‘On leave from: University of Ain Shams, Department of Mathematics, Cairo, Egypt. OCR Output

December l993

l\/IIRAMARE ~ TRIESTE

structure for the phase space and consequently the underlying symplectic structure.description of such systems due to Ostrogradsky, but unlike the later it provides a simplerorder Lagrangians is reached. The new Hamiltonian description recovers the conventional

Using Dirac’s analysis of second class constraints, a Hamiltonian formulation of higher

Abstract

International Centre for Theoretical Physics, Trieste, Italy.

MS. Rashid and S.S. Khalill

HAMILTONIAN DESCRIPTION OF HIGHER ORDER LAGRANGIANS

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

United Nations Educational Scientific and Cultural Organization

and

International Atomic Energy Agency

IC/93/420

x1 dt Ox(3) OCR Output-L ...... +(-i)m%{§,.}. dt O@QLQ

by methods of variation is given by the expressionfinite order say m . Then he showed that the equation of motion obtainedwhich is a function of the co-ordinates and their time derivatives up to a

analysis he considers systems described by the Lagrangian .C(;r, jc, cr, .... , x('”)),very problem was considered by Ostrogradsky a century ago [1],[2], in hisHamiltionian—to-be agrees with those obtained by methods of variation. Thistransformation and check whether the equations of motion derived from themotion, rather one ought to seek an appropriate generalisation of the aboveit is, since the resulting Hamiltonian does not produce the right equations ofblindly use the above expression for the Legender transformation, eq(2), as

However for systems described by higher order Lagrangian one cannot

nzi

H: } ip"<1n—£

is given by

6%<u- GP

conjugate momenta, which are defined asof freedom it entails i.e. the configuration space CO- ordinates qu and theirtransformation, which for a first order Lagrangian Lfqn, and the degreescourse, as it is well known, the corner stone in this description is the Legenderdescription of ordinary systems (described by first order Lagrangians). Ofintuitively would like to follow the normal procedure used for the Hamiltonian

ln order to reach to the Hamiltonian description of these systems one

matter.

variables, for instance, second order or any other finite order say m for thatin the sense they contain higher order derivatives of the configuration spacesystems are described by higher order Lgrangians, which are not first orderclass of dynamical systems which are oftenly ignored in the literature. Theseln this paper we are going to investigate the phase space structure of certain

1 Introduction

Such systems were extensively studied and a procedure for establishing a OCR Outputthat of Hamiltonian description of a system with second class constraints.surface of constraints i.e. {X", xm} qé 0. As a result the problem reduces toclass ones since the algebra of constraints does not vanish weakly on theconstraints imposed on the extended phase space of the system are secondfact it turns out, as we are going to show in the next section, that all thesethat the number of constraints equals the number of auxiliary fields. Inauxiliary fields qu, as well one has to introduce a set of constraints Xu such

Our procedure is based on introducing a new degrees of freedom calledthe canonical quantization of higher order Lagrangian easier to carry out.principle and gives a relatively simpler phase space structure, which makestivated by finding an alternate Hamiltonian description which is simpler inany attempt to quantize these systems hindered by technical difficulties. Moderlying symplectic geometry has a complicated structure, the thing makestrogradsky Hamiltionian description is quite intricate. Consequently the un

From the point of view of quantization the phase space structure of Osof such systems.grangian L . With this Ostrogradsky concluded his Hamiltonian descriptionidentical to the ones obtained by applying methods of variation to the Lato show that the equations of motion derived from the above Hamiltonian areWith the Hamiltonian 'H at our disposal, it is a matter of few lines of algebra

7`(:p]ii7+P2.;L;+ .......... +pm.I—L.(m)simple formLegender transformation which, in terms of these momenta, has the followingThe new generalised momenta, eqs (4), are then used to generalised the

Pm I %.

(4)p2= {2%}+ -··-· +<—1>"‘2%{%}pl = % — gg?} + ..... + (—}2;"·1j§$‘,§{%}

grangian, he first defined some generalised momenta asTo establish the Hamiltonian description of the above higher order La

U”'{><$/» xw} = 6Z~· (9) OCR Outputwith

Css’ : (A—l)S$/

such that Asp is a regular matrix, and let the inverse be defined as

Asa = {><$».><SI} (7)

the matrix of Poisson brackets of second class constraints

Let us denote all the second class constraints by X, raw 0. Then considerto use Dirac’s approach which in principle solves the problem in all cases.to implement in practice. However in our analysis of constraints we are goinging degrees of freedom are introduced. Such a procedure may be very difficultfreedom. By discarding them, new Poisson brackets, referring to the remainthese degrees of freedom can be solved for in terms of the other degrees ofgrees of freedom are redundant for the description of the system. In principle

ln fact second class constraints appear whenever some phase space de

classified as either first—class or second-class constraints.

class. Therefore in terms of these notations, all the constraints are thenPoisson brackets with at least one of the constraints, is said to be second—A quantity which is not first—class, namely which has a non weakly vanishing

{R xr} <—> {Rm} = ) lcfxt s¤ 0 (6)

with all constraints Xi is weakly vanishing:space quantity R(q,,, pn) is said to be first—class quantity ifits poisson bracketsfunction used to describe the system was introduced by Dirac[5],[9]. A phase

Acharactirization of constraints which is independent of the Lagrangianaccount of Dirac’s description of systems with second class constraints.For the sake of completeness, in the next few paragraphs we will give a briefHamiltonian description of which is explicitly given in refs [5],[6],[8]and[9].

quantization procedure. OCR Output

Dirac brackets of the phase space variables, since they lay at the heart of anytwo Lgrangians is investigated. ln section three we are going to compute thestraints is applied to the alternative Lagranian, then the equivalence of the

This paper is organised as follows, in section two Dirac`s analysis of conconsistently ignored.the redundant degrees of freedom related to the second—class constraints areis not really necessary ) one obtains a description of the system in which all

By using Dirac brackets and then solving the constraints ( this last stepprovided one uses Dirac brackets.imposed exactly, and not as weak equality, even before working out brackets,

included in the definition. As a result the second-class constraints can be

of freedom related to the second—class constraints XS x 0 are effectively notthat Poisson brackets are redefined in such a way that the redundant degreesDirac bracket achieve precisely the modification of Poisson bracket, namelysecond—class constraint X, is strictly zero i.e. X$}D : O. This shows thatMoreover, Dirac bracket of an arbitrary phase space quantity f with any

(U)if»H}D = {fit!} · } l{f,><$l}C“I5"{><$~»H}·

The equation of motion of an arbitrary phase space quantity f, is given

values of the fundamental brackets which are modified.

sense that they satisfy the same properties as Poisson brackets except thelt is easy to show that Dirac brackets are generalised Poisson bracket, in the

(10){f»y}D = {fig} - Z{f,><$}C”'{><S»»y}

The Dirac bracket of two phase space quantities is then given by

axI ——- : U. H1 + 15 ( ) OCR Output

OL

show that variations with respect to ar results in the following equationthese equations with the original equation of motion eq(3). lt is trivial tofunctional of the new Lagrangian, eq (14), to be stationary. Then comparethe multipliers i.e. (;v, qfs and Ms) and acquiring the new action, which is aas a result of varying the co-ordinates of the extended phase space includingdescriptions at the Lagrangain level, by first deriving the equation of motion

Next in our procedure is to investigate the equivalence between the two

time derivative.

first order in the sense it is only a function of the co-ordinates and their firstThis Lagrangian, unlike the original Lagrangian, has the advantage of being

(14)C = £¤(¤v,<1i» ---··»<1m»<1%¤) + } Quext

based on the new Lagrangian;

extended phase space. Consequently, one arrives to an alternate descriptionin fact it is not hard to see that eqs(l3) define the surface of constraints on the

xi =(<1i —<f) ~ 0; xz = (q2—<ii) ~ 0; -···- sxm = (qm—<im-i) ~ 0, (13)

will re—write the above equations as a set of m constraints;Since the new auxiliary fields q, are not true physical degrees of freedom we

qi =w; <12=qi; ···--· ;qm=qm~i (12)

by making the following identification in the original Lagrangian LThe Construction proceeds, first by introducing the new auxiliary fields

simpler, Hamiltonian description of these systems.and introducing some auxiliary fields one can establish an alternate, butsection we are going to show that on accounts of Dirac analysis of constraintsclass of dynamical systems described by higher order Lagrangians. But in thisln the previous section we have briefly given a Hamiltonian description of a

straints

2 Auxiliary Fields and Analysis of Con

straints in the new Hamiltonian description. In addition, the other set of OCR Outputit follows immediately that they correspond to the first set of primary con·By observing that the conjugate momenta in eqs(2())are time independent,

Qm—l Z Z `iumy (bm Z Qm—1'l' Hm z

(20)

Qi=f=-#2, @2=Qi+u2¤U»%:·Mi» (D1:P+/LlP'¤0a

set of canonical momenta

reproduces the original equation of motion. To do so, we define the followingthe Hamiltonian level, we are going to check whether the Hamiltonian-to-be

ln order to investigate the equivalence between the two descriptions atLagrangian level.tion, the thing shows the equivalence between the two descriptions at theWhich is identical to the original equation of motion as in the first descrip

m i-—(J)+T(4)+ ...... +(-1) -;(—%):0. (is) G1: dt Ox dt Ox dt Bxdm GLBL d OL d2 OL

and using the constraints, eq(l8), one obtains the expressionRemoving the multipliers in eqs(l5),(l6)and (17) by successive substitution

Y1 Z 0,Xg Z ......... Xm Z

straints

and variations with respect to the multipliers reproduces the primary con

6% dt3%17 ( )m I _? + — _? , M ( )

at rz arand

[km `i' Hm-] `i" : 0

(16)

pz + pi + $ = 0

Similarly variations with respect to qfs yield another set of equations

, Ai Z, 27 ( ) OCR Output

M Z (Qi · @-1)

then it follows trivially that

{nalin} Z(qe·<je-1), (26)and

(25){@2,,Ho} Z 5; + H2{@1,Ho} Z éi + Mi

to vanish i.e. in = ll) : O. Observing thatarbitrary multipliers by acquiring the time evolution of the above constraintsln this expression for the extended Hamiltonian, one determines the two

i=l i=1

HE : H0 + x,<1>i+ xm, (24)Z 2

while the form the extended Hamiltonian HE is

i:l{:1 {:1

Ho = xp + qiQ+ MW —£0($»¢1¢»<im)—> j/M(<1i —<1¢l1)» (23)Z i Z

The form of the basic Hamiltonian 'HO is

which shows that these constraints are second-class constraints.

{¢¢»7U} Z 6ey,(22)

{QSU I Ov {R-is7l-j} I 0

ln fact it easy to show that the above constraints satisfy the following algebra

GMHi Z R' O i: l,..»n. (21)

Lers;

primary constraints corresponds to the conjugate momenta of the multipli

the Lagrangian formulation. OCR Outputpected to appear since they are non other than the original constraints X, inThe last equation in eqs(31) is trivial, whereas the other equations are ex

Qm I lqm>HEl : qm_ qm—1 I {qm—l»HEl I qm

(31)

Qi Z l€l1»HEl I Q2_ JJ = {1*,,HE} Z qi

and

_ .. Qu

Z ,71 : - M Q2 {Q2 Q1 + @(12Q1;lQ1»HEl: _P ‘l‘ gilii I {RWE} I %%Q

In doing so one gets the following equationsthe phase space variables (;r,P,q,,Q,) with the extended Hamiltonian HE.of motion, following the normal procedure of evaluating Poisson brackets of

To check the validity of the above Hamiltonian, let us derive the equations

` HE : qmQm + qiP + q2Qi + + qmQm-i — Lo.HE = <imQm + qiQi—1£O

form

which after cancellations and rearrangements of some terms it has the new

’ `|r`I'l ' I HE I H0 +(<1¢— qi-1)(Q¢-1+ M) *· ml * MH (28)

the new expressionSubstituting the value of the two multipliers in the expression for HE gives

36 ( ) OCR Output—————·— = U, Gm dt(8q1)+Q1GLU d BLU

observing that one can re-write the first equation astonian level, we need to recover the equation of motion from eqs(30), by

To complete the equivalence between the two description at the Hamilof the expressions for the momenta.were determined by inspection, the analysis of constraints dictates the formdescription. But in this description, unlike the first one where the momentaThese equations are identical to the expressions for momenta in the first

QTY?. ; 3i€m)·(35)

I@1:%,-a;-$@5;)+ ....... +(~1>m·’;_;%%(§,$w)

arrives to the following expressionsApplying the same procedure to the rest of the momenta Q1, .... Qm, one

(34)m_ p Z 41 — 44) s T(-gi + ....... + (-1) *75%%) GJ: dt 82* dt Gm df Gmdm`1 OLOL ei 3L d2 OL

and then repeating this substitution m times one finally gets

33 ( )+‘···· 1 + gql d,(8q2) Q2

followed by the observation,

3%32 ( )P = —Q + I

eqs(30) to obtain,momenta are the same as those of the first description in disguise, one usesas the first description. To show that the above expressions for the generalisedthe original equation of motion and expressions for the generalised rnomenta

At this stage it is far from obvious whether the above equations reproduce

lf) OCR Output

non other than auxiliary fields. Furthermore, our calculation shows that theThe last equation, eq(4l), is expected since it suggests that the qi fields are

{qa QJ} I U

algebrawhile for the remaining degrees of freedom, they satisfy the following trivial

fxvPlD:{x>PlP:1=

instance

Now we are in a position to compute all the relevant Dirac brackets, for

A ‘ ( 9)(1... om ·but more relevant, is the inverse matrix

3 ( 8)Om Im -[m Om) »the constraints. Which in this case takes the simple formin the first section, one needs to determine the matrix of Poisson brackets ofphase space degrees of freedom. Following the procedure we have describedln this section we are interested in computing Dirac brackets of the extended

variables

3 Dirac brackets of the extended phase space

alent to that of Ostrogradsky description of higher order Lagrangians.eq(3). With this we have shown that the new description is classically equivReiterating this procedure m times yields the original equation of motion

————·—————·= 0. Bac dt(0q1)+dt2(8q2) Q2 37 ( )GLU d BLU d2 BLU

then substituting in the second equation one gets

11 OCR Output

Methods of Classical Mechanics, (Springer-Verlag, New York, 1978).(Benjamin, Reading, Massachusetts, 1978); V.I. Arnold, Mathematical

[2] R. Abraham and .l.E. Mardsen, Foundations of Mechanics, 2nd edition

[1] A.Ostrogradsky, Me’langes de l` Acad. de St. Pe’t. Oct. 1884.

References

for useful discussions.Physics, Trieste. One of the authors (MSR) would also like to thank Professor J. GovaertsEnergy Agency and UNESCO for hospitality at the lnternational Centre for Theoretical

The authors would like to thank Professor Abdus Salam, the International Atomic

5 Acknowledgements

lication in preparation, is easy to carry out.

structure. Consequently quantization of such systems, which is another puborder Lagrangians. In fact this description has a very simple phase spacenatural Hamiltonian description of dynamical systems described by higher

ln this paper we have shown that analysis of constraints provides a veryMills theories, .... etc. have been a fertile land for such analysis.singular systems. Finite dynamical systems on compact spaces, QED, Yangviable but effective methods of establishing a Hamiltonian description ofln the physics literature analysis of constraints is proven to be one of the

4 Conclusion

matter.

zation of systems described by higher order Lagrangians becomes a trivialDirac bracket. Equipped with the above symplectic structure, the quantifundamental Poisson bracket of (:c,P) is not modihed when generalised to

12 OCR Output

Press, 1991).ven Notes in Mathematical and Theoretical Physica, (Leuven UniversityJ. Govaerts, Hamiltonian Quantization and Constrained Dynamics, Leu[9]

(Springer—Virlag, Berlin, 1982).K. Sundermeyer Constrained Dynamics, Lecture notes in Physics 169,[8]

Systems (Academia Nazionale dei Lincei, Rome, 1976).A.J.Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian[7]

Boston, 1983).1980); N.E. Hurt, Geometric Quantization in Action, (D. Reidel,N.M.J. Woodhouse, Geometric Quantization (Oxford University Press,[6]

School of Science, Yeshiva University, New York, 1964).P.A.M. Dirac, Lectures on Quantum Mechanics, (Belfer Graduate[5]

ford Unoversity Press, 1985).P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edition (Ox[4]

Perspective (Jhone Wiley nd Sons, New York, 1974).E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: a Modern[3]