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Hamiltonian Description of Higher Order Lagrangians

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Hamiltonian Description of Higher Order Lagrangians IC/93/420 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS HAMILTONIAN DESCRIPTION OF HIGHER ORDER LAGRANGIANS MS. Rashid and S.S. Khalill International Centre for Theoretical Physics, Trieste, Italy. Abstract Using Dirac’s analysis of second class constraints, a Hamiltonian formulation of higher order Lagrangians is reached. The new Hamiltonian description recovers the conventional description of such systems due to Ostrogradsky, but unlike the later it provides a simpler structure for the phase space and consequently the underlying symplectic structure. l\/IIRAMARE ~ TRIESTE December l993 ‘On leave from: University of Ain Shams, Department of Mathematics, Cairo, Egypt. OCR Output 1 Introduction ln this paper we are going to investigate the phase space structure of certain class of dynamical systems which are oftenly ignored in the literature. These systems are described by higher order Lgrangians, which are not first order in the sense they contain higher order derivatives of the configuration space variables, for instance, second order or any other finite order say m for that matter. ln order to reach to the Hamiltonian description of these systems one intuitively would like to follow the normal procedure used for the Hamiltonian description of ordinary systems (described by first order Lagrangians). Of course, as it is well known, the corner stone in this description is the Legender transformation, which for a first order Lagrangian Lfqn, and the degrees of freedom it entails i.e. the configuration space CO- ordinates qu and their conjugate momenta, which are defined as - GP <u 6% is given by H: } ip"<1n—£ nzi However for systems described by higher order Lagrangian one cannot blindly use the above expression for the Legender transformation, eq(2), as it is, since the resulting Hamiltonian does not produce the right equations of motion, rather one ought to seek an appropriate generalisation of the above transformation and check whether the equations of motion derived from the Hamiltionian—to-be agrees with those obtained by methods of variation. This very problem was considered by Ostrogradsky a century ago [1],[2], in his analysis he considers systems described by the Lagrangian .C(;r, jc, cr, .... , x('”)), which is a function of the co-ordinates and their time derivatives up to a finite order say m . Then he showed that the equation of motion obtained by methods of variation is given by the expression @ QLQ-L ...... +(-i)m%{§,.}. dt O (3) OCR Output 1 dt Ox x To establish the Hamiltonian description of the above higher order La grangian, he first defined some generalised momenta as pl = % — gg?} + ..... + (—}2;"·1j§$‘,§{%} p2= {2%}+ -··-· +<—1>"‘2%{%} (4) Pm I %. The new generalised momenta, eqs (4), are then used to generalised the Legender transformation which, in terms of these momenta, has the following simple form 7`(:p]ii7+P2.;L;+ .......... +pm.I—L.(m) With the Hamiltonian 'H at our disposal, it is a matter of few lines of algebra to show that the equations of motion derived from the above Hamiltonian are identical to the ones obtained by applying methods of variation to the La grangian L . With this Ostrogradsky concluded his Hamiltonian description of such systems. From the point of view of quantization the phase space structure of Os trogradsky Hamiltionian description is quite intricate. Consequently the un derlying symplectic geometry has a complicated structure, the thing makes any attempt to quantize these systems hindered by technical difficulties. Mo tivated by finding an alternate Hamiltonian description which is simpler in principle and gives a relatively simpler phase space structure, which makes the canonical quantization of higher order Lagrangian easier to carry out. Our procedure is based on introducing a new degrees of freedom called auxiliary fields qu, as well one has to introduce a set of constraints Xu such that the number of constraints equals the number of auxiliary fields. In fact it turns out, as we are going to show in the next section, that all these constraints imposed on the extended phase space of the system are second class ones since the algebra of constraints does not vanish weakly on the surface of constraints i.e. {X", xm} qé 0. As a result the problem reduces to that of Hamiltonian description of a system with second class constraints. Such systems were extensively studied and a procedure for establishing a OCR Output Hamiltonian description of which is explicitly given in refs [5],[6],[8]and[9]. For the sake of completeness, in the next few paragraphs we will give a brief account of Dirac’s description of systems with second class constraints. Acharactirization of constraints which is independent of the Lagrangian function used to describe the system was introduced by Dirac[5],[9]. A phase space quantity R(q,,, pn) is said to be first—class quantity ifits poisson brackets with all constraints Xi is weakly vanishing: {R xr} <—> {Rm} = ) lcfxt s¤ 0 (6) A quantity which is not first—class, namely which has a non weakly vanishing Poisson brackets with at least one of the constraints, is said to be second— class. Therefore in terms of these notations, all the constraints are then classified as either first—class or second-class constraints. ln fact second class constraints appear whenever some phase space de grees of freedom are redundant for the description of the system. In principle these degrees of freedom can be solved for in terms of the other degrees of freedom. By discarding them, new Poisson brackets, referring to the remain ing degrees of freedom are introduced. Such a procedure may be very difficult to implement in practice. However in our analysis of constraints we are going to use Dirac’s approach which in principle solves the problem in all cases. Let us denote all the second class constraints by X, raw 0. Then consider the matrix of Poisson brackets of second class constraints Asa = {><$».><SI} (7) such that Asp is a regular matrix, and let the inverse be defined as Css’ : (A—l)S$/ with U”'{><$/» xw} = 6Z~· (9) OCR Output The Dirac bracket of two phase space quantities is then given by {f»y}D = {fig} - Z{f,><$}C”'{><S»»y} (10) lt is easy to show that Dirac brackets are generalised Poisson bracket, in the sense that they satisfy the same properties as Poisson brackets except the values of the fundamental brackets which are modified. The equation of motion of an arbitrary phase space quantity f, is given if»H}D = {fit!} · } l{f,><$l}C“I5"{><$~»H}· (U) Moreover, Dirac bracket of an arbitrary phase space quantity f with any second—class constraint X, is strictly zero i.e. X$}D : O. This shows that Dirac bracket achieve precisely the modification of Poisson bracket, namely that Poisson brackets are redefined in such a way that the redundant degrees of freedom related to the second—class constraints XS x 0 are effectively not included in the definition. As a result the second-class constraints can be imposed exactly, and not as weak equality, even before working out brackets, provided one uses Dirac brackets. By using Dirac brackets and then solving the constraints ( this last step is not really necessary ) one obtains a description of the system in which all the redundant degrees of freedom related to the second—class constraints are consistently ignored. This paper is organised as follows, in section two Dirac`s analysis of con straints is applied to the alternative Lagranian, then the equivalence of the two Lgrangians is investigated. ln section three we are going to compute the Dirac brackets of the phase space variables, since they lay at the heart of any quantization procedure. OCR Output 2 Auxiliary Fields and Analysis of Con straints ln the previous section we have briefly given a Hamiltonian description of a class of dynamical systems described by higher order Lagrangians. But in this section we are going to show that on accounts of Dirac analysis of constraints and introducing some auxiliary fields one can establish an alternate, but simpler, Hamiltonian description of these systems. The Construction proceeds, first by introducing the new auxiliary fields by making the following identification in the original Lagrangian L qi =w; <12=qi; ···--· ;qm=qm~i (12) Since the new auxiliary fields q, are not true physical degrees of freedom we will re—write the above equations as a set of m constraints; xi =(<1i —<f) ~ 0; xz = (q2—<ii) ~ 0; -···- sxm = (qm—<im-i) ~ 0, (13) in fact it is not hard to see that eqs(l3) define the surface of constraints on the extended phase space. Consequently, one arrives to an alternate description based on the new Lagrangian; C = £¤(¤v,<1i» ---··»<1m»<1%¤) + } Quext (14) This Lagrangian, unlike the original Lagrangian, has the advantage of being first order in the sense it is only a function of the co-ordinates and their first time derivative. Next in our procedure is to investigate the equivalence between the two descriptions at the Lagrangain level, by first deriving the equation of motion as a result of varying the co-ordinates of the extended phase space including the multipliers i.e. (;v, qfs and Ms) and acquiring the new action, which is a functional of the new Lagrangian, eq (14), to be stationary. Then compare these equations with the original equation of motion eq(3). lt is trivial to show that variations with respect to ar results in the following equation OL I ——- : U. H1 + 15 ( ) OCR Output ax Similarly variations with respect to qfs yield another set of equations pz + pi + $ = 0 (16) [km `i' Hm-] `i" : 0 and at rz ar m I _? + — _? , M ( ) 17 ( ) 6% dt3% and variations with respect to the multipliers reproduces the primary con straints Y1 Z 0,Xg Z ........
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