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Theory of Systems with Constraints Theory of systems with Constraints Javier De Cruz P´erez Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Abstract: We discuss systems subject to constraints. We review Dirac's classical formalism of dealing with such problems and motivate the definition of objects such as singular and nonsingular Lagrangian, first and second class constraints, and the Dirac bracket. We show how systems with first class constraints can be considered to be systems with gauge freedom. We conclude by studing a classical example of systems with constraints, electromagnetic field I. INTRODUCTION II. GENERAL CONSIDERATIONS Consider a system with a finite number, k , of degrees of freedom is described by a Lagrangian L = L(qs; q_s) that don't depend on t. The action of the system is [3] In the standard case, to obtain the equations of motion, Z we follow the usual steps. If we use the Lagrangian for- S = dtL(qs; q_s) (1) malism, first of all we find the Lagrangian of the system, calculate the Euler-Lagrange equations and we obtain the If we apply the least action principle action δS = 0, we accelerations as a function of positions and velocities. All obtain the Euler-Lagrange equations this is possible if and only if the Lagrangian of the sys- tems is non-singular, if not so the system is non-standard @L d @L − = 0 s = 1; :::; k (2) and we can't apply the standard formalism. Then we say @qs dt @q_s that the Lagrangian is singular and constraints on the ini- tial data occur. Basically a constrained system is one in Which can be rewritten as[1] which there exists relations between the system's degrees of freedom that holds for all time. Constraints should @2L @L @2L q¨s = − q_s (3) not be confused with constants of motion. Constant of @q_rq_s @qr @qsq_r the motion arise as a result of the equations of motions and constraints to be restrictions on the dynamics be- @2L Define the matrix Wrs = @q_ @q_ . Expression (3) can be fore equation of motion are even solved. For example in r s solved for the accelerationsq ¨s if and only if the electromagnetic field (which we will see later in more detail ), the time derivative of the A0 component of the det jWrsj 6= 0 (4) vector potential does not appear in the action of the sys- tem, therefore the momentum conjugate to A0 is always If this determinant vanishes, then the Lagrangian is sin- zero, which is a constraint . The first systematic discus- gular and the accelerations can't be solved in terms of sion of singular Lagrian systems was given by Dirac, who the positions and velocities. In Hamiltonian formalism, developed a standard technique to "Hamiltonize" a sin- this implies the existency of relations between p0s and gular Lagrangian and the new Hamiltonian formalism to q0s. obtain equations of motion for these systems.[2] In the following sections we will describe in more detail the concepts mentioned in this introduction and describe III. HAMILTONIAN TREATMENT-THE DIRAC the Dirac method. Finally we apply this formalism, to THEORY solve a classic example, the electromagnetic field. One of the most important feature of these systems is that The set up of a Hamiltonian formalism for a La- they present gauge invariance. Therefore it can be de- grangian starts with the Legendre transformation wich scribed by a gauge theory. These are a kind of field defines the momenta [3] theory in which the Lagrangian is invariant under a con- tinious group of local transformations. We will see the @L ps = s = 1; :::; k (5) relation between first class constraints and the gauge free- @q_s dom. Finally we will find the Dirac brackets between the field variables (potential 4-vector) and their conjugate Relation (5) is invertible if and only if det jWrsj 6= 0. momenta. Then they can be solved to obtain the velocitiesq _s as functions of coordinates and momenta. If the determi- nant vanishes the rank Wrs = R < k and only R veloci- ties can be obtained from (5) as functions of q's , p's and ∗Electronic address: [email protected] the remaining (k − R) velocities and also that there exist Systems with constraints Javier de Cruz P´erez (k − R) independent relations among p's and q's. These We can identify arbitrary functions with velocities that relations, wich are direct consequences of the definitions we can't obtain of (5). To write the equation of motion of p's and the structure of the lagrangian. more compactly. We introduce a mathematical opera- Supose that we solve the k − R constraints for the tor called Poisson bracket. For two arbitrary functions momenta [1] f(q; p) and g(q; p) the Poisson bracket is defined by pρ = ξρ(pj; qs) s = 1; :::; k; j = 1; :::; R; (6) @f @g @f @g ff; gg = i − i (8) ρ = R + 1; :::; k @qi @p @p @qi This relations are called "primary constraints", the word With this definition the equations of motion can be writ- primary meaning that the equations of motion were not ten as used to obtain them. If we started with 2k-dimensional ρ phase-space defined by 2k indpendent coordinates q0s q_i ≈ fqi;Hg + u fqi; φρg (9) 0 ρ and p s. The motion is going to be confined to a surface p_i ≈ fpi;Hg + u fpi; φρg (10) of lower dimensionality , defined by constraint equation. Dirac introduced the concept of "weak" and "strong" and, for an arbitrary function of the phase space, equations. Let the constrained submanifold in phase _ ρ space be called U, let f(q; p), g(q; p) be two functions f ≈ ff; Hg + u ff; φρg defined in a neighborhood of U. The values of f and g Now let us examine the consequences of these equation on U are obtained by replacing pρ by ξρ. If after this _ replacment f and g become equal, then we say that they of motion. We have the quantities φρ wich have to be are "weakly equal" and write [1] zero throughout all time. We can apply the equation of motion, taking f to one of the φ0s. f(qs; ps) ≈ g(qs; ps) ρ fφσ;Hg + u fφσ; φρg ≈ 0 (11) Both functions f and g have a 2k-dimensional 'gradient vector' at each point in phase space, with components We must examine these conditions to see what they lead ( @f ; @f ) and ( @g ; @g ) respectively. If f equals g on U to. It is possible for then to lead directly to an incon- @qs @ps @qs @ps and also the gradient of f agrees with that of g when the sistency. If that happens, it would mean that original arguments are restricted to U, we say that f and g are Lagrangian is such that the equation of motion are in- "strongly equal" and write consistent. If the equation(11) does not have an incon- sistencies, can be divided in three kinds. One kind of equation reduces to 0 = 0 and it is f(qs; ps) ≡ g(qs; ps) identically satisfied with the help of the primary con- The submainfold U is defined by a set of weak equation. straints. We have an another kind of equation if and Let us define a set of functions only if detfφσ; φρg 6= 0 then the matrix is invertible and we can find the arbitrary functions φρ(qs; ps) = pρ − ξ(qs; pj) (7) ρ ρσ u ≈ −C fφσ;Hg (12) then U can be defined by φρ ≈ 0. At this point adding ρσ the primary constraints via Lagrange mulipliers [2] Where C is the inverse matrix of fφσ; φρg. In this case we can define the following expression [1] j ρ L = p q_j − H(q; p) + u φρ j = 1; :::; R ρ = R + 1; :::; k ∗ ρσ fg; Hg = fg; Hg − fg; φρgC fφσ;Hg (13) The action of the system defined by This expression is called Dirac bracket and is a general- Z S = dt pjq_ − H(q; p) + uρφ ization of the Poisson bracket, allowing us to write the j ρ equations of motion as We apply the principle of least action δS = 0 and obtain d g(q; p) ≈ fg; Hg∗ (14) [1] dt @H @φρ No arbitrary function appears in the solution of the equa- q_ = + uρ i @pi @pi tion of motion. Let us now return to the analysis of equa- @H @φρ tion (11) In general the matrix fφσ; φρg is singular and p_i = − + uρ ρ @q @q the arbitrary functions u are not all determinate. If the i i rank of the matrix is M < (k − R) there are (k − R − M) ρ @H ρ a u = =q _ null eigenvectors λσ @pρ aσ φρ = 0 λ fφσ; φρg ≈ 0 a = 1; 2; :::; (k − R − M) (15) Treball de F´ıde Grau 2 Barcelona, January 2014 Systems with constraints Javier de Cruz P´erez Combining equations (11) and (15) we find the following For the properties of the first class constraints the terms further conditions on q0s and p0s which survive are aσ λ fφσ;Hg ≈ 0 a = 1; 2; :::; (k − R − M) (16) fφa;Hg ≈ 0 b0 These equations may or may not produce restrictions on fφb;Hg + κ fφb; φb0 g ≈ 0 the arbitrary functions. If they do not, we have produced 0 Now we can say that the matrix fφb; φbg is nonsingu- more constraints ζ ≈ 0 that restrict the motion in phase lar and therefore its inverse exists.
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