ,p ) , (C~) , , , - -C p ),, (t, m, p), (32)

M = = m: ~ cSm ~, (33)

and where m" is a stationary value of M". If the m~'s satisfy (30) and if in the Hamiltonian function H(t, m, p) the m~'s are substituted instead of the generalized coordinates, then the m~'s satisfy the following equations

Remark 8: The stationarity conditions (27) are differential equations of the fourth order. These stationarity conditions are meant in l~emark 6, above.


Remark 9: If we specify initial or boundary conditions to the equations of motion (1) the condition (29) will yield additional boundary conditons for the Eqs. (27).

Remark 10: From (1), (27), (28), (30) and (34) w e s e e that the solution of Eqs. (1) satisfies the stationarity conditions (27) and that actual values of the Lagrange's multipliers are zero. At the same situation, for arbitrary variations 6q ~ and 5~ at t ---- a and t ---- b, the condition (29) gives

m ~ - - q " for t ~ a and t - ~ b . (35)

Remark 11: I t is obvious, from (30), (34) and (35), that at the stationarity points of the functional (25) the generalized coordinates q~ are equal to m", i.e. m ~ = q". These quantities are equal but their meaning in the variational problem (24), (25) is different. For that reason, it would be erroneous to equate M ~ with Q" at the beginning of our considerations and to introduce such constraints in the variational problem (24), (25).

At the beginning of this chapter we mentioned the obvious difficulties in finding solution (12) of the Eqs.~(1.2). In many eases it is possible to have only an approximate solution, an approximate inversion of the problem. Then, for our further research, we need the influence of the approximate inversion on the value of functional I. Let the M ~ be exact solution which satisfies the Eqs. (24). If N ~ is an approximate solution to the algebraic inversion problem, then

0H (L:)~v ~-- P, -}- ~ (t, N, P), P: ---- v,(t, Q, (2) (36)

is the residual of the approximate inversion and in the general case, is different from zero. We will suppose that the exact and approximate inversions are two close curves, that is

57" = M" + AM ~, (37)

where AM" are small quantities.

Remark 12: We must underline a difference between the small quantities (Sin" and the small quantities AM ~. The 6m"'s are defined by (33), where m" is a stationary value of the M ~ and (24) and (30) are satisfied by M ~ and m ~, respectively. The AM"'s are defined by (37) but 2i" is no more in agreement with bhe relations like (24) and (30). So, the meaning of these two types of changes is completely different.

~Now, substituting (37) into (24), developing 9H/~Q" in series around N", using (36) and neglecting small quantities of the order (AM") ~ and higher, we obtain

AMy -~ (Ar")~ (L,)~v, (A~")~ = (ff,~")~(t, N, P), P, ---- v,(t, Q, (2). (38)

The value of G on the exact solution M" is

b

G(Q, M) ---- (p,Q,) b _ f [P~M" + H(t, M, P)] dr, (39) f t

where the P: are given by (5). Substituting (37) into (40), developing the Hamil-

172 Dj. S. Djukic and T. ~. Atanackovic:

tonian in series around an approximate solution N ~ and using (36) and (38) we have

r iV) = G(Q, M) -- Rx(Q, iV) + O,~((d~)~), (40) where

b ~N ~- ~(Q, .N) : (PaQ=)a b -- f [PaN c~ § N, P)] d~, (41)

is value of G on the approximate solution N ~, and where

b

Bx(Q,N) = ~ f (X~")x(Ly)x(L~)xdt (42) a

is weighted square residual of the approximate inversion N". The value of our functional I on the approximate inversion N: is

Ix(Q, N) = E(Q) -- Gx(Q, N) , (43)

where E(Q) and Gx(Q, N) are given by (6) and (41). Now, combining (25), (40) and (43) we have the following relation between

values of the functional I at the exact and the approximate inversion of the Eqs. (1.2)

I(q, M) = Ix(Q, N) - Rx(Q, N) + O~((~M"V). (44)

5. An Example

Here, we are going to solve approximately a nonlinear vibration problem as an illustration and application of the present theory. Let us consider a mechanicM system with two degrees of freedom, whose Hamiltonian function is

H = 2 P12 @ pu~ § (ql)2 1

where Z is a constant. The corresponding equations of motion are

~1 = 201, ~2 = p~ ; (46)

Pl = --)i~"[2q 1 d- (ql)S _ q2]; ib2 = --z2(q 2 -- ql). (47)

The formal solution of Eqs. (47) with respect to ql and i/2 is not suitable for practical use. Therefore, we will consider the equations as the constraints, and apply Lagrange's multiplier rule. :From (5), (6), (41), (43) and (45) we obtain the following functional

b

Ix(O, iv) -- -(01~} 1 + Q~}~)ob + g}l)~ + (~}~)~ _ ~ (Q~ _ Q~)~ g

(@?) (4s) (N1)~) -- z ~ ( ~ - ( @ ) ~ + -i- + Z~ ( ~ (ivl)~ + 2 -

e ] + lw~ l + iw~}~ + ~ (iw - iv~)~ dt,

An Extremum Variational Principle for Hamiltoni~n Systems 173

where N 1 and N ~ are app rox ima te solutions of (47) wi th respect to ql and q2 and fo r P l : Q1 and p2 = Q~.

Here, we will f ind an app rox ima te solution requir ing t h a t (Galerkin method)

b f + x=[ w + (:w)a] _ z (N= _ dt = O, (49)

a a

f {(2~ + z~(N ~ -- 2Vl)} lV~ dt = 0. (50) a

Now, f rom (11), (23) and (45) we conclude t h a t the functional (48) has a local m a x i m u m a t the solution of (46) and (47), and for the exact solution of the Eqs. (47). We assume approx ima te solutions of (46) and (47),

Q l _ a l s i n c o t ; Q ~ - ~ , s i n c o t ; N l : f l l s i n c o t ; N 2 fl~sincot, (51)

where al, a2, ill, f12, and co are constants . These tr ial funct ions we subst i tu te into (48), (49) and (50) and in tegra te f rom a ~-- 0 to b ---- ~/2co, we ob ta in

IN-----~--0~09 I~I2CO 2 -~-0/22CO2- ~2 (~12 0r 3 0r - - -~- (0~2 - - 0r

(52) [1 \ - - z~T - A)~I] -~ Z2 \Z'~- /~12+ 3 fl14'] lXlfll co2 (X2f12 co2 + (f12

( 3) _ ~ c o 2 + z2(fl~ _ / ~ , ) = 0. (54)

We extremize (52) wi th respect to the pa ramete r s cr ~2, fl,, ill, considering t h a t these pa ramete r s are constrained b y (53) and (54). The final results are

0 - 1 0r ~--- fll ; 0/2 ---- fi2 ; (/)2 __ Z 2 ; I N = 0 ;

0 (55)

where 0 = ~/~1, while the corresponding Lagrange ' s mul t ip l ie rs are equal to zero. The same result for this p rob lem is obta ined in [25, p. 685] as the first approx ima t ion b y a different method. I n this ease, the approx ima te solution will cause the weighted square residual (42), where

Xll = [z2(1 + 3(N1)~]-1; XI~ = [z~(1 + 3(_~1)~] 1;

.~22 _~ [2 -k 3(N1) 2] [~/e(1 -k 3(N1)U)]-l; (L~)~ = ~2 -k z2(N ~ - - N1); (56)

(L1) N : O 1 ~- Z2[N 1 Jr- (N1) ~] - - Z2(N 2 N1),

to be different f rom zero, Subst i tu t ing (51) into (56), using (42), (44) and (55), r emember ing t h a t a : 0 and b ~ ~/2co and assuming t h a t 08((AM~) 2) ~ O, we

13 Acta Mech. 50/3-4

174 Dj. S. Djukic and T. M. Atanackovic:

have

- { 1 - [1- v1 + / /(Q) = 12s~ }/i + 3(c(D 2 - 27(c,~) 6 ] , (57)

w h e r e 0~ 1 aIld (9 are given by (55). The value of I(Q) is a crude measure of the error contained in the approximate solution (51). For small values of the I(Q) we can except tha t the approximate solution (51) and (55) is of sat isfactory accuracy.

5. Discussion

The variat ional principle proposed in this paper is an ext remum principle if the quadratic form (23) is positive (negative) definite funct ion for every t C [a, b]. I f the quadrat ic form (23) changes its sign for t C [a, b], then the principle will be an ex t remum principle in subintervals of [a, b], where the form is positive (negative) definite function. The character of extremal i ty can be different in the two different subintervals, but in every subinterval the extremal value of the func- t ional is zero.

I n applying Ri tz method to obtain approximate solution, the functional I as a funct ion of undetermined parameters I ( a 1 . . . . ) has more than one ext remum and s ta t ionar i ty points. Then, a very impor tan t question a r i ses : how to choose the proper values of the parameters al, a2 . . . . . which corresponds to approximate solution of the problem? F rom the properties of I (Q) we know tha t it has a local max imum (minimum) on the exact solution of the same proplem. Hence, al, a2 . . . . mus t be choosen in such a way tha t I (al, a2 . . . . ) is in max imum (minimum) and tha t at max imum (minimum) I < 0 ( I > 0). The other ex t remum and s ta t ionar i ty points of the I t ha t do not satisfy the above conditions corresponds to s ta t ionar i ty points of I(Q) mentioned in Remark 6 (after Theorem 2). Fur the r applications of the variat ional principle presented here will be reported elsewhere.

References

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[2] Lurie, A. I. : Analytical mechanics (in Russian). Moscow: F. M. Liter 1961. [3] Tzyganova, N. Y. : Papers of Russian scientist from XIX century concerning principle

of least action and Hamilton-Ostrogradski principle (in Russian). Moscow: Akad. Nauk SSSR, Trudi instituta istorii estestovoznania i ~ehniki 19, (1957).

[4] Morse, M. : The calculus of variations in the large. Providence, R. I. : American Math. Soc. 1934.

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An Extremum Variational Principle for Hamiltonian S.ystems 175

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Di. S. D]ukic Faculty o/Technical Sciences

University o/ Novi Sad Yu- 21000 Novi Sad

Yugoslavia

T. M. Atanackovic* Hermann-F6ttinger-Institut

/i~r Thermo- uncl Fluiddynamik Technische Unwersitdt Berlin

D-JO00 Berlin 12

* On leave f.rom the UniT,er~ity o~ Novi Sad, Yugoslavia

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