David Lewis Meets Hamilton and Jacobi J. Butter�eld12 All Souls College, University of Oxford (This version submitted 7 Feb 2003) For the PSA02 special issue of Philosophy of Science: Symposium in memory of David Lewis Abstract I commemorate David Lewis by discussing an aspect of modality within an- alytical mechanics, which is closely related to his work on counterfactuals. This concerns the way Hamilton-Jacobi theory uses ensembles, i.e. sets of possible initial conditions. (A companion paper discusses other aspects of modality in analytical mechanics that are equally related to Lewis' work.) 1Send reprint requests to the author, All Souls College, Oxford OX1 4AL, England. 2Many thanks to: Peter Holland and Graeme Segal for conversations; Alexander Afriat, Robert Bishop, Larry Gould, Susan Sterrett and Paul Teller for comments on a previous version; and Gerard Emch and Klaas Landsman for technical help. 1 Introduction It is an honour to take part in commemorating David Lewis. He was a great philoso- pher, whose genius graced our lives in many ways: how much we miss his transcendent creativity and craftsmanship, his enormous intellectual generosity|and his sense of fun. I propose to commemorate him by discussing modality in analytical mechanics. Though this topic is not close to his interests, it will illustrate a view central to his metaphysical system, and to his in uence on analytical philosophy: that science, indeed all our knowledge and belief, is steeped in modality. Although philosophers seem not to have explored the modal involvements of an- alytical mechanics, they are both rich and subtle. There is much to explore here: as so often in the philosophy of physics, one can mine from a little physics, a lot of philosophy|at least, a lot more than one paper! To be brief enough, I shall have to be selective in various ways. The three main ones are:{ 1) I shall consider only a very limited class of classical mechanical systems, and as- sume knowledge of how analytical mechanics treats them (Section 2). This limitation is a matter of brevity and convenience: the philosophical discussion in Sections 3 and 4 applies much more widely. 2) Though elementary analytical mechanics is modally involved in many ways, I discuss just one such involvement, which arises in Hamilton-Jacobi theory; for it is closely connected to Lewis' theory of counterfactuals (1973). But several other modal involvements are at least as obvious, and worth pursuing, as this one: e.g. the way the Lagrangian and Hamiltonian approaches' variational principles state the law of motion by mentioning dynamical evolutions that violate the law. 3) This modal involvement, and others, are entangled, technically and philosophi- cally, with the fact that analytical mechanics provides general schemes for solving and representing problems. These general schemes hold philosophical morals, but here I must set them aside. (My (2003, 2003a) discuss several topics which are excluded here by limitations 1) to 3).) The plan of the paper is as follows. In Section 2, I introduce the elements of ana- lytical mechanics I need. In Section 3, I distinguish three grades of modal involvement, according to which kind of actual matters of fact are varied counterfactually. The rest of the paper concentrates on the �rst grade, which considers counterfactual ini- tial and/or �nal conditions, but keeps �xed the forces on the system and the laws of motion. It is strikingly illustrated by Hamilton-Jacobi theory's S-function, which rep- resents a structured ensemble of such conditions. The theory involves many di�erent S-functions, and so ensembles. So in Section 4, I discuss the structure of this set: in particular, there is an analogy with Lewis' spheres of possible worlds. 1 2 Technical preliminaries 2.1 Simple systems; Lagrangian and Hamiltonian mechanics I shall consider a mechanical system with n con�gurational degrees of freedom, whose evolution over time t is given by a curve in a �xed simply connected region G of (n+1)- dimensional real space IR n+1 , on which there are coordinates (q1; : : : ;qn; t) =: (qi; t) =: (q;t). If the system consists of N point-particles (or bodies small enough to be treated as point-particles), so that a con�guration is �xed by 3N cartesian coordinates, we may yet have n < 3N; for the system may be subject to constraints, and the qi 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).3 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 con�guration space Q � IR n with a time-interval [t � ; t+] � IR (where we allow t� = �1; t+ = +1); (ii) holonomic; and (iii) ideal. I also assume (iv): The system is to be conservative, and the La- grangian L, de�ned as the di�erence of the kinetic energy T and the potential energy V , L(q1; : : : ;qn; _q1; : : : ; _qn) := T � V , is to be a C 2 (twice continuously di�erentiable) 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 con�g- urations at times t0 and t1, makes stationary the action integral R 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 _qs does not vanish in G, i.e. the determinant j L _qi _qj j 6= 0 : (2.1) This yields (the simplest form of) Hamiltonian mechanics, and makes it equivalent to the previous Lagrangian mechanics; as follows. We de�ne canonical momenta, pi := L _qi; then use eq. 2.1 to solve this de�nition for the _qi as functions of qi;pi; t : _qi = _qi(qj;pj; t); then perform a Legendre transformation to introduce the Hamilto- nian 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 � coordinatized by the ps and qs. Finally, I assume (vi): the region G � IR n+1 is suÆciently small that between any two \event" points E1 = (q1i; t1);E2 = (q2i; t2) there is a unique extremal curve C. (I will sometimes suppress the i, writing E1 = (q1; t1);E2 = (q2; t2) etc.) This yields (the 3For further discussion of these features, cf. textbooks such as Arnold (1989) and Lanczos (1986). Some of these features, e.g. (iii), evidently involve modal notions; for discussion, cf. my (2003, 2003a). 2 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 �elds. 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 emphasise the extended con�guration space of Section 2.1, i.e. the region G � IR n+1 ; while it is equally illuminating to consider Hamilton-Jacobi theory in phase space. But this emphasis on G will suÆce 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 E1 = (q1; t1) to E2 = (q2; t2) is a function of the coordinates of the end-points. We call this function the characteristic function and write it as S(q1; t1; q2; t2) := Z t2 t1 L dt = Z t2 t1 (�ipi _qi � H) dt = Z �ipidqi � Hdt : (2.2) By considering arbitrary small displacements (Æq1;Æt1); (Æq2;Æt2) at E1;E2 respectively, one deduces that S, considered as a function of the n + 1 arguments (q2; t2) (i.e. with (q1; t1) �xed), obeys the equation|now rewriting t2;q2 as t;q| @S @t + H(q; @S @q ;t) = 0 : (2.3) This is the Hamilton-Jacobi equation. S also de�nes a family of hypersurfaces, which we can call `spheres' with centre E1 = (q1; t1): the sphere around (q1; t1) with radius R is given by the equation S(q1; t1; q2; t2) = R : (2.4) Every point E2 = (q2; t2) on this sphere is connected to the centre E1 = (q1; t1) by a unique extremal along which the action integral has value R. This is amusingly reminiscent of Lewis' spheres of worlds (1973, Chapter 1.3; 1986, Chapter 1.3): and more than amusingly|we will see in Section 4 that the analogy is deeper. 3 2.2.2 Hypersurfaces and �elds Of course, partial di�erential equations have many solutions: (the main contrast with ordinary di�erential 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 (1966, Chap. 2), who cites various masters of the last two centuries, especially Carath�eodory.) The result connects three diverse notions:| (a): Families of hypersurfaces in our region G of IRn+1 S(qi; t) = � (2.5) with � 2 IR the parameter labelling the family; where we assume that S is a C2 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 �ll 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 qi as C2 functions of n parameters u� and t qi = qi(u�; t) ; i = 1; : : : ;n ; (2.6) where each set of n u� = (u1; : : : ;un) labels a unique curve in the congruence. Such a congruence determines tangent vectors ( _qi; 1) at each (qi; t); and thereby also values of the Lagrangian L(qi(u�; t); _qi(u�; t); t) and of the momentum pi = pi(u�; t) = @L @ _qi : (2.7) (c): Fields, de�ned to be a set of 2n C2 functions qi;pi of (u�; t) as in eqs 2.6 and 2.7, i.e. with the qs and ps related by pi = @L @ _qi . So a congruence determines a �eld, and a �eld determines (by a Legendre transformation, using eq. 2.1) a set of tangent vectors, and so a congruence. Some jargon: (i) If all the curves of the congruence determined by a �eld are extremals, the �eld is called a �eld of extremals. (ii) We say a �eld (or its congruence) belongs to a family of hypersurfaces given by eq. 2.5 i� throughout the region G the pi = @L @ _qi of the �eld obey pi = @ @qi S(qi; t) = @ @qi S(qi(u�; t); t) : (2.8) (iii) We say that a �eld qi = qi(u�; t);pi = pi(u�; t) is canonical if the qi;pi satisfy Hamilton's equations: equivalently, if the curves of the congruence determined by the �eld are extremals. 4 So much for de�nitions; now the result. The following three conditions on a C2 function S : G ! IR are equivalent: (1): S is a C2 solution (throughout G) of the Hamilton-Jacobi equation @S @t + H(q; @S @q ;t) = 0 : (2.9) (2): The �eld belonging to the C2 function S : G ! IR, i.e. the �eld de�ned at each point in G by pi = @S @qi , is canonical. (3): The value of the action integral R L dt along the curve C of the congruence belonging to S, from any point P1 on the surface S(qi; t) = �1 to that point P2 on the surface S(qi; t) = �2 that lies on C, is the same for whatever point P1 we choose; and the value is just �2 � �1. That is: Z P2 P1 L dt = �2 � �1 : (2.10) In the light of eq. 2.10, we call a family of hypersurfaces S = � 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 E1 = (q1; t1) introduced above (eq. 2.4) 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 C2 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): any n-dimensional surface M is a level surface of a solution, and this solution is uniquely de�ned throughout G by its value on M (say S = 0 on M); and (2): for any such surface M and any suitably smooth function S : M ! IR, there is a uniquely de�ned 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 suÆces 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 o� a given value of the action integral R L dt. By varying the value, one de�nes a geodesically equidistant family and so a solution S. Returning �nally to mechanics: it is clear that each solution S of the Hamilton- Jacobi equation represents a kind of ensemble, i.e. a �ctitious population of mechanical systems (maybe including the actual system). Thus each solution S represents an en- semble with the feature that at all times t, there is a strict con�guration|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) := (@S @q ; t). So much by way of expounding Hamilton-Jacobi theory. I will return, after Section 3's introduction of modality, to discuss the structure of this set of ensembles (set 5 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. 3 Grades of modal involvement In the light of Section 2, I think it is natural to distinguish (in Quinean fashion!) three grades of modal involvement in analytical mechanics. So I shall write (Modality;1st) etc. Like Quine's three grades, the �rst is intuitively the mildest grade, and the third the strongest. But this order will not correspond to Section 2's (and the historical) order of the three approaches to analytical mechanics: Lagrangian, Hamiltonian and Hamilton-Jacobi. In particular, the �rst and arguably most intuitive approach, La- grangian mechanics, exhibits the third grade of modal involvement. The grades are de�ned in terms of the kind of actual matters of fact they allow to vary counterfactually.4 The �rst kind is, roughly, the state of the system. The second kind is the physical problem: which we can take as speci�ed by the number of degrees of freedom, and the Lagrangian or Hamiltonian which encodes all the forces on the system. A third kind is the laws of motion, as speci�ed by e.g. Hamilton's Principle or Lagrange's or Hamilton's equations. Thus we have the following grades. (Modality;1st): The �rst and mildest grade keeps �xed the actual physical problem and laws of motion. But it considers di�erent initial conditions and/or �nal conditions than the actual ones. And so it also considers counterfactual histories of the system; (since under determinism, a di�erent initial or �nal state implies a di�erent history, i.e. trajectory in state-space). This grade is evident throughout analytical mechanics. It arises from the postula- tion of a state-space, be it (in Section 2's notation) Q or � or G. For example, recall from our de�nition of a simple system: (i) the con�guration space Q is to have inde- pendently variable coordinates qi; and (ii) to de�ne ideal constraints, one needs the notion of a virtual displacement, i.e. a displacement that the system could undergo compatibly with the constraints and applied forces. But the most striking illustration of (Modality;1st) is Hamilton-Jacobi theory; cf. Section 4. (Modality;2nd): The second grade keeps �xed the laws of motion, but considers di�erent problems than the actual one (and thereby di�erent initial states). For ex- ample, it considers a counterfactual number of degrees of freedom, or a counterfactual potential function. Maybe no actual system is a simple system with 5,217 coordinates; or with a potential given (in certain units) by the polynomial 13x7 + 5x3 + 42. But 4I of course set aside the (apparent!) fact that the actual world is quantum, not classical; so I talk about e.g. an actual system obeying Hamilton's Principle. Since my business throughout is the philosophy of classical mechanics, it is unnecessary to encumber my argument with antecedents like `If the world were not quantum': I leave you to take them in your stride. 6 analytical mechanics continually considers such counterfactual cases: in Section 2, we generalized from the outset about the number n of degrees of freedom, and about what the Lagrangian or Hamiltonian was. Such generality of course pays o� in general theorems. (Modality;3rd): The third grade considers di�erent laws of motion, even for a given problem. This occurs even in the Lagrangian and Hamiltonian mechanics of simple systems; namely in their use of Hamilton's Principle. (And we saw in Section 2.2 that these variational principles are also involved in the Hamilton-Jacobi approach.) In all three approaches, the use of variational principles means|not that one explicitly states non-actual laws, much less calculates with them|but that one states the actual law by comparing the actual history of the system with counterfactual histories that do not obey the law. This is at �rst sight surprising, even mysterious. How can it be possible to state the actual law by a comparison of the actual history with possible histories that do not obey it? In particular, many philosophers hold that any actual truth is made true by an actual \truthmaker": a principle which such a statement of the laws of motion apparently violates. I take up these questions in Butter�eld (2003). To sum up, analytical mechanics gives many illustrations of all three grades: the subject is upto its ears in modality. But rather than multiplying examples, the remain- der of this paper focusses on Hamilton-Jacobi theory's illustration of (Modality;1st). There is no special philosophical diÆculty here: rather the situation presents an invi- tation to philosophers to study a new sort of modal structure. 4 On the set of ensembles Since the S-function, representing an ensemble of systems whose q and p are corre- lated by p = @S @q , stands at the centre of Hamilton-Jacobi theory, it is clear that the theory provides (Modality;1st) in spades. I stress that this use of ensembles involves no suspicious \possibilia power" (Lewis 1986a, p. 158). That is, there is no strange in uence (whether causal or constitutive) of the S-function, or the ensemble it rep- resents, on the actual system (or propositions about it). In particular, the evolution of a system (its trajectory in con�guration space or phase space) is �xed by, for ex- ample, the initial conditions|q; _q in Lagrangian mechanics and q;p in Hamiltonian mechanics|irrespective of which if any S-function we care to use.5 As mentioned at the end of Section 2.2, the structure of the set of ensembles (set of S-functions) is essentially given by the structure of the set of suitably smooth (say C2) 5I say `care to use' since, as mentioned at the start of Section 2.2, S-functions are principally used to solve otherwise intractable mechanical problems. Incidentally, the situation is di�erent in quantum theory. There, S has a close mathematical cousin (also written S) whose values do in uence the motion of the system. But again, this does not involve any weird \possibilia power". For this in uence is regarded as a strong, indeed the strongest, reason to take the quantum S-function as part of the actual physical state of the individual system; i.e. not as in classical mechanics, as \just" a description of an ensemble. 7 real functions on a n-dimensional manifold M; (M needs to \lie across" the region G so as to be transverse to a congruence of extremals). For since there is a locally unique solution to the Hamilton-Jacobi initial value problem, each such function determines| as well as is determined by|a solution throughout G of the Hamilton-Jacobi equation. So one infers that the set of solutions (ensembles) is some kind of in�nite-dimensional set. This set has various kinds of structure, and a full discussion would take us into those aspects of Hamilton-Jacobi theory that we had to set aside at the outset of Section 2.2. But even with just the results of Section 2.2, we can discern two kinds of structure|which bear on Lewis' account of counterfactuals and modality. These two kinds of structure arise from two di�erent choices about what to take as the analogue, in Hamilton-Jacobi theory, of a Lewisian possible world. 4.1 Con�gurations as worlds Let us think of an event (i.e. instantaneous con�guration) (qi; t) 2 G as like a possible world. Then Hamilton's characteristic function eq. 2.2, and the geodesic spheres it de�nes eq. 2.4, yield a neat analogy with Lewis' theory of counterfactuals. For recall Lewis' proposed truth-conditions for a counterfactual `If A were the case, then C would be the case', which I will write as A ! C (1973, Chap. 1.3). Lewis wants to avoid the assumption that there is a set of A-worlds all tied for �rst equal as regards similarity to the actual world @ (the Limit Assumption). He also allows the counterfactual to be vacuously true: namely i� no world in the union of nested spheres around @, [ $@, makes A true. So Lewis proposes that the counterfactual A ! C is true at @ i�:{ 1) no A-world belongs to any sphere S in the system $@ of spheres around @; or 2) some sphere S in the system $@ contains at least one A-world, and A � 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 (qi; t) 2 G as worlds and R 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" (q1; t1) and the \closest A-world", say (q;t), that the evaluation of the conditional carries us to, could be actual con�gurations of the system. For simplicity I will ignore the vacuous case, 1) above. This yields the following truth-condition, relative to a given con�guration (q1; t1): a) A is true at a possible con�guration (q;t), to which the given con�gura- tion (q1; t1) could evolve (i.e. would evolve for some p1 at t1) with t > t1; and b) for every possible con�guration (q0; t0) to which (q1; t1) could evolve with t > t1, and such that R q 0 ;t 0 q1;t1 L dt � R q;t q1;t1 L dt: A � C is true at (q0; t0); (i.e. if A is true at (q0; t0), so is C). 8 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 ! 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)) implies that this dis- cussion of counterfactuals can be generalized so as to de�ne similarity of worlds using S-functions other than Hamilton's characteristic function. For example, we could take a n-dimensional surface M that is topologically a sphere surrounding some given point (q1; t1) 2 G, de�ne M to be a surface of constant S, say S = 0, and consider the (locally unique) solution of the Hamilton-Jacobi equation thereby de�ned outside M. That is, we could de�ne the dissimilarity of our worlds (q;t) from the base-world (q1; t1), 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 (qi;pi; t). This is perhaps a more natural choice than Section 4.1's instantaneous con�gurations (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 de�ne conditionals A ! C by using a solution S of the Hamilton-Jacobi equation to de�ne dissimilarity. But for the sake of variety, I shall not pursue this. I shall instead describe how an S-function enables us to de�ne various sets of possible worlds which are \preferred" relative to our choice of S; in fact, the last of these de�nitions 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) 2 G has an associated canonical momentum, viz. p := @ @q S(q;t), and so an associated world in our sense, viz. (q;p � @S @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 de�ning the associated worlds (q;p � @S @q ; t) only for (q;t) 2 M. There are two obvious ways to specify such an M; both make M n-dimensional. First, we can de�ne M as the level surface of S that passes through some given (q;t) 2 G. This de�nition will connect M with Section 2.2.2's discussion of geodesi- cally equidistant hypersurfaces. And thinking of (q;t) as the system's actual present con�guration, M de�nes a preferred set of counterfactual events, i.e. instantaneous con�gurations (which are in general not simultaneous with (q;t)). 9 Secondly, we can �x t. For the chosen t, we consider the gradient @ @q S(q;t) of S as a function on Q. The preferred worlds are then given by all (q;p � @ @q S(q;t)) for q 2 Q. So the worlds are given as before, except that the �xed value of t is now implicit in the de�nition of p. This de�nition gives us a suitable note to end on. For it turns out that this second de�nition 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 (1989, Chap.s 7,8)). So I like to think of Lewis' genius|always so creative, insightful and generous| giving us a philosophical perspective on the deep and beautiful structures of classical mechanics. 5 References Arnold, V. (1989), Mathematical Methods of Classical Mechanics, New York: Springer- Verlag (second edition). Butter�eld, J. (2003), \Some Aspects of Modality in Analytical Mechanics", forthcom- ing in Formal Teleology and Causality, ed. M. St�oltzner, P. Weingartner, Paderborn: Mentis. Butter�eld, J. (2003a), \Postulating all States, Solving all Problems: some philosoph- ical morals of analytical mechanics". In preparation. Lanczos, C. (1986), The Variational Principles of Mechanics, New York: Dover (4th edition). Lewis, D. (1973), Counterfactuals, Oxford: Blackwell. Lewis, D. (1986), On the Plurality of Worlds, Oxford: Blackwell. Lewis, D. (1986a), Philosophical Papers, volume I, New York: Oxford University Press. Rund, H. (1966), The Hamilton-Jacobi Theory in the Calculus of Variations, New York: Van Nostrand. 10