Let us now consider the quantity pnqn − L . (Whenever there is a repeated suffix I assume a summation over all values of that suffix.) Let us make variations in the variables q and q, in the coordinates and the velocities. These variations will cause variations to occur in the momentum variables p. As a result of these variations,
(1-5)
by (1-3). Now you see that the variation of this quantity pnqn − L involves only the variation of the q’s and that of the p’s. It does not involve the variation of the velocities. That means that pnqn L— can be expressed in terms of the q’s and the p’s, independent of the velocities. Expressed in this way, it is called the Hamiltonian H.
However, the Hamiltonian defined in this way is not uniquely determined, because we may add to it any linear combination of the φ’s, which are zero. Thus, we could go over to another Hamiltonian
(1-6)
where the quantities cm are coefficients which can be any function of the q’s and the p’s. H* is then just as good as H ; our theory cannot distinguish between H and H*. The Hamiltonian is not uniquely determined.
We have seen in (1-5) that
This equation holds for any variation of the q’s and the p’s subject to the condition that the constraints (1-4) are preserved. The q’s and the p’s cannot be varied independently because they are restricted by (1-4), but for any variation of the q’s and the p’s which preserves these conditions, we have this equation holding. From the general method of the calculus of variations applied to a variational equation with constraints of this kind, we deduce and
(1-7)
or
(1-8)
with the help of (1-2) and (1-3), where the um are unknown coefficients. We have here the Hamiltonian equations of motion, describing how the variables q and p vary in time, but these equations involve unknown coefficients um .
It is convenient to introduce a certain formalism which enables one to write these equations briefly, namely the Poisson bracket formalism. It consists of the following: If we have two functions of the q’s and the p’s, say f (q, p) and g (q, p), they have a Poisson bracket [ f, g ] which is defined by
(1-9)
The Poisson brackets have certain properties which follow from their definition, namely [f, g] is antisymmetric in f and g :
(1-10)
it is linear in either member:
(1-11)
and we have the product law,
(1-12)
Finally, there is the relationship, known as the Jacobi Identity, connecting three quantities:
(1-13)
With the help of the Poisson bracket, one can rewrite the equations of motion. For any function g of the q’s and the p’s, we have
(1-14)
If we substitute for qn and pn their values given by (1-7) and (1-8), we find that (1-14) is just
(1-15)
The equations of motion are thus all written concisely in the Poisson bracket formalism.
We can write them in a still more concise formalism if we extend the notion of Poisson bracket somewhat. As I have defined Poisson brackets, they have a meaning only for quantities f and g which can be expressed in terms of the q’s and the p’s. Something more general, such as a general velocity variable which is not expressible in terms of the q’s and p’s, does not have a Poisson bracket with another quantity. Let us extend the meaning of Poisson brackets and suppose that they exist for any two quantities and that they satisfy the laws (1-10), (1-11), (1-12), and (1-13), but are otherwise undetermined when the quantities are not functions of the q’s and p’s.
Then we may write (1-15) as
(1-16)
Here you see the coefficients u occurring in one of the members of a Poisson bracket. The coefficients um are not functions of the q’s and the p’s, so that we cannot use the definition (1-9) for determining the Poisson bracket in (1-16). However, we can proceed to work out this Poisson bracket using the laws (1-10), (1-11), (1-12), and (1-13). Using the summation law (1-11) we have:
(1-17)
and using the product law (1-12),
(1-18)
The last bracket in (1-18) is well-defined, for g and φm are both functions of the q’s and the p’s. The Poisson bracket [g, um ] is not defined, but it is multiplied by something that vanishes, φm . So the first term on the right of (1-18) vanishes. The result is that
(1-19)
making (1-16) agree with (1-15).
There is something that we have to be careful about in working with the Poisson bracket formalism: We have the constraints (1-4), but must not use one of these constraints before working out a Poisson bracket. If we did, we would get a wrong result. So we take it as a rule that Poisson brackets must all be worked out before we make use of the constraint equations. To remind us of this rule in the formalism, I write the constraints (1-4) as equations with a different equality sign ≈ from the usual. Thus they are written
(1-20)
I call such equations weak equations, to distinguish them from the usual or strong equations.
One can make use of (1-20) only after one has worked out all the Poisson brackets which one is interested in. Subject to this rule, the Poisson bracket (1-19) is quite definite, and we have the possibility of writing our equations of motion (1-16) in a very concise form:
(1-21)
with a Hamiltonian I call the total Hamiltonian,
(1-22)
Now let us examine the consequences of these equations of motion. In the first place, there will be some consistency conditions. We have the quantities φ which have to be zero throughout all time. We can apply the equation of motion (1-21) or (1-15) taking g to be one of the φ’s. We know that g must be zero for consistency, and so we get some consistency conditions. Let us see what they are like. Putting g = φm and g = 0 in (1-15), we have:
(1-23)
We have here a number of consistency conditions, one for each value of m. We must examine these conditions to see what they lead to. It is possible for them to lead directly to an inconsistency. They might lead to the inconsistency 1 = 0. If that happens, it would mean that our original Lagrangian is such that the Lagrangian equations of motion are inconsistent. One can easily construct an example with just one degree of freedom. If we take L = q then the Lagrangian equation of motion (1-2) gives immediately 1 = 0. So you see, we cannot take the Lagrangian to be completely arbitrary. We must impose on it the condition that the Lagrangian equations of motion do not involve an inconsistency. With this restriction the equations (1-23) can be divided into three kinds.
One kind of equation reduces to 0 = 0, i.e. it is identically satisfied, with the help of the primary constraints.
Another kind of equation reduces to an equation independent of the u’s, thus involving only the q’s and the p’s. Such an equation must be independent of the primary constraints, otherwise it is of the first kind. Thus it is of the form
( 1-24)
Finally, an equation in (1-23) may not reduce in either of these ways; it then imposes a condition on the u’s.
The first kind we do not have to bother about any more. Each equation of the second kind means that we have another constraint on the Hamiltonian variables. Constraints which turn up in this way are called secondary constraints. They differ from the primary constraints in that the primary constraints are consequences merely of the equations (1-3) that define the momentum variables, while for the secondary constraints, one has to make use of the Lagrangian equations of motion as well.
If we have a secondary constraint turning up in our theory, then we get yet another consistency condition, because we can work out ẋ according to the equation of motion (1-15) and we require that ẋ ≈ 0. So we get another equation
(1-25)
This equation has to be treated on the same footing as (1-23). One must again see which of the three kinds it is. If it is of the second kind, then we have to push the process one stage further because we have a further secondary constraint. We carry on like that until
we have exhausted all the consistency conditions, and the final result will be that we are left with a number of secondary constraints of the type (1-24) together with a number of conditions on the coefficients u of the type (1-23).
The secondary constraints will for many purposes be treated on the same footing as the primary constraints. It is convenient to use the notation for them:
(1-26)
where K is the total number of secondary constraints. They ought to be written as weak equations in the same way as primary constraints, as they are also equations which one must not make use of before one works out Poisson brackets. So all the constraints together may be written as
(1-27)
Let us now go over to the remaining equations of the third kind. We have to see what conditions they impose on the coefficients u. These equations are
(1-28)
where m is summed from 1 to M and j takes on any of the values from 1 to J . We have these equations involving conditions on the coefficients u, insofar as they do not reduce merely to the constraint equations.
Let us look at these equations from the following point of view. Let us suppose that the u’s are unknowns and that we have in (1-28) a number of non-homogeneous linear equations in these unknowns u, with coefficients which are functions of the q’s and the p’s. Let us look for a solution of these equations, which gives us the u’s as functions of the q’s and the p’s, say
(1-29)
There must exist a solution of this type, because if there were none it would mean that the Lagrangian equations of motion are inconsistent, and we are excluding that case.
The solution is not unique. If we have one solution, we may add to it any solution Vm (q, p) of the homogeneous equations associated with (1-28):
(1-30)
and that will give us another solution of the inhomogeneous equations (1-28). We want the most general solution of (1-28) and that means that we must consider all the independent solutions of (1-30), which we may denote by Va m (q , p ), a = 1, . . . , A. The general solution of (1-28) is then
(1-31)
in terms of coefficients va which can be arbitrary.
Let us substitute these expressions for u into the total Hamiltonian of the theory (1-22). That will give us the total Hamiltonian
(1-32)
We can write this as
(1-33)
where
(1-33)
and
(1-34)
In terms of this total Hamiltonian (1-33) we still have the equations of motion (1-21).
As a result of carrying out this analysis, we have satisfied all the consistency requirements of the theory and we still have arbitrary coefficients v. The number of the coefficients v will usually be less than the number of coefficients u. The u’s are not arbitrary but have to satisfy consistency conditions, while the v ’s are arbitrary coefficients. We may take the v’s to be arbitrary functions of the time and we have still satisfied all the requirements of our dynamical theory.
This provides a difference of the generalized Hamiltonian formalism from what one is familiar with in elementary dynamics. We have arbitrary functions of the time occurring in the general solution of the equations of motion with given initial conditions. These arbitrary functions of the time must mean that we are using a mathematical framework containing arbitrary features, for example, a coordinate system which we can choose in some arbitrary way, or the gauge in electrodynamics. As a result of this arbitrariness in the mathematical framework, the dynamical variables at future times are not completely determined by the initial dynamical variables, and this shows itself up through arbitrary functions appearing in the general solution.
We require some terminology which will enable one to appreciate the relationships between the quantities which occur in the formalism. I find the following terminology useful. I define any dynamical variable, R, a function of the q’s and the p’s, to be first-class if it has zero Poisson brackets with all the φ’s:
(1-35)
It is sufficient if these conditions hold weakly. Otherwise R is second-class. If R is first-class, then [R, φj ] has to be strongly equal to some linear function of the φ’s, as anything that is weakly zero in the present theory is strongly equal to some linear function of the φ’s. The φ’s are, by definition, the only independent quantities which are weakly zero. So we have the strong equations
(1-36)
Before going on, I would like to prove a
Theorem: the Poisson bracket of two first-class quantities is also first-class. Proof. Let R, S be first-class: then in addition to (1-36), we have
(1-36)
Let us form [[R, S], φ j ]. We can work out this Poisson bracket using Jacobi’s identity (1-13)
by (1-36), (1-36), the product law (1-12), and (1-20). The whole thing vanishes weakly. We have proved therefore that [R, S] is first-class.
We have altogether four different kinds of constraints. We can divide constraints into first-class and second-class, which is quite independent of the division into primary and secondary.
I would like you to notice that H given by (1-33) and the φ a given by (1-34) are first-class. Forming the Poisson bracket of φa with φ j we get, by (1-34), Va m [φ m , φ j ] plus terms that vanish weakly. Since the Va m are defined to satisfy (1-30), φa is first-class. Similarly (1-28) with Um for um shows that H is first-class. Thus (1-33) gives the total Hamiltonian in terms of a first-class Hamiltonian H’ together with some first-class φ’s.
Any linear combination of the φ’s is of course another constraint, and if we take a linear combination of the primary constraints we get another primary constraint. So each φa is a primary constraint; and it is first-class. So the final situation is that we have the total Hamiltonian expressed as the sum of a first-class Hamiltonian plus a linear combination of the primary, first-class constraints.
The number of independent arbitrary functions of the time occurring in the general solution of the equations of motion is equal to the number of values which the suffix a takes on. That is equal to the number of independent primary first-class constraints, because all the independent primary first-class constraints are included in the sum (1-33).
That gives you then the general situation. We have deduced it by just starting from the Lagrangian equations of motion, passing to the Hamiltonian and working out consistency conditions.
From the practical point of view one can tell from the general transformation properties of the action integral what arbitrary functions of the time will occur in the general solution of the equations of motion. To each of these functions of the time there must correspond some primary first-class constraint. So we can tell which primary first-class constraints we are going to have without going through all the detailed calculation of working out Poisson brackets; in practical applications of this theory we can obviously save a lot of work by using that method.
I would like to go on a bit more and develop one further point of the theory. Let us try to get a physical understanding of the situation where we start with given initial variables and get a solution of the equations of motion containing arbitrary functions. The initial variables which we need are the q’s and the p’s. We don’t need to be given initial values for the coefficients v . These initial conditions describe what physicists would call the initial physical state of the system. The physical state is determined only by the q’s and the p’s and not by the coefficients v.
Now the initial state must determine the state at later times. But the q’s and the p’s at later times are not uniquely determined by the initial state because we have the arbitrary functions v coming in. That means that the state does not uniquely determine a set of q’s and p’s, even though a set of q’s and p’s uniquely determines a state. There must be several choices of q’s and p’s which correspond to the same state. So we have the problem of looking for all the sets of q’s and p’s that correspond to one particular physical state.
All those values for the q’s and
p’s at a certain time which can evolve from one initial state must correspond to the same physical state at that time. Let us take particular initial values for the q’s and the p’s at time t = 0, and consider what the q’s and the p’s are after a short time interval δt. For a general dynamical variable g, with initial value g0, its value at time δt is
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