Mixed variational principles for some non-self-adjoint dynamical systems

MECHANICS RESEARCH COMMUNICATIONS 0093-6413/84 $3.00 + .00

Vol. ii(4),253-263, 1984. Printed in the USA Copyright (c) 1984 Pergamon Press Ltd.

MIXED VARIATIONAL PRINCIPLES FOR SOME NON-SELF-ADJOINT DYNAMICAL SYSTEMS Leon Y. Bahar and Harry G. Kwatny Department of Mechanical Engineering and Mechanics Drexel University Philadelphia, Pennsylvania 19104

(Received 9 April 1984; accepted for print 16 April 1984)

Introduction Time-lndependent, generalized, bilinear Lagrangians have been recently used to study the behavior of some linear non-self-adJoint dynamical systems. The construction of such Lagrangians is accomplished by doubling the size of the original state space. In the physics literature such Lagrangians are known as Morse and Feshbach Lagrangians, and in the mechanics literature as Leipholz Lagrangians. In a recent paper, the authors generalized the bilinear Lagrangians described to dynamical systems with multiple degrees of freedom, and studied their features when they are used to generate dissipative and gyroscopic systems and their first integrals. An interesting aspect of such systems is their variational formulation leading to the governing equations of motion, and conversely, the construction of the bilinear Lagrangians starting from the equations of motion, using the methodology of the inverse problem of Lagrangian dynamics. As these facets of the problem have not been previously addressed, they have been made the subject of the present investigation. Time-independent Bilinear La~rangians Soma time ago Morse and Feshbach [i] introduced a bilinear Lagrangian for a linear, viscously damped oscillator, which in addition to the equation of motion governing the behavior of that oscillator, also furnished the equation of motion of a "mirror-lmage" oscillator characterized by "antifriction".

In subsequent discussion the latter equation was ignored.

In

[I] it was specifically cautioned that "This is not very satisfactory if

253

254

L.Y.

BAHAR and H. G. ~4ATNY

another method of solution is known ...".

Another Lagrangian similar in

structure was derived by Leipholz [2], who used it extensively in his investigations [3-5].

It should be emphasized that the germ of the idea will be

found in a paper by Bateman [6], published much earlier.

Such Lagrangians

have continually been used by physicists as evidenced by the recent contributions of Feshbach et. al. [7], Tikochinsky [8], and Greenberger [9].

In

mechanics, this technique has been extended to continuous systems by introducing the complex conjugate mirror field into the Lagrangian density.

An

analogous Lagrangian has been recently used by Mitchell and Wait [i0] as a basis for the finite element modeling of the one-dimensional heat conduction equation.

Bilinear Lagrangians have also been used by Gladwell [Ii, 12] to

obtain equations for damped vibrations of structures, and mechanical transmission

lines.

Recently, Bahar and Kwatny [13] gave a simple presentation of how the bilinear time-independent Lagrangian can be used to obtain the well known quadratic time-dependent Lagrangian for the damped harmonic oscillator, as well as the corresponding time-dependent energy-like first integral. In another contribution the same authors [14] gave a generalization of their study in [13] to systems with multiple degrees of freedom, and showed how the different bilinear time-independent Lagrangians in use are equivalent. This paper explores the variational formulation of the problem studied in detail in [14]. Consider the Lagrangians LI = y~x

- yTcx - yTKx

(Leipholz)

(i)

L2 = y~

+ yTcx - yTKx

(Modified Leipholz)

(2)

L 3 = yTM~ + [(yTcx - yTcx)/2] - yTKx (Morse and Feshbach)

(3)

In (i), (2), and (3) the variables x and y are vectors whose n components represent the generalized coordinates and their adjoints respectively; M, C, and K denote constant square matrices of order n associated with the mass, damping and stiffness of the system, and the superscript T designates the transpose of a matrix.

Note that in contrast to

quadratic Lagrangians

customarily used, the bilinear forms (i), (2), and (3) provide an avenue for

NON-SELF-ADJOINT SYSTEMS

255

the study of systems in which M, C, and K are not necessarily sylmuetric. The equations of motion resulting from any of the three Lagrangians LI, L 2 or L 3 are given by:

d'-T and

- ~:M~+C~+Kx:O

~d (~)-

(4a)

~x~--L" MTy + cTy + KTy = 0

(4b)

Note that the second members of (4) have been obtained directly by carrying OUt the operations indicated in the first member without the introduction of a generalized force or the Rayleigh dissipation function.

It should be kept

in mind, for future reference, that the Lagrange equation with respect to the adJoint vector y produces the equation of motion in the physical vector x, while the Lagrange equation with respect to the physical vector x, leads to the equation of motion governing the adJoint vector y. It should be pointed out that despite the fact that the Lagrangians represented by (1), (2), and (3) are coupled in the coordinate as well as the velocity vectors, the final equations of motion are uncoupled because cancellations eliminate the appearance of any coupling terms from the final equations of motion. The fact that the three Lagranglans

(i), (2) and (3) give rise to iden-

tlcal equations of motion should not be surprising.

Indeed, a simple calcu-

lation shows that the Lagrangians described by (l), (2) and (3) are related by _

L 2 - L I = 2(e 3

d

e I) = 2(e 2 - L 3) :

~

(yTcx)

(5)

and hence differ from each other by a gauge term (the time derivative of an arbitrary function of the generalized coordinates and the time).

It is

well known that all such Lagrangians yield identical equations of motion, the proof of which is given in Saletan and Cromer [15]. The system of equations of motion represented by (4) can be written in matrix form as:

I°

I- -I +

,,"o I-i-I +[o

I-:-I :

(6)

256

L.Y.

BAHAR and H. G. KWATNY

The vector equation of motion because its coefficients

(6) is formally self-adjoint

(FSA),

satisfy the Helmholtz conditions for FSA as

recently developed by Santilli [16]. It is now clear that the success of bilinear Lagrangians a mere happenstance of an ad hoc nature.

is not due to

They are useful precisely because

they produce a set of equations that complement the original set of governing equations of motion,

thus producing an overall system which is FSA when

the equations in the original physical variable, and the adjoint variable are considered together as a singlw system.

Thus, as is well known, the

billnear Lagranglans lead to equations of motion that are FSA, and conversely sets of equations of motion that are FSA as given

(without having to use a

multiplier matrix to make them FSA) are derivable from bilinear Lagrangians. Uses of such Lagranglans to study the properties of some dissipative systems, gyroscopic systems, and the associated conservation laws have been considered in [14]. It is, therefore, natural to attempt to obtain the equations of motion related to the various Lagrangians by variational methods, particularly as the notion of FSA naturally arises in such a context, and the possibility of extending the present work to problems with variable end points, constrained systems which require non-contemporaneous holonomic constraints),

variations (such as non-

etc. is enhanced.

As will be seen, the variational method to be used is not a trivial extension of Hamilton's principle,

as it requires the simultaneous variation

of the physical and adJoint vector subject to fixed end points in both variables to produce the desired results. Mixed Variation_al Principles If the problem is now approached from a variational standpoint,

it is

necessary to consider Hamilton's principle with fixed end-polnts in space and time, and the time is not allowed to vary between the dynamical trajectory and the comparison path (contemporaneous variation).

The vector of

the actual physical variables x, as well as the vector of the adjoint variables are to be varied simultaneously.

NON-SELF-ADJOINT SYSTEMS

257

For completeness, the explicit results for the various Lagranglans developed are given By the expressions:

/,

Lldt

=

{"

~yT(H~ + Cx + Kx)dt

tI

-

/"

tI

~xT(MT~ - cT# + KTy)dt

tI

. T~T ~t2 + [~y~

t2

+ ~

(7)

- o~ ~ yJtl t2

_/t2 6yT(Mx + Cx + Kx)dt

L2dt = tI

l

~xT(MT~; - cT~ + KTy)dt tI

tI

T. ~t2

(8)

+ [~yTMx + exTMTy + Qy ~XJtl

t2

Lsdt -

tI

_ l t2 ~yT(M.~ +

Cx + Kx)dt

-

f t2 ~xT(MTy

Jt I

- cTy + KTy)dt

tI

+ [SyT~ + ~xT~T~+ (~yTcx _ 6xTcTy)/2]t2

(9)

In establishlng the relations (7) through (9), use has been made of the commutation between variation and differentiation or integration, which is permissible for a contm.poraneous variation.

Also, for convenience, some

scalar terms have been written as their own transposes. The expressions given by (7) through (9) show that, although the integrated t e r m s are different in each case, they vanish as soon as w-miltonVs principle is invoked, which requires that the variation of the original vector as well as its adJoint be zero at the initial and final times.

For the purpose of considering the equations of motion, it is

therefore sufficient to concentrate on the variation of the Lelpholz Lagranglan alone, which reduces to

7"

Lldt =

tI

-

f" tI

6yT(Mx + C~ + Kx)dt

-

f"

tI

8xT(MT$ - cTy + KTy)dt

(10)

258

L.Y.

BAHAR and H. G. KWATNY

Recalling that the variations of the original vector and its adjoint are a priori independent at the formulation stage of the problem, and appealing to the fundamental lemma of the calculus of variations,

the rela-

tion (i0) leads in the usual manner to the equations of motion given by (4). It should be pointed out that if there are external forces acting in the direction of the original generalized coordinate vector and its adjoint, the Leipholz Lagrangian can be modified to incorporate

them as follows:

L 1 = yTMx - yTcx - yTKx + yTfx(t) + xTfy(t)

(ii)

which leads to the equations of motion Mx + Cx + Kx = f (t)

(12a)

X

MTy - cTy + KTy = fy(t)

(12b)

where the f

and f are the forcing function vectors corresponding to the x x y and y directions respectively. The $ priqri independence of the x and y vectors shows that it is easier to obtain the variations indicated in theexpressions

(8) through

(i0) by varying only one of those variables at a time, and obtain the same equations of motion individually.

Due to the linearity of the variation of

each individual vector the variation with respect to both variables can be obtained by simply adding the partial results. Determination of the Various Lagrangians The various Lagrangians LI, L2, and L 3 used in this paper can be easily derived by an "inverse problem" type approach, based on the identification of an arbitrary combination of terms that appear in the equations of motion in the x variable,

as the time rate of change of the generalized momentum in

the y variable, and vice versa. the non-uniqueness i.

This arbitrariness

is permitted in view of

of the Lagrangian.

Explicit Determination of the Leipholz Lagrangian L 1 a) Method Based on the Equation of Motion In the Lagrangian

(4a) set arbitrarily

~L I .... Mx

(13)

NON-SELF-ADJOINT

SYSTEMS

259

Then, it follows that 8L I • ~y Integrating

=

-C~

-

Kx

(14)

(13) yields L 1 = y'T'M%:+ F(x, x,y,t)

where F is a scalar function of its arguments.

(15) Substituting the expression

(15) into (14) followed by integration results in F = -yTcx - yTKx + G(x,x,t)

(16)

which leads to the Lagrangian L I - yTM~ - yTcx - yTKx + G(x,x,t)

(17)

Clearly, if the equation of motion with respect to x is generated by applylng the Lagranglan

(4b) to (17), there results

MTy - cTy + KTy + ~ Using (4h), the expression

- ~x

= 0

(18)

(18) reduces to

d (gGI d-~ -

8G

(19)

8"~ = 0

which shows Chat G(x,~,t) has zero Lagrangian derivative, and can be discarded, as it is in fact a gauge term. b) Method Based o~ the AdJoi_nt Equation of Motion When the identifications

Mz _ CZy

8x

(20)

and 3L 1 ~x = -KTy

(21)

are used in (4b), upon integration of Eq. (20), and matchlng of the results with Eq. (21) produces LI = y~x where the Lagranglan

- yTcx - yTKx

(22) is defined up to a gauge function.

(22) The details of

the derivation parallel directly those of the previous section, and are

260

L.Y.

BAHAR and H. G. KWATNY

omitted here and in the sequel. It is worthwhile to notice the separation of the role of each variable in the defining equations (20) and (21), which leads to the bilinear form of the mixed Lagrangian in (22). 2.

Explicit Determination of the Modified Leipholz Lagrangian L 2 a) Method Based on the Equatiqn of Motion The following choice of the defining relations ~L 2

. = Mx+

Cx

(23)

and ~L 2

3y

= -Kx

(24)

in (4a) leads to, upon integrating (23), and making the result satisfy (24), the Lagrangian L 2 = yTMx + yTcx - yrKx

(25)

which is defined up to a gauge term. b) Method Based on the AdJoint E~uation of Motion Again, setting in (4b) ~L2

T.

(26)

=My and 3L2 ~x =

CTy - KTy

(27)

and integrating the expression (26) subject to the condition stipulated by the relation (27) leads to the Lagrangian given by L 2 = yTMx + yTcx - yTKx to within a gauge function.

(28)

To write Eq. (28) in the form given, the trans-

pose of each scalar term that occurs in that equation was taken.

NON-SELF-ADJOINT SYSTEMS

3.

261

Ex2licit Determinatlon of the Morse and Feschbach Lagran~lan L3 a) MethOd Based on the ERuation ofMotlon In the Lagranglan (4a) introduce the definitions 8L3 ffi ~ x + Cx 8y 2

and

8L3

Cx

8y

Z

(29)

Kx

(30)

Integrating Eq. (29), and making the result conform wlth Eq. (30) leads to the Lagrangian L3 = ~x

(31)

+ [(yTcx - yTcx)/2] - yTKx

which is defined up to a gauge function. b) Method Based on the__Ad~olnt E~uetion of Motlo.n In (4b), if the terms are. separated in such a manner that BL

• 3 =M~-C-~ 2

(32)

8x and

8L3 = C 2 ~ ~x

" KTy

(33)

again, integration of (32) subject to the conditions expressed by (33) leads to the Lagrangian L 3 ffi yTM~ + [ ( ~ T c x - y T c ~ ) / 2 ] - yTKx defined

to within

Clearly,

a gauge function.

the above derivations

I t may b e a r g u e d t h a t tion

of all

completeness, Lagrangians various tion

it

in order

to close

by researchers

show t h a t

is not necessary

the above Lagrangians.

definitions

(34)

to display

the explicit

They w e r e d e v e l o p e d f o r

some g a p s i n t h e u t i l i z a t i o n

in physics

of canonical

L 1 = L1, L 2 = L2, and L 3 ffi L3.

and e n g i n e e r i n g

construc-

the sake of of the various

mechanics.

momenta u s e d a b o v e a l l o w a n a t u r a l

Also,

the

transi-

from the Lagranglan to the corresponding Hamiltonlan, and it is impor-

tant to be able to trace them back to their natural origin.

262

L.Y.

BAHAR and H. G. KWATNY

Concludin~ Remarks A variational approach to time-independent bilinear Lagrangians was developed by means of mixed variational principles.

Conversely, all the

bilinear Lagrangians currently used by various authors have been constructed starting from the equations of motion, by utilizing the methodology of the "inverse problem" of Lagrangian dynamics. Passage from time-independent bilinear Lagrangians to time-dependent quadratic Lagrangians has been investigated in detail in [14], which can be considered a companion paper to the study presented herein. Acknowledgement This research has been supported in part by the U.S. Department of Energy under Contracts ET-78-C-01-2092 and ET-78-S-01-3088, which the authors acknowledge gratefully. References [i]

P.M. Morse and H. Feshbach, '~ethods of Theoretical Physics~ Vol. I", McGraw-Hill Book Co., Inc., New York, N.Y. 1953, pp. 298-299.

[2]

H.H.E. Leipholz, "Application of a Generalized Principle of Hamilton to Nonconservative Problems", Solid Mechanics Division, University of Waterloo, Waterloo, Ontario, Canada, Report No. 38, March 1970. Published in Ingenieur-Archiv, Vol. 40, 1971, pp. 55-67.

[3]

H.H.E. Leipholz, "Di!ect Variational Methods and Ei~envalue Problems in Engineering", Noordhoff Int. Publ. Leyden, Holland, 1977, pp. 207239.

[4]

H.H.E. Leipholz, "Analysis of Non-Conservative Nonholonomic Systems", Proc. 15th Int. Congress of Theoretical and Applied Mechanics, Toronto, Ontario, Canada, North-Holland Publ. Co. Amsterdam, Holland, 1980, pp. i-ii.

[5]

H.H.E. Leipholz, "Six Lectures on Variational Principles in Structural EngineerinK", Third Edition, Mechanics Division, University of Waterloo, Waterloo, Ontario, Canada, 1982, pp. 71-90.

[6]

H. Bateman, "On Dissipative Systems and Related Variational Principles", Physical Reyiew, Vol. 38, 1931, pp. 815-819.

[7]

H. Feshbach et al., "Quantization of the Damped Harmonic Oscillator", Contribution to A Festschrift for I.I. Rabi, Transactions of the New York Academy of Sciences, New York, N.Y., Vol. 38, 1977, pp. 44-53.

NON-SELF-ADJOINT SYSTEMS [8]

Y. Tikochinsky, "Exact Propagators for Quadratic Hamiltonians", J. Math. Phys., Vol. 19, No. 4, April 1978, pp. 888-891.

[9]

D.M, GreenSerger, "A Critique of the Major Approaches to Damping in Quantum Theory", J. Math. Phys., Vol. 20, No. 5, May 1979, pp. 762770.

263

[I0] A.R. Mitchell and R. Wait, "The Finite Element Method in Partial Differential Equations", John Wiley and Sons, Inc., N.Y., 1978, pp. 32-34. [ii] G.M.L. Gladwell, "A Variational Formulation of Damped Acoustostructural Vibratlonal Problems", J. Sound Vlb., Vol. 4, No. 2, 1966, pp. 172-186.

[12] G.M.L. Gladwell, "Variational Calculation of the Impedance of a Lossy Transmission Line", J. Sound Vib., Vol. 7, No. 2, 1967, pp. 200-219. [13] L.Y. Bahar and H.G. Kwatny, "Generalized Lagrangian and Conservation Law for the Damped Harmonic Oscillator", Am. J. Phys., Vol. 49, No. Ii, Nov., 1981, pp. 1062-1065. [14] L.Y. Baharand H.G. Kwatny, "Exact Lagrangians for Linear Nonconservatlve Systems", Hadronlc J., Vol. 2, No. i, 1979, pp. 238-260. [15] E.J. Saletan and A.H. Cromer, "Theoretical Mechanics", John Wiley and Sons, Inc., New York, N.Y., 1970, pp. 40-41. [16] R.M. Santilli, "Foundations of Theoretical Mechanlcs~ Part I~ The Inverse Problem in Newtonlan Mecha.nlcs", Springer-Verlag, New York, N.Y., 1978, pp. 116-119.