2.1. Simple Systems; Lagrangian and Hamiltonian Mechanics

I shall consider a mechanical system with n configurational degrees of freedom, whose evolution over time t is given by a curve in a fixed simply connected region G of $(n+1) $-dimensional real space $\mathstrut{\Bbb R} ^{n+1}$, on which there are coordinates $(q_{1},\,\ldots,\,q_{n},\,t) =\mathrel{\mathop:} (q_{i},\,t) =\mathrel{\mathop:} (q,\,t) $. If the system consists of N point-particles (or bodies small enough to be treated as point-particles), so that a configuration is fixed by 3N cartesian coordinates, we may yet have $n< 3N$; for the system may be subject to constraints, and the $q_{i}$ are to be independently variable in the region G. I shall also assume that the system is simple, in the sense that it has the following six features, (i)–(vi).Footnote ¹

Any constraints on the system are to be: (i) scleronomic, i.e. time-independent, so that the region G is a cartesian product of a configuration space $Q\subset \mathstrut{\Bbb R} ^{n}$ with a time-interval $[ t_{-},\,t_{+}] \subset \mathstrut{\Bbb R} $ (where we allow $t_{-}=-\infty $, $t_{+}=+\infty $); (ii) holonomic; and (iii) ideal. I also assume (iv): The system is to be conservative, and the Lagrangian L, defined as the difference of the kinetic energy T and the potential energy V, $L(q_{1},\,\ldots,\,q_{n},\,\dot{q}_{1},\,\ldots,\,\dot{q}_{n}) \mathrel{\mathop:}= T-V$, is to be a $C^{2}$ (twice continuously differentiable) function in all 2n arguments.

These four features yield (the simplest form of) Lagrangian mechanics. For they imply that the laws of motion of the system are given by the Euler-Lagrange (also known as, Lagrange) equations, and that these are equivalent to Hamilton's Principle: that the motion in Q of the point representing the system, between prescribed configurations at times $t_{0}$ and $t_{1}$, makes stationary the action integral $\int L\, dt$. A curve in G satisfying these laws of motion is called an extremal.

I also assume (v): that the Hessian of L with respect to the $\dot{q}$s does not vanish in G, i.e. the determinant

This yields (the simplest form of) Hamiltonian mechanics, and makes it equivalent to the previous Lagrangian mechanics; as follows. We define canonical momenta, $p_{i}\mathrel{\mathop:}= L_{\dot{q}_{i}}$; then use (1) to solve this definition for the $\dot{q}_{i}$ as functions of $q_{i}$, $p_{i}$, t: $\dot{q}_{i}=\dot{q}_{i}(q_{j},\,p_{j},\,t) $; then perform a Legendre transformation to introduce the Hamiltonian function and so render the laws of motion, i.e. the Euler-Lagrange equations, equivalent to the canonical (also known as: Hamilton's) equations. Accordingly, we take the system's state-space to be, not G or Q, but the 2n-dimensional phase space Γ coordinatized by the ps and qs.

Finally, I assume (vi): the region $G\subset \mathstrut{\Bbb R} ^{n+1}$ is sufficiently small that between any two “event” points $E_{1}=(q_{1i},\,t_{1}) $, $E_{2}=(q_{2i},\,t_{2}) $ there is a unique extremal curve C. (I will sometimes suppress the i, writing $E_{1}=(q_{1},\,t_{1}) $, $E_{2}=(q_{2},\,t_{2}) $ etc.) This yields (the simplest form of) Hamilton-Jacobi theory; as follows.

2.2. Hamilton-Jacobi Theory

I will describe Hamilton-Jacobi theory in more detail than Lagrangian and Hamiltonian mechanics: for we will need this detail in order to explore its modal involvements. In Section 2.2.1, I introduce the Hamilton-Jacobi equation via Hamilton's characteristic function; then in Section 2.2.2, I discuss hypersurfaces, congruences, and fields. Even so, these details will give only a limited view of a very rich theory. In particular:

1. I will ignore aspects to do with problem-solving (especially the use of separation of variables, leading on to action-angle variables and Liouville's theorem) since—though obviously crucial for physics—they are not illuminating about modality.

2. I will ignore the integration theory of the Hamilton-Jacobi equation, which involves the theory of generating functions and complete integrals; (though this deep and beautifully geometric theory is illuminating about modality.)

3. Both Sections 2.2.1 and 2.2.2 will emphasize the extended configuration space of Section 2.1, i.e. the region $G\subset \mathstrut{\Bbb R} ^{n+1}$; while it is equally illuminating to consider Hamilton-Jacobi theory in phase space. But this emphasis on G will suffice for our purposes—to reveal some distinctive modal involvements.

2.2.1. The Characteristic Function and the Hamilton-Jacobi Equation

Assumption (vi) implies that the value of the action integral along the unique extremal C from $E_{1}=(q_{1},\,t_{1}) $ to $E_{2}=(q_{2},\,t_{2}) $ is a function of the coordinates of the end-points. We call this function the characteristic function and write it as

By considering arbitrary small displacements $(\delta q_{1},\,\delta t_{1}) $, $(\delta q_{2},\,\delta t_{2}) $ at E ₁, E ₂ respectively, one deduces that S, considered as a function of the $n+1$ arguments $(q_{2},\,t_{2}) $ (i.e. with $(q_{1},\,t_{1}) $ fixed), obeys the equation—now rewriting t ₂, q ₂ as t, q—

This is the Hamilton-Jacobi equation.

S also defines a family of hypersurfaces, which we can call ‘spheres’ with centre $E_{1}=(q_{1},\,t_{1}) $: the sphere around $(q_{1},\,t_{1}) $ with radius R is given by the equation

Every point $E_{2}=(q_{2},\,t_{2}) $ on this sphere is connected to the center $E_{1}=(q_{1},\,t_{1}) $ by a unique extremal along which the action integral has value R. This is amusingly reminiscent of Lewis's spheres of worlds (Reference Lewis1973, chap. 1.3; Reference Lewis1986a, chap. 1.3): and more than amusingly—we will see in Section 4 that the analogy is deeper.

2.2.2. Hypersurfaces and Fields

Of course, partial differential equations have many solutions: (the main contrast with ordinary differential equations being that typically, the solution contains an arbitrary function (or functions) rather than an arbitrary constant (or constants)). So Hamilton-Jacobi theory studies the whole space of solutions of the Hamilton-Jacobi equation. I need to report the main classical result of this study. (For details, a good reference is Rund (Reference Rund1966, chap. 2), who cites various masters of the last two centuries, especially Carathéodory.) The result connects three diverse notions:

a. Families of hypersurfaces in our region G of $\mathstrut{\Bbb R} ^{n+1}$

with $\sigma \in \mathstrut{\Bbb R} $ the parameter labelling the family; where we assume that S is a $C^{2}$ function in all $n+1$ arguments, and that the family foliates the region G simply in the sense that through each point of G there passes a unique hypersurface in the family.

b. Congruences of curves that: (i) cross the hypersurfaces and fill G simply in the corresponding sense that through each point of G there passes a unique curve in the congruence; and (ii) may be parametrically represented by n equations giving $q_{i}$ as $C^{2}$ functions of n parameters $u_{\alpha }$ and t:

where each set of n $u_{\alpha }=(u_{1},\,\ldots,\,u_{n}) $ labels a unique curve in the congruence. Such a congruence determines tangent vectors $(\dot{q}_{i},\,1) $ at each $(q_{i},\,t) $; and thereby also values of the Lagrangian $L(q_{i}(u_{\alpha },\,t),\,\dot{q}_{i}(u_{\alpha },\,t),\,t) $ and of the momentum

c. Fields, defined to be a set of 2n C ² functions q _i, p _i of $(u_{\alpha },\,t) $ as in (6) and (7), i.e. with the qs and ps related by $p_{i}=\partial L/ \partial \dot{q}_{i}$. So a congruence determines a field, and a field determines (by a Legendre transformation, using (1)) a set of tangent vectors, and so a congruence.

Some jargon: (i) If all the curves of the congruence determined by a field are extremals, the field is called a field of extremals. (ii) We say a field (or its congruence) belongs to a family of hypersurfaces given by (5) iff throughout the region G the $p_{i}=\partial L/ \partial \dot{q}_{i}$ of the field obey

(iii) We say that a field $q_{i}=q_{i}(u_{\alpha },\,t) $, $p_{i}=p_{i}(u_{\alpha },\,t) $ is canonical if the q _i, p _i satisfy Hamilton's equations: equivalently, if the curves of the congruence determined by the field are extremals.

So much for definitions; now the result. The following three conditions on a $C^{2}$ function $S\thinspace{:}\thinspace G\rightarrow \mathstrut{\Bbb R} $ are equivalent:

1. 1. S is a $C^{2}$ solution (throughout G) of the Hamilton-Jacobi equation

   

2. 2. The field belonging to the $C^{2}$ function $S\thinspace{:}\thinspace G\rightarrow \mathstrut{\Bbb R} $, i.e., the field defined at each point in G by $p_{i}=\partial S/ \partial q_{i}$, is canonical.

3. 3. The value of the action integral $\int L\, dt$ along the curve C of the congruence belonging to S, from any point $P_{1}$ on the surface $S(q_{i},\,t) =\sigma _{1}$ to that point $P_{2}$ on the surface $S(q_{i},\,t) =\sigma _{2}$ that lies on C, is the same for whatever point $P_{1}$ we choose; and the value is just $\sigma _{2}-\sigma _{1}$. That is:

   

In the light of (10), we call a family of hypersurfaces $S=\sigma $ satisfying any, and so all, of these three conditions geodesically (or geodetically) equidistant (with respect to the Lagrangian L). So the concentric spheres centred on $E_{1}=(q_{1},\,t_{1}) $ introduced above (4) are an example of a geodesically equidistant family.

This result leaves it an open question which n-dimensional surfaces M in G are level surfaces of a $C^{2}$ solution S of the Hamilton-Jacobi equation. In fact it can be shown, subject to some mild conditions about non-vanishing determinants etc., that:

1. 1. any n-dimensional surface M is a level surface of a solution, and this solution is uniquely defined throughout G by its value on M (say $S=0$ on M); and

2. 2. for any such surface M and any suitably smooth function $S\thinspace{:}\thinspace M\rightarrow \mathstrut{\Bbb R} $, there is a uniquely defined solution on all of G which restricts on M to the given S.

(So (2) generalizes (1) by M not having to be a level surface.)

In the jargon, the initial value problem for the Hamilton-Jacobi equation has locally a solution, that is unique given suitably smooth prescribed values of S. But I shall not go into details about this. It suffices to state the intuitive idea for the case where M is a level surface. The solution is “grown” from the given surface by erecting a congruence of curves, transverse to the surface, and passing along them to mark off a given value of the action integral $\int L\, dt$. By varying the value, one defines a geodesically equidistant family and so a solution S.

Returning finally to mechanics: it is clear that each solution S of the Hamilton-Jacobi equation represents a kind of ensemble, i.e., a fictitious population of mechanical systems (maybe including the actual system). Thus each solution S represents an ensemble with the feature that at all times t, there is a strict configuration-momentum, i.e,. $q-p$, correlation given by the gradient of S. That is, S prescribes for any given $(q,\,t) $, a unique $(p,\,t) \mathrel{\mathop:}= (\partial S/ \partial q,\,t) $.

So much by way of expounding Hamilton-Jacobi theory. I will return, after the introduction of modality in Section 3, to discuss the structure of this set of ensembles (set of S-functions)—and so the modal involvements of Hamilton-Jacobi theory. I end this section by emphasising that, as mentioned in Section 1, my restriction to simple systems is a matter of brevity and expository convenience, not of substance. Much of the formalism above, and the philosophical morals below, apply much more widely.

4.1. Configurations as Worlds

Let us think of an event (i.e. instantaneous configuration) $(q_{i},\,t) \in G$ as like a possible world. Then Hamilton's characteristic function (2), and the geodesic spheres it defines (4), yield a neat analogy with Lewis's theory of counterfactuals. For recall Lewis's proposed truth-conditions for a counterfactual ‘If A were the case, then C would be the case’, which I will write as $A\rightarrow C$ (Reference Lewis1973, chap. 1.3). Lewis wants to avoid the assumption that there is a set of A-worlds all tied for first equal as regards similarity to the actual world @ (the Limit Assumption). He also allows the counterfactual to be vacuously true: namely iff no world in the union of nested spheres around @, ∪ $_@, makes A true. So Lewis proposes that the counterfactual $A\rightarrow C$ is true at @ iff:

1. 1. no A-world belongs to any sphere S in the system $_@ of spheres around @; or

2. 2. some sphere S in the system $_@ contains at least one A-world, and $A\supset C$ is true at every world in S; (i.e. C is true at every A-world in S).

We can easily transplant this kind of truth-condition to geodesic spheres; i.e., taking points $(q_{i},\,t) \in G$ as worlds and $\int L\, dt$ as the measure of distance (dissimilarity) between such worlds. However, the resulting conditionals hardly deserve the name ‘counterfactual’, since both the “base-world” $(q_{1},\,t_{1}) $ and the “closest A-world,” say (q, t), that the evaluation of the conditional carries us to, could be actual configurations of the system. For simplicity I will ignore the vacuous case, (1) above. This yields the following truth-condition, relative to a given configuration $(q_{1},\,t_{1}) $:

1. a. A is true at a possible configuration (q, t), to which the given configuration (q ₁, t ₁) could evolve (i.e. would evolve for some $p_{1}$ at $t_{1}$) with $t> t_{1}$; and

2. b. for every possible configuration $(q^{\prime },\,t^{\prime }) $ to which $(q_{1},\,t_{1}) $ could evolve with $t> t_{1}$, and such that $\int_{q_{1},t_{1}}^{q^{\prime },t^{\prime }}L\, dt\leq \int_{q_{1},t_{1}}^{q,t}L\, dt$: $A\supset C$ is true at $(q^{\prime },\,t^{\prime }) $; (i.e. if A is true at $(q^{\prime },\,t^{\prime }) $, so is C).

In the abstract, this truth-condition seems a mouthful. But in fact mechanics provides countless examples of such conditional propositions, though of course in a much less formal guise! A very simple example is given by a bead sliding on a wire that lies in a vertical plane; (to be a simple system, the bead must slide frictionlessly). We can take A to say that the bead is at the lowest point of the wire, and C to say that its potential energy is at a minimum. Then $A\rightarrow C$ can be expressed informally as ‘Whenever the bead is next at the lowest point of the wire, its potential energy will then be at a minimum’. Similarly, with C saying instead that the kinetic energy is at a maximum; and so on.

Finally, the results in Section 2.2.2 (especially condition 3) imply that this discussion of counterfactuals can be generalized so as to define similarity of worlds using S-functions other than Hamilton's characteristic function. For example, we could take an n-dimensional surface M that is topologically a sphere surrounding some given point $(q_{1},\,t_{1}) \in G$, define M to be a surface of constant S, say $S=0$, and consider the (locally unique) solution of the Hamilton-Jacobi equation thereby defined outside M. That is, we could define the dissimilarity of our worlds (q, t) from the base-world $(q_{1},\,t_{1}) $, and so the truth-conditions of counterfactuals, in terms of the value of $S(q,\,t) $.

4.2. States as Worlds

On the other hand, let us take as the analogue of a Lewisian world an instantaneous state in the sense of a $2n+1$-tuple $(q_{i},\,p_{i},\,t) $. This is perhaps a more natural choice than Section 4.1's instantaneous configurations (events), since it determines a history, i.e. a phase space trajectory, of the system, our “toy-universe.” There are various constructions one could make with this concept of world. In particular, one could define conditionals $A\rightarrow C$ by using a solution S of the Hamilton-Jacobi equation to define dissimilarity. But for the sake of variety, I shall not pursue this. I shall instead describe how an S-function enables us to define various sets of possible worlds which are “preferred” relative to our choice of S; in fact, the last of these definitions is important for physics.

Here again, the S-function can be any solution of the Hamilton-Jacobi equation. Given such an S, every point $(q,\,t) \in G$ has an associated canonical momentum, viz. $p\mathrel{\mathop:}= (\partial / \partial q) S(q,\,t) $, and so an associated world in our sense, viz. (q, $p\equiv \partial S/ \partial q,\,t$). If we wish, we can also pick out subsets so that not every event (q, t) is included in a world “preferred” by our S. For example, we could do this by picking out a sub-manifold M of G, and defining the associated worlds (q, $p\equiv \partial S/ \partial q,\,t$) only for $(q,\,t) \in M$.

There are two obvious ways to specify such an M; both make M n-dimensional. First, we can define M as the level surface of S that passes through some given $(q,\,t) \in G$. This definition will connect M with Section 2.2.2's discussion of geodesically equidistant hypersurfaces. And thinking of (q, t) as the system's actual present configuration, M defines a preferred set of counterfactual events, i.e. instantaneous configurations (which are in general not simultaneous with (q, t)).

Secondly, we can fix t. For the chosen t, we consider the gradient $(\partial / \partial q) S(q,\,t) $ of S as a function on Q. The preferred worlds are then given by all (q, $p\equiv (\partial / \partial q) S(q,\,t) $) for $q\in Q$. So the worlds are given as before, except that the fixed value of t is now implicit in the definition of p.

This definition gives us a suitable note to end on. For it turns out that this second definition is crucially important for the mathematics and physics of Hamilton-Jacobi theory in phase space. It leads to the mathematics of Lagrangian submanifolds, and the physics of focussing and caustics (and even to the quantum-classical relation!). In fact, we can analyse the structure of the set of possible preferred sets by studying the set of all Lagrangian manifolds; (for some more details, cf. e.g. Arnold (Reference Arnold1989, Chaps. 7, 8)). So I like to think of Lewis's genius—always so creative, insightful and generous—giving us a philosophical perspective on the deep and beautiful structures of classical mechanics.