SPEC & GEN RELATIVITY
SPEC & GEN RELATIVITY PHZ 6607
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This 12 page Class Notes was uploaded by Maureen Roob IV on Friday September 18, 2015. The Class Notes belongs to PHZ 6607 at University of Florida taught by Bernard Whiting in Fall. Since its upload, it has received 23 views. For similar materials see /class/206871/phz-6607-university-of-florida in Physics 2 at University of Florida.
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Date Created: 09/18/15
CONSTRAINTS notes by BERNARD F WHITING Whether for practical reasons or of necessity we often find ourselves considering dy namical systems which are subject to physical constraints In such situations it is possible to consider rede ning the dynamical variables in a theory so that constrained and un constrained degrees of freedom become decoupled For situations where that separation cannot be carried out explicitly or it is undesirable to do so Dirac has introduced a con struction which acts as an effective Poisson bracket on the physical phase space in which the constrained degrees of freedom can be essentially eliminated In these notes we shall look basically at methods for categorizing and dealing with constraints in dynamical sys tems noticing an important distinction between constraints of different type and illustrate Dirac7s procedure with a simple example In fact as we shall see most points which need to be made can be demonstrated very well with simple examples A complete understanding of constrained dynamical systems requires a thorough knowledge of classical Hamiltonian theory By the use of a number of carefully selected examples I will attempt to show how this knowledge is used and how it may be usefully extended without deviating from the spirit of the traditional Hamiltonian approach In order to illustrate the difference in the kinds of constraints that arise we can begin with two examples of problems which should be quite familiar Then a minimal set of skills necessary to proceed will be brie y described Finally simple examples showing how these procedures can be put into practice will be given Examples distinguishing constraint types i Constraints form canonical pairs This example concerns particle motion confined to the surface of a sphere We suppose that the Lagrangian for the system is given by km 2 i2 Q2 22 7 V7y727 subject to the constraint bl xx2y22 27a0 where m 0 implies is zero when the constraint 1 is imposed which fixes the radius of the sphere The conjugate momenta for this system are defined by pxmia pymya pzmia which give rise to the canonical Hamiltonian 1 Hcpq7 pip pi Vyz However for this constrained system we take the canonical generator of time translations to be the primary Hamiltonian given by Hp He A451 5 where A is taken as a Lagrange multiplier Note A is not considered to be a dynamical variable and it has no canonically conjugate momentum but variation of Hp with respect to A enforces the constraint bl 0 Before proceeding we need to determine whether there are any additional constraints arising from the requirement that 151 0 be maintained in time To this end we compute lt51 451 H mg ypy 21 7 1 mm where by definition 810 By 810 By W 9191 7 9191 W 39 01105 yqp E Since 131 does not automatically vanish we must impose it as an additional constraint 452 0 W 22 22 mg ypy 21 Then 39 Z52 152 7Hp 452 ch l Af 2 7451 in which we can always impose 13952 0 by solving for A which in general will be time dependent since 1 2 l is non zero In fact we see that even though 151 and 152 should be zero they satisfy the conditions of a pair of canonical coordinates on phase space but in this formulation 151 and 152 are not independent of the other canonical variables at y z and pmpy and 1 Can we effect a canonical transformation which disentangles constraint degrees of freedom from dynamically non trivial degrees of freedom We could have eliminated all necessity of constraints in this problem by starting in spherical polar coordinates with 7 a and 739 0 imposed explicitly in the Lagrangian so the answer to our question should be an obvious yes However to get a feel for what the question involves its useful to reformulate this particular problem in spherical polar 2 coordinates while maintaining the original constraint Thus for the Lagrangian we would have m 2 2 392 2 2 392 3 7 7 6 7 sin 6 gt7Vr6 and for the constraint bl 7 7 a 0 The canonical momenta are now given by p mf p9 mr26l and 19 m7 sin2 6d so that HC becomes 1 2 p p 7 7 7 V a 2m pT r2 r2 sin26l 70 45 while the definition of Hp in terms of HC and 151 remains unchanged Similarly 43951 17Hppr a so that 152 simplifies considerably to 452 Pr Then 432 Mic A which can again be solved for A Now however we see that our canonical variables can be split into two groups 6199 b and 19 which are now unconstrained and 151 7 7 a 152 1 which have decoupled from the remainder and are both constrained to vanish In other words we have found the answer to the question posed above We now consider another familiar example ii Conjugate of constraints totally unconstrained The most important observation here is that the constraints will be of a different type This second example is electromagnetism for which we consider the Lagrangian density to be cartesian coordinates in at space will be assumed throughout 1 7 iZFWFW xjt AL in which FM BMAV 7 EVA and A is now a coupling constant NOT a Lagrange multiplier In this Lagrangian density A0 has no time derivative so the definition H will introduce the constraint 15 H0 m 0 3 The canonical Hamiltonian density can be shown to be 1 l V HG ininl nag10 1131 7 WA while the Hamiltonian density generating canonical time translations is Hp He 19 Then 43950 057117 Bil Ii joa which requires us to introduce the additional constraint IBHij 0 For the time derivative we now compute 43951 ainsz A0030 mm In this case we cannot solve for the Lagrange multiplier Mac but must instead restrict our attention to sources whose current is conserved Note in particular that 07 1 0 5 So now 15 and 151 do not form a canonical pair In fact if we decompose A and Hi into orthogonal transverse and longitudinal components A A AiL and Hi H H where V satisfies BiVTi 0 AiTHTi represent two dynamical degrees of freedom while A0 H0 and A BiHLi where AiL 2A represents decoupled degrees of freedom in which H0 15 and BiHLi 151 7 MO are given by constraints while canonical time evolution can tell us nothing at all about the time dependence of A0 nor A if we follow the gauge theory approach indicated below With this decomposition in other relevant Poisson brackets vanish and up to integration by parts the Hamiltonian density can be broken into decoupled disjoint parts H HT HL The constraints here are of a different type to those in the previous example because they no longer form conjugate pairs instead their conjugates dynamically decouple from the 4 physical degrees of freedom and their time dependence may no longer be governed by the dynamical equations of the theory Handling constraints in dynamical systems In the first example considered above we could use our knowledge of the problem to achieve a definite split between the dynamical and non dynamical degrees of freedom This splitting once obtained essentially allows us to discard the constrained canonical variables Practitioners have seen that simple problems like this could indicate a general principle to pursue with constraints of this type In the second example the split was perhaps much less obvious it is actually considerably clearer in momentum space and the different character of the constraints is related to the deep rooted property that electromagnetism couples to conserved currents Since it occupies such an important place in Physics electromagnetism as a gauge theory is often taken as the archetypical model for systems with this kind of constraints Although electromagnetism is a field theory the theory discussed in Example 2a below has the same kind of constraints with finite degrees of freedom In all situations the principle of disentangling will be the same ie one tries to so categorize the dynamical variables and the relations between them that a subset of them can be regarded as obeying an unconstrained dynamics while the remainder decouple and become dynamically if not physically irrelevant at least in classical mechanics Fortunately Dirac has put forward a formulation which deals with the situation where the split is too dif cult to achieve explicitly or where it may even be technically impossible Before going on to consider Dirac7s formulation it will be useful to establish more clearly the distinction between the two types of constraint which we have already encountered Although usually not described quite this way the classification of constraints I will give makes most sense if we can imagine that the dynamical and constrained degrees of freedom have already been separated and that each group can really be re organized into distinct canonical pairs Then when a pair of canonical variables pq st 11 l are both constraints we describe the constraints as being second class When in a canonical pair only one variable is a constraint we describe that constraint as being rst class Only true dynamical degrees of freedom commute with all the constraints Up to this point what has been said is consistent with the usage given by most authors in the field Even the terminology l have used so far is not standard in the literature but beyond this point also the general procedure to follow is not universally accepted Some 5 would have us introduce additional gauge fixing7 conditions O qp 0 to convert all first class constraints into second class constraints In other situations we would be persuaded to introduce extra degrees of freedom so that all second class constraints can be transformed into first class constraints this may lead to some modification of the effect of the original constraints see the comments concerning Example 3 below and then the whole problem can be treated as an extended gauge theory In either case the purpose is to allow eventually for a uniform treatment of the adjusted system For concreteness I will follow a development compatible with Dirac7s though not always following his exquisite logic As one can see the main objective indicated is to come up with a well defined set of quantities the true dynamical variables of the reduced phase space which commute with all the constraints When such a split is not possible in the second class case we shall follow Dirac directly and introduce a modified bracket it is actually a Poisson bracket on the reduced phase space which serves to govern the dynamics in the same way that the Poisson bracket does for unconstrained systems To define Dirac7s bracket we must first define the matrix7 of Poisson brackets for all the second class constraints 01 i j which is non singular and therefore invertible In terms of its inverse 04 we can then define the Dirac bracket AaBDB AaB Ag i071i1 i53 a This does not depend for its definition on the constraints being isolated from the remaining dynamical variables nor does it require that they already be organized into canonical pairs However we do have to be able to distinguish the first from the second class constraints since the former would prevent the matrix of Poisson brackets from being invertible Furthermore this definition has the obvious property that the Dirac bracket of a constraint with anything else will always be zero Thus all physical quantities Dirac bracket commute7 with all constraints a condition which renders the constraints classically irrelevant even if they cannot be separated explicitly from the true degrees of freedom Nest we discuss the first class case There is a general tendency to attempt to interpret first class constraints as generates of gauge transformations though Example 2a below provides a counterexample Since gauge theories do occur frequently in physics we need to be able to handle them and since one is often required to fix a gauge even in classical theory I will consider this approach brie y using electromagnetism in Example 4 6 below Normally we would attempt to decouple first class constraints and true dynamical degrees of freedom whenever it is possible to do so When that is not possible resort to gauge fixing may become desirable but then we often have to tackle the problem of the non existence of a global gauge condition Locally gauge xing conditions 0 0 which really serve as additional constraints are generally required to satisfy two criteria i given any set of canonical variables there must exist a gauge transformation which brings it into the chosen gauge and ii the chosen conditions must fix the gauge completely Together these two criteria imply that there must be just as many gauge conditions as there are first class constraints and that the commutators Oaa 12 form an invertible matrix where 151 are the original first class constraints by this construc tion turned into second class constraints Some authors define a number of additional Hamiltonians Typically the first and second class constraints are separated with the multiplier conditions for the latter being incorporated back into the Hamiltonian As I will illustrate in Example 3 this can lead to changes though they are unphysical in the value of the constrained degrees of freedom Another modification is employed in the first class case since those of interest are generally gauge theories if there are additional such constraints which were not present in the pri mary Hamiltonian this modification results in extra Lagrange multipliers being introduced for the subsidiary first class constraints leading to an extended7 Hamiltonian I prefer to understand the situation in the first place without these additional measures since they depend for their validity on further information about the true nature of the physical system being considered But for typical physical applications they may be essential Before going on to consider specific examples several further comments are in order First an important point which is often neglected is that for the various definitions to be workable the constraints must satisfy certain regularity properties These can be best exemplified by saying that in any variation for which 6g and 6 are 06 then 6 qp must also be 06 Thus of 190 1920 0 only the first is acceptable while for pi 1 0 which implies p1 0 p2 0 7 only the latter two constraints are acceptable Finally none of this at all is necessary if we simply wish to know a set of equations to solve for the dynamics However it does serve a purpose if we wish to know how to remove non dynamical degrees of freedom from the formulation This in particular may be essential if we are to avoid quantizing non dynamical aspects of the problem and is the main area where current interest in the subject lies Illustrative examples Example 1 This first example stresses that any quantity depending upon 2N degrees of freedom can not arbitrarily be treated as a canonical Hamiltionian In fact this example also deals with a situation in which a purported Hamiltonian has an odd number of arguments a circumstance which arises frequently for coherent states based on generalized group rep resentations A canonical Hamiltonian is properly defined as a functional of paired sets of variables the coordinates of phase space which generate its symplectic geometry Thus the function Fa b c Vc gm b2 is not strictly speaking a Hamiltonian since no pairing of the variables is indicated which would require them to satisfy the Possion bracket relation 11 1 In the way by which interest in this particular problem arises its author was really looking at a situation which can be best described by a Lagrangian in a non traditional first order form 1 dcoscl9sinc7Vc 7 i a2 b2 It is obvious that if we now try to identify pa cosc pb sine pa 0 7 then each of these must be regarded as giving rise to a separate constraint 15 pa 7 cos c 151 pl 7 sin c I52 190 Thus the primary Hamiltonian becomes 1 Hp VC l i012 b2 l A0ltPa COS 0 l A1ltPb Sin 0 l 2Pc 8 As previously we now look to see if their are any additional constraints 13950 OHp 7a sin CA2 13951 17117 7b 7 cosc2 g3 27p 7 Aosinc 7 A1 cosc These give the one additional constraint 11 acosc bsinc for which we find 1 11Hp A0 cosc Alsinc A27a sinc b cosc Now we have a set of equations suf cient to find all the Xs As none of the Xs remain undetermined all the constraints will turn out to be second class Thus only two dynamical variables will remain corresponding to one unconstrained canonical degree of freedom We can tell that there is only one unconstrained canonical degree of freedom in the problem without having found it explicitly at this stage In this problem the constrained and unconstrained degrees of freedom can be de coupled To carry that out it is helpful to proceed in several discrete steps We consider then the set of variables Aa coscb sinc1110 PA 1 coscpb sinc7 l 15 cosc 1 sinc 0 0 B 7a sincb cosc PB 71 sincpl2 cosc 7 sinc bl cosc1 0 00 pc 20 Note Ame B 5 Papa PB 19190 A a 133190 PA 71 Finally we introduce Pcpc7BPA1APB and BBPc 271 3 w1 31 which has become a constraint replacing 190 Now we have A m 0 PA 0 B m 0 amp PB 0 as second class constraints while 0 amp PC survive as an unconstrained canonical pair ie 9 we have conveniently arranged our degrees of freedom into two canonical pairs composed entirely of constraints and one remaining pair representing the only true dynamical degree of freedom in the problem A judicious change of variables at the beginning and perhaps with hindsight can greatly simplify the ensuring analysis Since it will provide us with material for a later example we now carry this out With the new definitions A a cosc b sinc B7a sincb cosc we find that the Lagrangian can be rewritten as 1 AiBOiVc7 A2B2 in which the A7 degree of freedom decouples completely and the further identification PC E 73 even eliminates the remaining pair of second class constraints bringing us to the point we finished with above regarding the single true7 dynamical degree of freedom Example 2 This second example shows that sometimes whether constraints are first or second class can depend on properties of the Hamiltonian We shall also use this example to give a specific illustration of Dirac7s procedure the example is provided by the Lagrangian 1 geyacz 7 Vy It is clear that in this case there is one primary constraint 450 pg 7 so that the primary Hamiltonian becomes 1 Hp e yp Vac y Apy For the time derivative of 15 we find 1 8V 0 0 7 7y 2 7 7 45 H 26 I 8y indicating that we must introduce an additional constraint which we assume to be WW y y 151 pg 7 facy where f2y 26y 10 The time derivative of 151 leads to an equation for A 8V Bfw 391 1 77 7N gt i Hi 8A8y 0 while the Possion bracket 0 1 does not vanish so these contraints are second class Now one reason why this example is interesting is that without speci c knowledge of Vxy we cannot explicitly decouple the constraints from the dynamical degrees of freedom This is where Dirac7s procedure comes to our rescue Using the definition given above note that we here have 9649be 96191 96 l 1 1px 1 1 71 m 1 a as does 20 fxyDB so in particular we now have yDB 31 0 because via 1512 is no longer independent of 1 and an Example 2a A second reason why the above example is so interesting is that it seems to take on an entirely different character when the potential Vx y vanishes since the constraints then become equivalent to be pya bl pm and they are now both first class The solution for the dynamics is given by x constant y any function of time This solution can certainly be well understood but the problem itself does not fit easily into the mold people have tried to cut out for it since the general inclination to interpret first class constraints as generates of gauge transformations clearly is inappropriate for 151 Example 3 The third example is derived from our first by taking from it the decoupled constrained degree of freedom A alone and shows that solving and substituting for the Lagrange multipliers can change the value of the constrained variables We take for the Lagrangian 1 A 7 7A2 2 The details of the calculation can be done as an exercise the results of which are easily summarized The constraints are found to be PA lOaIldA0 11 and the solution for the Lagrange multiplier is A 0 However if we substitute this back into the Hamiltonian the subsequent solution for the constrained variables becomes 1401 andPAclT02 Which is very different unless we use the original solution to the constraints as part of the initial data Frequently in the treatment of gauge theories for physical systems differences like this are often treated as irrelevant but one may sometimes be too cavalier in dismissing such changes to the theory especially if one has quantization in mind Example 4 For the final example I refer again to the original example of electromagnetism It is clear there that taking the additional constraints gauge fixing conditions A0 0 and A 0 gives us a situation With entirely second class constraints and completely fixes the gauge On the other hand the condition 8M4 0 does not it is not enough conditions and is insensitive to the change A a A 84 for any b Which satisfies V2 0
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