Discussion

Investing capital rentals to sustain periodic motion

in classical mechanics by John Hartwick

Thomas Russell*

Department of Economics, Santa Clara University, 300 Kenna Hall, Santa Clara, CA 95053, USA

Received 4 April 2003; received in revised form 1 December 2003; accepted 8 December 2003

It is a great pleasure to be asked to discuss this paper by Professor Hartwick. Professor

Hartwick’s summary (1977) of Professor Solow’s well-known paper, Solow (1974), in the

form of what is now called ‘‘Hartwick’s Rule’’ is a classic which, like a fine wine, simply

improves with JSTOR-age.

Let me say at once that I am not an expert in the area of resource economics. There are a

number of such experts in the room, so I will simply make some general remarks then hand

the discussion back to the floor.

The central question examined by Professor Hartwick is the relationship between capital

theoretic resource economics on the one hand and classical mechanics on the other. I share

Professor Hartwick’s view that these two disciplines are intimately related in a natural and

unforced way. The reason for this, it seems to me, is very simple. The phase space of

resource economics and the phase space of classical mechanics have the same geometry.

They are both Poisson manifolds. This term is due to Lichnerowicz (1977) and, in the

special case of symplectic manifolds, will be discussed further below. As Arnold (1981,

p. 161) has put it, ‘‘Hamiltonian mechanics is geometry in phase space,’’ (emphasis added),

to which the economist might add ‘‘and so is resource economics.’’

That these two systems have a common geometry, however, seems to me to be largely a

historical accident. Hamilton developed what is now called the Hamiltonian approach as a

way of showing that optics, his source of income as Astronomer Royal of Ireland, could be

given the same D’Alembert/Lagrange mathematical structure as the then considerably

more prestigious field of mechanics. When he was done, he noticed almost serependipi-

tously that all of the optics argument could be reversed to derive the propositions of

classical mechanics from his new fangled Hamiltonian. It should be noted that this added

nothing to our understanding of mechanics, and if physics had ended in 1850, the

Hamiltonian would have been little more than a curiosity. Kepler in Linz in 1620 knew

Japan and the World Economy

16 (2004) 359–362

* Tel.: þ1-408-554-6953; fax: þ1-408-554-2331.

E-mail address: trussell@scu.edu (T. Russell).

0922-1425/$ – see front matter # 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.japwor.2003.12.001

as much about planetary motion without the Hamiltonian as Hamilton in Dublin in 1830

knew with it.

However, physics did not end in 1850, and what brought the Hamiltonian fame and

fortune was its ‘‘sudden and unexpected starring role in Schrodinger’s wave mechanics

theory of quantization’’ (Olver, 1993). Also in classical mechanics the Hamiltonian was

perfectly suited to the Lie Poisson Noether method of solving differential equations by

finding first integrals and reducing dimension.

In economics the Hamiltonian has a completely different history. Hamiltonians came to

prominence in the late-1960s and early-1970s as a representation of the dual functions of

minimization problems. Cass and Shell (1976), for example, pioneers in this area, cite

approvingly Lau’s use of the term ‘‘Restricted profit function’’ to describe the Hamiltonian.

Underlying this work was Ramsey’s use of the calculus of variations and the Euler

Lagrange equation in optimal growth theory.

But geometry is geometry no matter how it is arrived at, and it would be very surprising if

the now mature development of Poisson brackets in mechanics did not contain nuggets

waiting to be found by economic prospectors. Surprisingly to me, very few economists

have chosen to enter this mine. The most notable exception is our host Professor Sato (see,

for example Sato and Kim, 2002). He has shown that the resource extraction problem

discussed by Solow and Hartwick has a time invariant Hamiltonian from which Hartwick’s

Rule just falls out.

The underlying geometric problem couched in the language of Poisson manifolds is well

described by Weinstein (2002). Since this approach is likely to be unfamiliar to economists,

here is an extended quote setting out the details.

The relation between conserved quantities and symmetries becomes very simple when

expressed in terms of Poisson brackets. On a symplectic manifold P, the time evolution

of a function F under the Hamiltonian flow of a function H is given by the formula

dF

dt¼ fF;Hg:

Antisymmetry of this bracket implies immediately that F is a conserved quantity for the

Hamiltonian flow of H if and only if the Hamiltonian flow of F consists of symmetries of

H. If F and G are both conserved quantities, then so is {F, G}, by Poisson’s theorem,

which is most conveniently proved via the Jacobi identity. If F1, . . ., Fk are conserved

quantities, then so is any function g(Fi, . . ., Fk), so the process of building new conserved

quantities from old ones naturally terminates if we arrive at a list F1, . . ., Fk of functions

such that, for each pair of functions in the list fFi;Fjg ¼ pijðF1; . . . ;FkÞ for functions pij.

Lie [13] refers to such a list of functions as generating a function group, the function

group itself consisting of all the g(F1, . . ., Fk). If F1, . . ., Fk are functionally independent,

then the pij are uniquely determined smooth functions on Rk and define on the set of

all smooth functions on Rk the structure of what Lie [13] calls an abstract function

group.

The passage above clearly has major implications both for resource economics and for

Professor Hartwick’s goal of recasting classical mechanics as a problem in resource

360 T. Russell / Japan and the World Economy 16 (2004) 359–362

conservation. For example, if we know that consumption is conserved, then we know

that

dC

dt¼ fC;Hg ¼ 0

so C is constant (sustainability) when the Poisson bracket of C with H is zero. Other

constants of the motion say K must satisfy fK;Cg ¼ 0, because of the Jacobi identity. This

gives a way of uncovering new constants.

This is the way modern mechanics proceeds. Perhaps it is also the way economics should

proceed. That is to say, the question of how to extract coal from a pit may benefit greatly

from a careful study of how Poisson geometry is applied in mechanics.

This, however, is not what Professor Hartwick is about. He wants to reverse the process.

Professor Hartwick is interested not in the intellectual debt which the economic pit owes to

the mechanical pendulum. Rather he is interested in the debt which the pendulum owes to

the pit. On the value of this obligation I am afraid I am very skeptical.

Of course pits and pendula share the same geometry, so we know it can be done.

Moreover, in some sense it is clear that in the quarter circle in which the pendulum is losing

potential energy and gaining kinetic energy something like a sustainability criterion must

be satisfied. Professor Hartwick has shown exactly how this can be done and to do so is

not easy.

But so what? What does framing it as a resource sustainability problem tell us about the

pendulum? More generally what do we learn about mechanics if we define it in economic

terms? My sense is not much, but time will tell and time is pressing so I will conclude

with one extended and early quotation on whether the exercise conducted by Professor

Hardwick has any general validity. Writing on the question of whether or not nature was an

economist, the author said this

But although the law of least action has thus attained a rank among the highest theorems

of physics, yet its pretensions to a cosmological necessity, on the ground of economy in

the universe, are now generally rejected. And the rejection appears just, for this, among

other reasons, that the quantity pretended to be economized is in fact often lavishly

expended. In optics, for example, though the sum of the incident and reflected portions

of the path of light, in a single ordinary reflection at a plane, is always the shortest

of any, yet in reflection at a curved mirror this economy is often violated. If an eye be

placed in the interior but not at the center of a reflecting hollow sphere, it may see itself

reflected in two opposite points, of which one indeed is the nearest to it, but the other on

the contrary is the furthest; so that of the two different paths of light, corresponding to

these two opposite points, the one indeed is the shortest, but the other is the longest

of any.

In mathematical language, the integral called action, instead of being always a

minimum, is often a maximum; and often it is neither the one nor the other: though

it has always a certain stationary property, of a kind which has been already alluded

to, and which will soon be more fully explained. We cannot, therefore, suppose the

economy of this quantity to have been designed in the divine idea of the universe: though

simplicity of some high kind may be believed to be included in that idea.

T. Russell / Japan and the World Economy 16 (2004) 359–362 361

And though we may retain the name of action to denote the stationary integral to which it

has become appropriated which we may do without adopting either the metaphysical or

(in optics) the physical opinions that first suggested the name yet we ought not (I think)

to retain the epithet least: but rather to adopt the alteration proposed above, and to speak,

in mechanics and in optics, of the Law of Stationary Action.

This author was, of course, Hamilton (1833).

References

Arnol’d, V.I., 1981. Mathematical Methods of Classical Mechanics. Springer, New York.

Cass, D., Shell, K. (Eds.), 1976. The Hamiltonian Approach to Dynamic Economics. Academic Press, New

York.

Hamilton, W.R., 1833. On a general method of expressing the paths of light, and of the planets, by the

coefficients of a characteristic function. Dublin University Review and Quarterly Magazine 1, 795–826.

Hartwick, J.M., 1977. Intergenerational equity and the investing of rents from exhaustible resources. American

Economic Review 67, 972–974.

Lichnerowicz, A., 1977. Les varietes de Poisson et leurs algebras de Lie associes. Journal of Differential

Geometry 12, 253–300.

Olver, P.J., 1993. Applications of Lie Groups to Differential Equations. Springer, New York.

Sato, R., Kim, Y., 2002. Hartwick’s Rule and economic conservation laws. Journal of Economic Dynamics and

Control 26, 437–449.

Solow, R.M., 1974. The economics of resources or the resources of economics. American Economic Review 64,

1–14.

Weinstein, A., 2002. The Geometry of Momentum. http://arxiv.org/abs/math.SG/0208108.

362 T. Russell / Japan and the World Economy 16 (2004) 359–362