Hamiltonian systems with linearly dependent constraints (irregular systems),
are classified according to their behavior in the vicinity of the constraint
surface. For these systems, the standard Dirac procedure is not directly
applicable. However, Dirac's treatment can be slightly modified to obtain, in
some cases, a Hamiltonian description completely equivalent to the Lagrangian
one. A recipe to deal with the different cases is provided, along with a few
pedagogical examples.Comment: To appear in Proceedings of the XIII Chilean Symposium of Physics,
  Concepcion, Chile, November 13-15 2002. LaTeX; 5 pages; no figure

Miskovic, Olivera

Zanelli, Jorge

English

arXiv

Hamiltonian systems with linearly dependent constraints (irregular systems), are classified according to their behavior in the vicinity of the constraint surface. For these systems, the standard Dirac procedure is not directly applicable. However, Dirac's treatment can be slightly modified to obtain, in some cases, a Hamiltonian description completely equivalent to the Lagrangian one. A recipe to deal with the different cases is provided, along with a few pedagogical examples

Miskovic, O

Zanelli, J

CERN Document Server

Irregular Hamiltonian SystemsOlivera Miˇskovic´1,2 and Jorge Zanelli11Centro de Estudios Cient´ıficos (CECS), Casilla 1469, Valdivia, Chile.2Departamento de F´ısica, Universidad de Santiago de Chile,Casilla 307, Santiago 2, Chile.olivera@cecs.cl, jz@cecs.clAbstractHamiltonian systems with linearly dependent constraints (irregular systems), are classi-ed according to their behavior in the vicinity of the constraint surface. For these systems,the standard Dirac procedure is not directly applicable. However, Dirac’s treatment can beslightly modied to obtain, in some cases, a Hamiltonian description completely equivalentto the Lagrangian one. A recipe to deal with the dierent cases is provided, along with afew pedagogical examples.1 IntroductionDirac’s Hamiltonian analysis provides a systematic method for nding the gauge symmetriespresent in a theory. The analysis identies and classies the constraints, which are local func-tions of the phase space coordinates. Consistency requires that the constraints be preserved induring the evolution (for the review of the Hamiltonian analysis see Ref.[1]{[4]). However, if theconstraints are not functionally independent, then Dirac’s procedure is not applicable. The testof functional independence are the so-called regularity conditions, and those systems which failthe test are said to be irregular.Irregular systems are not necessarily intractable or exotic. A simple example is a relativisticmassless particle (pµpµ = 0), which is irregular at the origin of momentum space (pµ = 0). Thispoint in phase space is exceptional, as it is unclear whether this would be an observable statefor a photon, say. On the other hand, we know the conguration pµ = 0 to be a very importantone: the ground state. There are other physical circumstances in which regularity is violated,and not only for isolated states but on large portions of the region in phase space where thesystem evolves. Chern-Simons theories in 2n + 1 spacetime dimensions are examples where, forsome initial congurations, regularity can fail at all times and one is forced to live with thisproblem.Here we discuss the possible ways in which the constraints can fail the test of functionalindependence, and how the Hamiltonian treatment of Dirac must be modied in each case.12 Regularity ConditionsIf we call zi  (q, p) (i = 1, . . . , 2n) the coordinates in the phase space Γ, the constraintsφr(z)  0 (r = 1, ...R) dene the constraint surface  given by = fz 2 Γ j φr(z) = 0 (r = 1, . . . , R) (R  2n)g . (1)Regularity Conditions (RCs): The constraints φr  0 are regular if and only if theirsmall variations δφr, evaluated on , are R linearly independent functions of δzi.To rst order in δzi, the variation of the constraints are δφr = Jri δzi, where Jri  ∂φr∂ziΣ.Consequently, the RCs can also be dened as [2]:The set of constraints φr  0 is regular if and only if the Jacobian Jri  ∂φr∂ziΣhas maximalrank: <(J) = R.A simple classical mechanical example of functionally dependent constraints occurs in a 2-dimensional phase space with coordinates (q, p) and constraints φ1  q  0 and φ2  pq  0.In this case, J ="1 p0 q#Σand <∂(φ1,φ2)∂(q,p)q=0= 1. A system of just one constraint can alsofail the test of regularity. Consider the constraint φ = q2  0 in a 2-dimensional phase space.In this case, J ="2q0#q2=0= 0 and < (J) = 0. The same problem occurs with the constraintq7  0, which has a zero of seventh order at the constraint surface, or with any other constraintwhich is not linear in zi − zi.3 Classification of Irregular ConstraintsIrregular constraints can be classied according to their approximate behavior near the surface.A. Linear constraints. The Jacobian has constant, non-maximal rank throughout , andφr  Jrizi − zi 0, <(J) = R′ < R. (2)These are regular systems in disguise. Regularity fails simply because R − R′ constraints areredundant and should be discarded. The regular system gives the correct description.B. Multilinear constraints. In the vicinity of , the constraints are of the formφ  1m!Si1...imzi1 − zi1  zim − zim 0 (m  2n) , (3)where the coecients Si1...im vanish if any two indices are equal. Thus, φ has simple zeros onsurfaces of dimension 2n−1, and zeros of higher order occur at the intersections of these surfaces.2The RCs fail at the points of intersection, where φ has multiple zeros. At those intersections, φcan be replaced by the equivalent1 set of constraints,ϕ1  zi1 − zi1  0,    , ϕm  zim − zim  0. (4)At the intersections, the setϕ1, ..., ϕm}is regular and therefore this provides a recipe forsubstituting the irregular multilinear constraint φ by a regular set of linear constraints. Forexample, the constraint φ  qp  0 is irregular at (0, 0) because it admits a linear approximationeverywhere except at this point. Replacement φ by the linear constraints fq  0, p  0g at (0, 0)regularizes the system.C. Higher order constraints. In the vicinity of , the constraints do not possess a linearapproximation:φ  C (zs − zs)k  0 (k > 1) . (5)The Jacobian vanishes everywhere on the constraint surface. Naively, it would seem possible tochoose zs − zs  0 as an equivalent regular constraint, but it turns out that this could changethe dynamics of original theory, as we show below.These three classes are generic and, in general, there can be combinations of these threetypes occurring simultaneously near a constraint surface. By selecting a suciently small neigh-borhood of a point on , one can always expect to be in one and only one of the three situationsjust described.In correspondence with the three generic cases in which regularity can fail, the nature of theconstraint surface  falls into one of three categories:A. The RCs are satisfied on the whole constraint surface. These are regular systems, eitherdesguised or not.B. The RCs fail on a submanifold of : < (J) = R except on a submanifold 0   where< (J) = R′ < R .C. The RCs fail everywhere on : J has constant lower than R rank on .The rst case is treated in the standard texts and will not be further discussed here.In the second case, the constraint surface can be decomposed into two non-empty submani-folds, 0 and R such that  = 0 [R and 0 \R is empty. Then, the rank of the Jacobianjumps from < (J) = R on R, to < (J) = R′ on 0. As mentioned above, in this case it ispossible to replace φ at 0 by a set of regular constraintsϕ1  0,    , ϕm  0} which regularizethe system at 0.The important question is how to proceed in the third case. If the RCs fail everywhere on, the previous approach is not applicable, because there is no guarantee that the resultingHamiltonian dynamics will be equivalent to that of the original Lagrangian system.This can be seen in the example of Lagrangian in a three-dimensional conguration space,L (x, y, z) = _x _z + yz2. (6)1Two sets of constraints are said to be equivalent if they define the same constraint surface Σ.3The general solution of Euler-Lagrange equations describes a system with one degree of freedom{ a free particle, whose time evolution x = p0 t + x0 is determined by two initial conditions p0and x0. The remaining elds are y(t), a Lagrange multiplier with indeterminate evolution, andz(t), with trivial evolution, z(t) = 0.The Hamiltonian approach gives just two (rst class) constraints, py  0, z2  0 (R = 2),and one degree of freedom, as expected from the Lagrangian approach. However, this is onlysupercially correct, because the Jacobian has rank 1, and not 2. This is because the constraintfunction φ = z2 has no linear approximation at z = 0.On the other hand, if we naively take z  0, which is regular and equivalent to z2  0,then the Hamiltonian analysis generates three rst class constraints py  0, z  0, px  0,which leave zero physical degrees of freedom. This result is not consistent with the Lagrangiandescription where there is one degree of freedom.As seen in the previous example, even if two constraints are equivalent in dening the sameconstraint surface, they may yield dierent dynamics and should be treated more carefully.Suppose φ  0 is a constraint of the form (5), which is equivalent to the regular constraintχ  φ1/k  0. (7)The question is whether χ  0 gives also the correct dynamics. The answer to this questiondepends on whether χ is a rst or second class constraint. Namely, it makes a dierence whetherthe linear constraint χ can generate a transformation in phase space that leaves the Hamiltonianaction unchanged or not . As shown in [5], if the linearized constraint χ is second class, thenit is not only geometrically equivalent to φ in the sense that it denes the same , but thesubstitution also yields the same dynamical description as the Lagrangian approach. On theother hand, if χ is rst class, then the subtitution generates a system whose dynamics is dierentfrom the one obtained from the Euler-Lagrange equations.4 ConclusionsThe recipe for treating the non-regular constraints is: Every linear or multi-linear set of constraints can be exchanged by an equivalent regularset. It allows to carry out Dirac’s procedure in the standard way. A higher order constraint φ = C (zs − zs)k  0, can be exchanged by the equivalent linearconstraint χ = φ1/k  0. If χ is a second class constraint, the dynamics of the new systemis equivalent to the Lagrangian one. If χ is rst class, the substitution yields a systemwhich is not dynamically equivalent to the Lagrangian one. In this latter case, one shouldview the original Lagrangian as an incomplete, if not a totally inconsistent description fora dynamical system.4AcknowledgmentsWe are deeply grateful to Ricardo Troncoso for many enlightening insights and discussions.This work is partially funded by grants FONDECYT 1020629, 7020629, 1010450, 2010017, andMECESUP USA 9930 scholarship. The generous support of Empresas CMPC to the CECS isalso acknowledged. CECS is a Millennium Science Institute and it is funded in part by grantsfrom Fundacion Andes and the Tinker Foundation.References[1] P. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964).[2] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton UniversityPress, Princeton, 1992).[3] M. Blagojevic, Gravity and Gauge Symmetries (Institute of Physics Publishing, London,2001).[4] N. P. Chitaia, S. A. Gogilidze and Yu. S. Surovtsev, Phys. Rev. D 56 (1997) 1135; Phys.Rev. D 56 (1997) 1142.[5] O. Miskovic and J. Zanelli, Dynamical Structure of Irregular Constrained Systems (in prepa-ration).5

Irregular Hamiltonian Systems

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