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Equivalent Lagrangians: Multidimensional caseS. Hojman and H. Harleston

Citation: J. Math. Phys. 22 , 1414 (1981); doi: 10.1063/1.525062 View online: http://dx.doi.org/10.1063/1.525062 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v22/i7 Published by the American Institute of Physics.

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Equivalent Lagrangians: Multidimensional case a)S. Hojman and H. HarlestonCentro de Estudios Nucleares. Universidad Nacional Aut6noma de Mexico, Apdo. Postal 70-543, Mexico 20,D. F., Mexico

(Received 27 May 1980; accepted for publication 17 September 1980)

~ generalize a theorem known for one-dimensional nonsingular equivalent Lagrangians L and

L ) to the multidimensional case. In particular, we prove that the matrix A, which relates the lefthand sides of the Euler- Lagrange equations obtained from Land L, is such that the trace of all itsinteger powers are constants of the motion. We construct several multidimensional examples inwhich the elements of A are functions of position, velocity, and time, and prove that in some casesequivalence prevails even i f detA = O

P ACS numbers: 03.20. + i

I. INTRODUCTION

The aim of this work is to generalize some results obtained by Currie and Saletan in the study of one-dimensional nonsingular equivalent Lagrangians to the multidimensional case and illustrate them with the explicit constructionof nontrivial multidimensional examples.

A Lagrangian L = L (q ,i/,t ) is said to be nonsingularwhen

detW #0,

We say that two Lagrangians Land E are s-equivalentwhen the sets of all solutions to the differential equationsobtained from them coincide. Notice that it is not necessaryfor the equations of motion obtained from both Lagrangiansto be exactly the same, so that situations more interestingthan E pL - dF q,t )ldt (P#O) arise.

We are interested in different Lagrangian descriptionsofa system that give rise to the same set of solutions because,after all, the trajectories (and not the equations of motionthemselves) are related to observation and experiment, atleast within the realm of classical physics 2 3:

Let

L ~ JL _ JL, dt Jq Jq

JEd dLL r

dt Jqr

we will prove tha t s-equivalence implies

Lr Ar (q,q,t )L

and that tr(A are constants of the motion for any positiveinteger k, generalizing a theorem given in Ref. 1 for onedimension. Furthermore, we will show through some examples that even when detA = 0, s-equivalence prevails; i.e., itis possible that L being regular gives rise to an s-equivalentLagrangian E which may be singular. As a ma tter offact, forthe two-dimensional examples we obtain s-equivalence evenwhen the rank of A is zero.

We will work out examples where the elements of A arefunctions of position, velocity, and time (not just numbers),

'Part of this work was presented by one of the authors (H. H.) as a thesis inpartial fulfillment of the requirement s to obtai n a B.Sc. in Physics at Un -versidad Nacional Autonoma de Mexico.

also avoiding the repeated use of the one-dimensional resultsof Ref. 1. These Lagrangians represent the two- and threedimensional harmonic oscillators and are of fourth degreerather than quadratic.

We should mention that there is a closely related(though more general) problem called the inverse problemof the calculus of variat ions, which we will not discuss here.

t consists essentially in trying to find all the Lagrangianstha t upon var iation will give rise to a given system of differential equations. A large amount of very interesting work inthat direction has been published lately.4-12

The impact of the equivalent description of classicalsystems at the quantum level is still not completely understood. The study of the problem in gauge as well as fieldtheories also remains to be done.

In Sec. III we prove the theorem of multidimensional s-equivalence, in particular that tr(A k are constant for anypositive integer k.

In Sec. IV we work out three examples of s-equivalentLagrangians in some detail, and, finally, in Sec. V we discussbriefly what has been done and point out some problemswhich remain to be solved.

II. THE ONE DIMENSIONAL CASEThe problem of determining whether a (second-order)

differential equation can be derived from a variational principle (the inverse problem of the calculus of variations) wassolved by Darboux in 1891 13 see also Ref. 14). He provedthat in the case of one second-order differential equat ion forone variable it is always possible to cons truc t infinite La

grangians that will yield the desired equation upon variation,and provided a way to const ruct them. The two-dimensionalcase was treated by Douglas. IS In the work presented here,however, we will not consider the inverse problem but will bemostly interested in dealing with the problem of equivalentLagrangians in the case of dynamical systems with n degreesof freedom.

The study of equivalent nonsingular Lagrangians forthe one-dimensional case in classical mechanics was bothproposed and solved in Ref. 1. We will briefly review themain concepts and results here to facilitate the understanding of the multidimensional case, in which we are mainlyinterested. We will use a slightly different wording and ap-

1414 J. Math. Phys. 22 7). July 1981 0022-2488/81/071414-06 1.00 1 981 American Institute of Physics 1414



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proach but both the ideas and results are equivalent to thoseof Ref. 1.

Two Lagrangians L (q,q,t ) and L(q,q,t )will be called sequivalent iff the two sets of all solutions to the Euler-Lagrange equations obtained from them coincide or, in o therwords, when every solution to the equations of motion corresponding to one of the Lagrangians satisfies the equations ofmotion obtained from the other Lagrangian, and conversely.The task of finding the general rela tionship between s-equivalent (one-dimensional) Lagrangians was undertaken in Ref.1, and it was proved that s-equivalence implies that the lefthand side of the equations of motion are proportional toeach other, the proportionality factor (which must neithervanish nor become infinite) being a function of the coordinate q, the velocity q, and time t, then, the condition forL (q,q,t) and L(q,q,t) to be s-equivalent can be written as

d a L a L . [d a L ) aL ]dt aq - a;; = /(q,q,t) dt aq - a;; , 1)

with

f=Lqq/Lqq, 0=1-/=1-00. 2)Currie and Saletan proved that / is a constant of the

motion, and, given any constant of the motion of the dynamics due to L, they provided a way to construct L explicitly.

An interesting example of the above mentioned theorem is

L = 0 2 - X2,L = XX2 cost - iX3 sint - iX3 cost,

-. aL _ aL = x + x = 0,dt ax ax

d a L a L ( . . )(.. ) 0- . - - = x cost - x smt x+

x = ;dt ax ax

therefore,

/ = x cost - x sint

and

3)

4)

5)

6)

7)

j = 0 , 8 )as it can be easily verified. The set of all solutions to theequation of motion 6), i.e.,

(x cost - x sint) = 0, 9)

and/or

x +x) = 0, 10)

can be written as

x(t ;A,a = A sin(t + a , 11)A,a being constant, which is the general solution to Eq. 6).Therefore, it is not necessary to require/ =1-0to prove theequivalence of Land L; i.e., the solutions to the equation ofmotion/(q,q,t) = 0 are particular solutions ofEq. 6) fora=O.

The fact tha t one still has s-equivalence even when considering/ = 0 constitutes a slight (but significant, especiallyfor quantum purposes) generalization of the cases considered in the literature. We will elaborate this point further inthe multidimensional case.

1415 J. Math. Phys., Vol. 22, No.7 July 1981

III THE MULTIDIM ENSIONAL CASE

Even though the one-dimensional case is of great interest in itself, it is appealing to consider more realistic caseswhere the number of dimensions could be appropriate todescribe nature. In particular, when discussing the impact ofthese results in quantum theory, it is relevant to ask whetherinteresting examples exist in higher dimensions.

We will discuss in this section the generaliza tion of theone dimensional problem to n dimensions.Consider the Lagrangian L = L (qi,qi,t), i = l, .. ,n,

such that

12)

Define

L = -. aL) _ aL _ W ij' + a2

L i/ _ aL 13)r - dt aqr aqr - rs ail aq aqr

(We drop the explicit time dependence of L, but all the results which will be obtained hold for an arbitrary explicitly

time-dependent Lagrangian as well.)The LagrangianL = L(qi,qi,t ) withdetW =1-0 is said tobe subordinate to L iff

14)

We will now prove that if Lis subordinate to L, then Lissubordinate to Land that

L, = A/(q,q,t )L 15)

with detA I O. Moreover, the t race of all integer powers of Aare constants of the motion.

The equations of motion for Land L read

16)

17)

Let U and Ube the inverse matrices to Wand W respectively; then from Eq. 17)

ijs = U p ( ; ~- a : : ~q ). 18)and we have assumed that Eq. 16) implies Eq. 17). Therefore, t he expression 18) can be inserted back in Eq. ( 16), from

which we getW Usp a2L a

2L . ,) _ aL+ qs aqP a(jPaq Jqr

and Lr can be written as

19)

L = W.Usp W q. - - q - - = WrsU'PL , 20)(a2L aL) - -

r r, P aqP Jq aqP P

i.e.,

21)

s. HOiman and H. Harleson 1415



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with

A/_Wps(q,q,t)Usr(q,q,t), (22)

and detA 0, i.e.,ULr = O j ~ ~= O J ~ U L r= O j q ~= OJ). (23)

We will now get a differential equation for A that willallow us to prove that triA m is constant for any integer m.

Equation 19) can be written asa2J: q' _ a L = A r( a

2L q' _ aL) . (24)

aqSaq' aqs s a i l q' aq' '

when using the definition (22). Differentiating with respectto qU, one gets

a l L

and, using the equation of motion 16), one gets

a 2L . aL = _ W '

aqraq' q - aqr q .

Using the fact that

A ,s W s, = Wr"one can prove that

aAs' a 2L aAs' a 2Laqu aq' aq' = aq' aq' aqu .

Furthermore

A' a3L . a3L .

S

q - qJq'artaq' aqUaqSaq,

_ a A s ' ~aq' q ai l aqu .

Therefore, Eq. 25) becomes

a2L a2L dA r_____ _ _____ = _ _ _ _ s WaqSaqU aqUaqS dt ru

+ A r ~ _ ~ ) .S aq'aqU aqUaq

i.e., defining

a2L a LT = = T'U - ailaqu aqUaq' u"

_ a 2L a2L -T - - - = - T

su - aqSaqU aitaqS us'

one gets, finally,

A = - TU+ATU.I t can be directly proved that

(tr(A rr = 0

25)

26)

27)

28)

29)

30)

31a)

31b)

(32)

(33)

for any integer m, using the fact that Tand Tare antisymmetric, U and Ware symmetric, and A = Wu. Of course, notall these constants are functionally independent because, forany n X n matrix A, triA k can be writt en algebraically in

1416 J. Math. Phys., Vol. 22, No.7, July 1981

terms oftr(A V i = 1,... ,n for k>n + 1. Furthermore, it mayhappen that even the trace of the first n powers of A are notfunctionally independent. This result agrees with the oneobtained when discussing the one-dimensional easel and canbe interpreted by saying that all the invariants (or eigenvalues) of A are constants of the motion.

The theorem we have just proved will be helpful in theconstruction of examples in Sec. V. It is worth noting thatthe result is a natural generalization of the analogous theorem discussed in Sec. II.

We should anticipate that in some of the examples thecondition detA 0 may be relaxed without losing s-equivalence.

We will now define as 1as the difference between LandpL 1p#0):

1= L(q,q,t) - pL (q,q,t). 34)

We have that Ir = ~ - pL but = A/Ls ' so that

Ir = (Ar' - po/)Ls - f l / Lwhere

f l = A - p I .

35)

36)

We have that detA 0, but this fact does not implydetfl =10 so that, although {Ls = 0] implies IIr = OJ{Ir = OJ does not imply fLs = 0] in general; i.e., the equations of motion of are valid whenever the ones for L are, butnot conversely. For instance, if L can be written as

such that LI and L2 have no variables in common, then

II =AIL I and 12 = A 2L 2 , - p A l i=0i=A 2= 1 - p ,

are two examples of such an I; i.e.,

37)

LI = p L +A\L I and L2 = p L + A zL 2 (38)are both equivalent to Land

{Lr = O)===> lla = 0] but la = O]=:i>{Lr = OJL r = O j ~ { l 2 b = 0 ]but l2b=Oj=:i>[Lr=Oj.

One explicit example is

L = (x2 + ;2) - (x 2 + y2), (39)II = 01(X 2 - x 2 ), 12 = 02Cl- y2), (40)L 2 = U X 2 - j ) _ ( X 2 _ y2)], p = 1 2 AI = 0 , 2

A2 = 2. 41)

[The case I = - dF(q,t )ldt is, of course, trivial with Ir = 0

being identities.]It is interesting to note that in some of the examples wewill consider detA = s a possible equation of motion, andin spite of this fact we will still obt ain s-equivalence; i.e.,det Wmay vanish even if det W 0 and this will not give riseto solutions of the equations of motion ~ = 0, not alreadycontained in the set of all solutions of L, = 0. A theoremdiscussing in which cases this situation is possible is currently under investigation.

IV EXAMPLES

Using the results ob tained in Sec. III we will now construct three examples of s-equivalent Lagrangians; two for

S. Hojman and H. Harleson 1416



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the bidimensional simple harmonic oscillator BSHO), andanother one for the same mechanical system, but in threedimensions TSHO).

Example 1: Consider the usual Lagrangian associatedwith the BSHO:

L = T - V = ~ q ~+ q; - ~ q ~+ ~ ); 42)the related equations of motion for L are

L j = i j i + qi = 0, i = 1,2. (43)Then, since Eq. 35) is an identity, linear in the accelerations(ij ), it can be separated into two identities: one for the termscontaining accelerations and another one for the rest of theterms; that is,

44a)

a l - f l= , qaq'

44b)

where we have used L given in Eq. 42).Since the trace of f l and f l 2 must be constant, and since

f l is symmetric, according to Eq. 44a) we propose f l to havethe following structure :

f l = E CC E

(45)

where E = Mt + ~ + t + q ~ )is the mechanical energy ofthe BSHO and the quantity C is a constant of motion whosedependence on qi and 1/ is to be determined from Eq. (44).

With this choice of f l we can rewrite Eq. 44) explicitly;thus.

a21- - = E 46a)aqlaql - ,

aZI- - = E 46b)aq2 a q2 - ,

a21- - = C , 46c)aqlaqz

46d)a 21 . a 2/ aZI a l

- - q l - - q 2 + - - = C q l + E q z 4 6 e )aq2 a ql aq2 a q2 aq2 at aq2

One solution to these equations is

- ~ q i q ~- k(qi + qi)The equations of motion obtained from I are then

II = E(i j l + ql) + C(ij2 + qz = 0,lz = C(ijl + q.J + E(ij2 + q2 = 0,

or

I, = f l /Ls = 0, r,s = 1,2,

47)

48)

49a)

49b)

50)

which is the expected result, with f l given in (45) and Lsgiven in (43).

On calculating the determinant of fl, we obtain

1417 J. Math. Phys., Vol. 22, No.7, July 1981

deW = (E2 - C Z) = (E - C)(E + C), 51)which is always a quantity greater than or equal to zero. Inorder to see this clearly, define the vectors

u i = (l y2)( 'Lqi) ' i= 1,2,with which we have

E = ui + uLC = 2u l o U 2 ,

so that

52)

53a)

53b)

deW = (E - C)(E + C) = (u l - U 2 )2 U I + U 2 )2;;;.0. (54)I t is interesting to note that if

deW = 0, 55)

the solutions to this Eq. 55) are particular solutions to Eqs.(43) since deW = 0 implies U I = U z and/or U I = - u 2 IfU I = U2, then E = C, the rank of f l is 1 whenever E #0, andEqs. 49) both reduce to 2(qi + qi)(ijl + ql) = 0; that is, wehave two equations whose solutions are contained in the setof solutions to Eqs. 49). If U I = - U 2 , then E = - C, therank of n is 1 whenever E 0), and again we have two equations whose solutions are contained in the set of solutions toEqs. 49). Finally, iful = U2 and U I = - U 2, thenqi = qi = 0, and we have the oscillator at rest. Thus, we haves-equivalence even when we relax the condition deW 0.

Ifwe now consider relation (34), i.e.,

56)

the equations of motion related to r r e

57)

where

A = E + C P C)E + p

58)

and we have

detA = E 2 - C 2 + 2 p + p2 > 0, 59)so that we have s-equivalence among Land r, because A isinvertible. The Hamiltonian obtained from the Lagrangian48) is

h 1['4+'4 4 4] 3 2 2 2 2)= g q l q 2 + q l + q 2 + 4 q l q 2 + q l q 2+ (qi + q ~ ) q i+ q ~ )+ qlq2qlq2

where

with

ql = ~ S I+ S2 + TI + T2 ,qz = ~ S I- S2 + TI - T 2 ),

SI = P(PI + P2 + [(ql + q2( + 9(pl + P2)ZJl/2) 113S2 = [3(pl - pz) + [(ql - qZ 6 + 9(pl - P2f]1/2) 113TI = [3(pl + P2 - [(ql + q2 6 + 9(pl + P2)2]1/2) 113T2 = P(PI - P2 - [(ql - q2 6 + 9(pl - P2)2j1/2) 1/3,

a l a lPI = aql ' P2 = aq2




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Example 2: If we now chose f l to be given by

(61e)

62)

The equations of motion obtained from this Lagrangian arethen

I, = C(ii, + qtl + E(ii2 + q2) = 0,12 = E(ii, + qtl + C(ii2 + q2) = 0,

which is again the expected result for f l given in 60).

(63a)

(63b)

With arguments similar to those stated in Example 1,we conclude here that we also haves-equivalence, even whenwe relax the condition deW 0 . Also, with relation 34)E = pL + 1, p> 0) we obtain L which is s-equivalent to L.

The Hamiltonian obtained from the Lagrangian 62) ish = (q,Q2 + q,q2)(4i + 4i + qf + ~ ) ,where

with

q, = ~ ( S ,+ S2 + T, + T2 ),q2 = ~ ( S , S 2 + T, - T2 ),

S, = l3 p, + P2) + [(q, + q2)6 + 9(p + h)2] /2] /3,S2 = [ - 3(p, - P2) + [(q - q2)6 + 9(p - P2)2]1/2j1/ 3,T, = P p, + P2) - [(q, + q2)6 + 9(PI + P2)2]'/2] '/3,Tl = [ - 3(p, - Pl) - [ ql - q2)6 + 9(p, - P2f]'/Z] 113,

al alp, = - . P2 = - aql aq2

Example 3: Consider now the usual Lagrang ian associated with the TSHO

L = (q7 + ~ + ~ ) - ( q ~+ ~ + ~ ) . 64)The related equations of motion fo L are now

Li==ii, + qi = 0, i = 1,2,3. (65)The method we used in the last two examples for con

structing both s-equivalent Lagrangians for the BSHO caseis also useful in the three-dimensional case, although thecalculations become much more tedious. Nevertheless, we


can take advantage of the symmetry in the Lagrangian obtained in the first example and given by Eq. 48) from whichwe will construct the three-dimensional s-equivalent Lagrangian by just symmetrizing this expression to include thethird coordinate (q3)' After adding the necessary terms, weobtain

1= i.Mi + qi + qj + ( q i q ~+ ~ ~+ ~ q ~ )+ (qi + ~ + ~ ) ( q ~+ ~ + ~ )+ q,qlq,qZ + q,q3q,Q3 + q2q3q2q3- i(qi ~ + ~ q ~+ ~ q ~)- A(qi + qi + qj). 66)

The equations of motion obtained from this Lagrangian arethen:

I, = E(ii, + ql + Cdii2 + q21 + C'3(ii3 + q3) = 0,Iz = C2 Ml + qtl + E (ii2 + q2) + C23(ii3 + q3) = 0,13 = C 3 Ml + qtl + Cd42 + q2) +E 43 + q3) = 0,which can be written concisely in the form

{ ~ ~ ) = ~ I~ 2~ 3 C 31 C 32

(67a)

(67b)

(67c)

(67 )

where E = ( c j ~+ j ~ + ~ + qi + ~ + ~ ) is the total energy of the oscillator and the quantities Cij are all constantsand are given by

C ij = qiqj + qiQj 68)In this case we then have

f l = ( ~ I C ~ ~ ~ : ) 69)C 3 , C 32 E

We can define again the vectors U i as in the previous twoexamples [cf. Eq. 52)]. If E = 0, then U i = 0, for every i, andwe have the oscillator at rest. Now, taking E> 0, the determinant of n can be written:

2u,ou2 l E

deW = E3 2u 20u,IE 12u30u,IE 2u 30u21E

Using the inequalities

u, - u)f + ~ ; ; ; . O ,- u, + Uj )2 - u ~


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p > 0), once again, we have

73)

from which we get

detA = deW + 3E2 - Ci2 - Ci3 - C ~ 3 P+ 3E p 2p3>O,

becauseE 2 >Ct due to the fact that -l ;CijIE..;l. Thismeans the family of Lagrangians Lis s-equivalent to L because A is invertible.

v CONCLUSIONS ND OUTLOOK

We have generalized the results of Ref. 1 to the case ofseveral dimensions. The theorem which states that the traceof all integer powers of A are constants of the motion allowsone to cons truct examples in which the elements of A arefunctions of qj> qi and t (without reiterating the one-dimensional procedure). A particularly interesting result is that itis possible to retain s-equivalence even when detA vanishes,

and this fact is important when the quantum theory is considered. For the two-dimensional examples we have shownthat s-equivalence prevails even when the rank of A is zero.A general theorem which would allow one to determine inwhich cases this is possible is currently underinvestigation. 16

Some problems which remain unsolved are:i) the general solution of Eqs. 35) (for I and n ) n any

number of dimensions;


ii) the consequences in quan tum theory of the fact thatclassical realistic systems can be described equally well inmore than one way be means of s-equivalent Lagrangians 2 ,3;

iii) the study of the problem discussed here within therealm of gauge and field theories. 17

CKNOWLEDGMENTS

We would like to thank Luis de la Perra, Santiago Lopez

de Medrano, and Michael P. Ryan Jr. for fruitful discussionsand encouragement. We also thank the referee for usefulsuggestions.

'D. F. Currie and E. J. Saletan, J. Math. Phys. 7, 967 1966).2G. Marmo and E. J. Saletan, Hadronic J. 1,955 1978).3S. Hojman and R. Montemayor, Hadronic J. 3,1644 1980).4R. M. Santilli, Ann. Phys. (N. Y.) 103, 354 1977).'R. M. Santilli, Ann. Phys. (N. Y.) 103,409 1977).OR M. Santilli, Ann. Phys. N. Y. 105, 227 1977).7R. M. Santilli, Foundations o f Theoretical Mechanics. I (Springer, NewYork,1978).

SR. M. Santilli, Hadronic J. 1, 223 1978).oR. M. Santilli, Hadronic J. 1, 574 1978).lOY. Gelman and E. J. Saletan, Nuovo Cimento B 18, 62 1973).

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G. Darboux, Lec,ons sur l tMorie generale des surfaces (Gauthier-Villars,Paris, 1891).

'40. Bolza, Lectures on the Calculus o f Variations (Dover, New York, 1961).15J Douglas, Solution of the Inverse Problem of the Calculus ofVari

ations, Trans. Am. Math. Soc. 50, 71 1941).'Some work along these lines has been done, for the one-dimensional case,

by S. Lopez de Medrano and M. Lara Aparicio (private communication).17S. Hojman and L. Shepley, in preparat ion.


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