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REFER Hivu:- IC/89/61 INTERNATIONAL CENTRE for THEORETICAL PHYSICS REFER hivU:- IC/89/61 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THEORY OF SINGULAR LAGRANGIANS Jose F. Carinena INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/89/61 1. INTRODUCTION International Atomic Energy Agency There are different approaches in Classical Mechanics and one usually speaks of New- tonian, Lagrangian and Hamiltonian Mechanics. The main guide for Newtonian mechan- and ics is the determinism principle, according to which the knowledge of the positions and United Nations Educational Scientific and Cultural Organization velocities of the points of a mechanical system at a fixed time determines their future positions and velocities [1]. The idea is then to use a system of second order differential INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS equations in normal form i- = /U*J>), (l.i) because the theorem of existence and uniqueness of the solution of such systems fits well to the deterrninism principle. In the following, for the sake of simplicity, we will restrict ourselves to the study of cases in which the functions /' do not depend on the time t. Such a study may be reduced to that of an associated Erst order system THEORY OF SINGULAR LAGRANGIANS • (1.2) Jose F. Carinena ** International Centre for Theoretical Physics, Trieste, Italy. and these equations can be geometrically interpreted as a set of differential equations locally determining the integral curves of a vector field (1.3) which is a special kind of vector field in the tangent bundle of the configuration space, to be studied later on. The introduction of the Lagrangian mechanics is based on the observation that, at least for conservative systems, the equations of motion may also be Been as the Euler equations determining the solution curves of a variational problem [1-3]; the action J Ldi, with L = T ~ V, is extremal for the actual path when compared with other fixed-endpoint paths. More specifically, if the functions /* depend only on x} and there exists a function V{x') such that /' = — Jp-, the equations of motion (1.1) reduce to the MIRAMARE - TRIESTE Euler-Lagrange equations ar j t ar \ (1-4) April 1989 dtydx' where £ = j £r=i (*') — V(x>), and it is well known that these equations can be seen as furnishing the curves with fixed ends i'(ti) = x\, I'((J) = x\ for which the action 5= L(x',i>)dt (1.5) is an extremal. Submitted for publication. Permanent address: Departamento de Fi'sica Te6rica, Universidad de Zaragoza, 50009 Zaragoza, Spain. -1- There are very many interesting advantages in the use of this "Hamilton principle" Consequently, if the function V(i',wJ) is defined by because the extremal principle does not depend on a particular choice of coordinates and moreover it may incorporate the holonomic constraints with a particular choice, leading in this way to a considerable simplification. v-,(*-f A), In fact, if we have k independent holonomic constraints 4>"{x') = 0, a = 1,..., i, the freedom in the choice of coordinates allows one to choose the last k coordinates yi by then Fi can be written yTi-t+a _ ^"»(a-J) gjjd to complete this set to form a new set of (generalized) coordinates. SV d(8V The constraint forces can be simulated [1] by the introduction of a potential and a Lagrangian for the description of the motion of a charged particle is then given by ^) (1.7) and taking the limit of the real numbers TJ° going to infinity The last k Euler-Lagrange equations only mean that the motion is on the surface defined by the constraints <f>" = 0; where T is the kinetic energy T = Jm(v • v). and n — k Euler-Lagrange equations, where the values of the last k y's are fixed to be Notice that the Lagrangian for a system (1,1) can be nonuniquely defined. In the zero, remain. particular case of (1.7), if we change from (A, ^) to a gauge equivalent description (A' = The use of Hamilton's principle has been shown to be very efficient for attacking A + Vx, 4>' — 4> - iff), we will get a different (but gauge equivalent) Lagrangian different problems in both Classical Mechanics and Classical Field Theory. The problem of what kind of second order differential equations can be obtained as the Euler-Lagrange equations of a Lagrangian function, is known as the "inverse problem", and it has received (1-8) a lot of attention [4,5] till the last few years in which geometric conditions, substituting for the well-known Hehnholtz conditions, have been given [6]. where x(x,t) is an arbitrary but differentiable function. The inverse problem can also be considered in a more generalized sense [7], namely, The Euler-Lagrange equations in generalized coordinates the question of whether there exist functions /ij (such that det jij ^ 0) and a Lagrangian function L such that the Euler-Lagrange equations defined by L are dtyd'J 8q' (1.9) are a set of second order differential equations or in other words, what is physically relevant is not the system (1.1) but its set of solutions, the system (1.6) being then equivalent from this point of view. (1.10) A particulary interesting case of equations (1.1) admitting a Lagrangian description is that of the motion of a particle with an electric charge q in an electromagnetic field. We recall that such equations are given by the Lorentz force (summing over repeated indices is understood), which can be expressed in normal form (1.1) if the Hessian matrix: F = ,[E+lvAB]. The electric and magnetic fields can be expressed in terms of the vector and scalar potentials A,<t>, by means ofE = -V<^-l^,B = rotA, and therefore, with some is regular. The function L is then said to be a regular Lagrangian. The Hamiltonian vectorial calculations we find that formulation is then introduced as follows [1-3]. Since det W / 0, the Bet of equations Pi ~ 5-rr-1 = „0 (1.12) dq • -2- -3- and the time-evolution of a dynamical variable f(q,p) can be written, making use of can be used, by means of the implicit function theorem, to define q* as a function Hamilton's equations, as I -&••&• -«»>• (1.17) and the correspondence from (q, q) to (q, p) BO defined is called the Legendre transfor- 2. SINGULAR LAGRANGIANS: DIRAC'S THEORY mation. We can then define a function in the phase space T of the g' and pj by If the Hessian matrix W given by (1.11) is not everywhere regular, the Lagrangian function X will be called a singular Lagrangian and for technical reasons, to be explained (1.13) later, we will assume that the rank of W is constant. As far as the Euler-Lagrange equations (1.10) are concerned, we remark that they cannot be expressed in a normal •=] form because the matrix of the coefficients of the accelerations is singular. This implies called the Hamiltonian function. And then, that the theorem for existence and uniqueness of solutions of systems of differential equations does not apply; and in fact there can be some initial conditions for which there dH exists no solution at all, for only some initial positions and velocities are compatible with such equations. Indeed, if f (q, q) denotes an eigenvector function for W with zero Qqf Oof eigenvalue, then when left-multiplying (1.10) by the row £ we obtain as a necessary condition for compatibility: Therefore, if (q'(i),pj(t)) is the curve corresponding to a solution of (1.9), the Hamil- ton equations: (2.1) hold, and conversely, if the curve (q>(.t),pj(t)) satisfies (1-14), the curve q\t) is a solution and therefore in order to have a solution with particular initial values of q' and ?*, a set of (1.9). The Lagrangian and Hamiltonian formulations for a regular Lagrangian are in of restrictions as given by (2.1) will appear. Of course, some of these restrictions may be this way put in a one-to-one correspondence. in some cases trivial identities. Moreover, particular initial values of g' and g' tor which a solution of (1.10) exists In the phase space T it is possible define an R-bilinear map associating with a couple do not uniquely determine only one solution, but a set of possible solutions. This is in of functions / and g on T, a new function, to be denoted {/,ff}, as follows [1-3]: conflict with the determinism principle assumed in the foundations of mechanics, and it is evidence for the existence of spurious degrees of freedom, namely gauge degrees (1.15) of freedom. One may think that a way of overcoming this problem is to go to the Hanultonian formalism which was proved to be, for regular L&grangians, equivalent to the Lagrangian description. However, we soon find that if L is singular, the Legendre Such a function {f,g} is called the Poitsen bracket of both functions, and it is easy transformation is not invertible because the implicit function theorem does not work, and to check that the following properties hold: then we cannot invert relation (1.12), and consequently a Hamiltonian function of q and p only is not well defined. Indeed, the fact of X being singular means that the image of the velocity phase space under tbe Legendre transformation is not all the phase space but only a subset of it, and the coordinates ?' and momenta p< are not independent but are related by some constraint functions <l>a(g,p) = 0, where a = 1,...,M, called by Dirac "primary constraints".
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