## Info

Data from Murray and Dermott (1999).

5 See Murray and Dermott (1999), Section 8.17.

Cohen and Hubbard (1965) discovered the resonance, not by observation but via a numerical integration of the system over a period of 120 000 years. Their calculation was extended to cover 4.5 million years by Williams and Benson (1971) and the results confirmed.

orbit of Pluto is highly eccentric (e = 0.25) and, for a significant proportion of its 248-year orbit, is actually closer to the Sun than Neptune, but the two planets are never close together because the orbital resonance maximizes their separation at conjunction.7

When it comes to analysing resonant orbital motion as illustrated by the examples above, the mathematical techniques available at the beginning of the nineteenth century run into immediate problems. The thorn in the side of perturbation theory is the presence in the expansions for the elements of small divisors, and these are caused precisely by the presence of resonances between orbital periods. In orderto make theoretical progress it was necessary to develop new techniques, and the pioneers were the mathematicians Carl Gustav Jacobi and William Rowan Hamilton, from Germany and Ireland, respectively, who took the analytical mechanics begun by Euler and Lagrange to a new level.

### Hamilton and Jacobi

Hamilton developed his general methods for dynamics from a study of optical systems. At the beginning of the nineteenth century, optics mostly was treated geometrically. Hamilton developed an analytic theory much as Euler and Lagrange had done for mechanics. He generalized his results subsequently to coverthe theory of dynamics and published two fundamental papers in 1834/5. Hamilton's respect for Lagrange is clear from the introductory remarks to the first of these:

Lagrange has perhaps done more than any other analyst... by showing that the most varied consequences respecting the motions of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem.

Hamilton's dynamics revolved around what he called the 'principal function S' - now usually referred to as the 'action' - that is the integral of the Lagrangian function L = T - V (i.e. S = f L dt), and his fundamental dynamical principle was that the actual motion between two given points at two given times is such that this action is stationary. Hamilton's principle of stationary action can be shown to be equivalent to Lagrange's equations d dL dL

7 The minimum separation is about 2.6 billion km (Stern and Mitton (1998), p. 141).

where, in general, L = L (qa, qa, t), with t appearing explicitly and L no longer necessarily equal to T(qa, qa) - V(qa). Following Poisson, Hamilton introduced new independent variables pa, where dL dL

dqa dqa the second equation in Eqn (11.2) following from the first because of Eqn (11.1). The variable, pa is the called the 'momentum conjugate' to qa. In principle, we can solve for the N quantities qa in terms of the pas, the qa s, and t, and then construct a new function H(pa, qa, t), which is now referred to as 'the Hamiltonian', defined by

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