Stability Proofs and Consistency Proofs: A Loose Analogy

Abstract

A loose analogy relates the work of Laplace and Hilbert. These thinkers had roughly similar objectives. At a time when so much of our analytic effort goes to distinguishing mathematics and logic from physical theory, such an analogy can still be instructive, even though differences will always divide endeavors such as those of Laplace and Hilbert.

Appendix

Demonstrations of the stability of dynamical systems have been fraught with perplexities, despite all the elegance. If one has a set of simultaneous linear differential equations for n variables, and then eliminates these equations in succession so as to favor one variable, the process can terminate in a differential equation for that one. Substitute e^δt for it: the result is then an equation for δ; if there is a repeated root then there will be a second solution for te^δt. Should such a thing happen in the theory of small oscillations it would transpire that a repeated value of δ will lead to terms of the form t cos Kt, t sin Kt (K = iδ) and, save for specially designed initial conditions, a small oscillation will grow indefinitely. This is another way of putting Newton’s and Laplace’s problem: the planetary system is always representable as such a set of simultaneous linear differential equations. What is to prevent a small external disturbance from provoking a small internal oscillation of this sort (as when a rock in the road is responsible for an oscillatory vibration in a car’s speedometer-needle) such that the latter grows to disruptive proportions? This has never been found to happen in our system. But could it? Newton felt that the Deity would not allow it. But, if it did come about it would contradict the fundamental principle that if the planetary system’s potential energy is a minimum in the position of equilibrium (the Lagrange-Laplace assumption), and the initial displacements and velocities were sufficiently small (but not zero!), then there will of necessity be a limit that no displacement can ever exceed. Otherwise engineers could, by clever constructions, ‘feed’ small oscillations into a suitably designed mechanism generating thereby any amount of oscillatory energy at the terminus of the power train – a conclusion which conflicts with every Conservation Principle known. Whatever it is that prevents the construction of a perpetuum mobile (of the First Type) also prevents extra-systematic disturbances from generating intra-systematic dislocations any greater than the original disturbance. But what then of the theory of small oscillations which allows for progressive enlargements of oscillatory amplitudes?

Laplace was deeply troubled by this. Routh ultimately found a resolution in 1877 (Routh 1877, and compare Heaviside 1902, 529). If the system in question is not dissipative and the roots are unequal, the zeros of the minor of any element in the leading diagonal separate those of the original determinant (Cf. Weierstrass 1858. Cf. also Lamb 1920, 222–226, and Bromwich 1906). If, then, the determinant Δ has a factor (p² + α²)^k, every first minor contains the factor (p² + α²)^k − 1. When we evaluate the contribution from the initial conditions to the operational solution

$$ \left[\mathrm{i}.\mathrm{e}.\kern0.5em {X}_m=\frac{Erm}{\varDelta }\ {a}_{rs}\left({p}^2{u}_s+p{v}_s\right)\right] $$

a factor (p² + α²)^k − 1 will cancel out. We will be left only with a single factor (p² + α²) in the denominator. This will happen with every repeated root; the interpretation will contain only terms of the forms cos αt and sin αt. Thus varying u_s and the v_s will alter ratios of the coefficients of these trigonometric factors for different coordinates; but they will never introduce terms like t cos αt or t sin αt – the trouble makers.

Consider now not a general initial disturbance, but one confined to the period equation. Suppose the equations of motion are these:

$$ {\ddot{x}}_1+f{\ddot{x}}_1+{c}_1{x}_1=b{\dot{x}}_2=0 $$

$$ {\ddot{x}}_2+f{\dot{x}}_2+{c}_2{x}_2+b{\dot{x}}_1=0 $$

(where c₁ < 0, c₂< 0, f ≥ 0, and b is large enough to ensure stability when f = 0)

Assume solutions proportional to e^δt; δ must then satisfy

$$ {\delta}^4+2f{\delta}^3+{\delta}^2\left({c}_1+{c}_2+{b}^2+{f}^2\right)+ f\delta \left({c}_1+{c}_2\right)+{c}_1{c}_2=0, $$

from which it follows at once that

$$ \varSigma \delta =-2f\le 0 $$

and that

$$ \varSigma \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\delta $}\right.\kern0.5em =-f\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${c}_1$}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${c}_2$}\right.\right)\ge 0. $$

The roots for f = 0 may now be chosen as ±in₁, ±in₂ [where n₁ > n₂]. For f > 0 (but very small) let the roots be ±in₁ − α₁, −in₂ − α₂, to order f.Then:

$$ {\alpha}_1+{\alpha}_2=f>0 $$

and

$$ \frac{1}{i{n}_1-{\alpha}_1}+\frac{1}{-i{n}_1-{\alpha}_1}+\frac{1}{i{n}_2-{\alpha}_2}+\frac{1}{i{n}_2-{\alpha}_2}\doteqdot -\frac{2{\alpha}_1}{n_1^2}-\frac{2{\alpha}_2}{n_2^2}>0 $$

Such inequalities are consistent only if α₁ and α₂ have opposite signs: in fact, since n₁ > n₂, α₁ > 0 and α₂ < 0.

Hence any system which is kept stable by gyroscopic action only is such that small friction will always produce instability. The fastest free vibrations will be damped, but the slower ones will increase in amplitude with time (to achieve which the system’s ‘gyroscopic energy’ must be rechanneled into the growing slow oscillations, upsetting thereby the system’s ‘gyroscopic stability’).

If, in the case of our planetary system, we postulate a near-gyroscopic configuration, and neglect friction altogether, then all the roots are purely imaginary: hence the system is stable! If c_rsx_rx_s ≥ 0, in the presence of very little friction ${a}_{rs}{\dot{x}}_r{\dot{s}}_s+{c}_{rs}{x}_r{x}_s$ will decrease and all oscillations will be gradually damped down! Such a system will then be called secularly stable: as with our planetary system, imbalance-oscillations will become periodic over long periods, and their amplitudes will decrease.

But if the quadratic form is not ≥0 and the system is kept stable only by the gyroscopic terms, the slower oscillations must increase in amplitude, leading ultimately to profound changes in the character of the overall motion. This kind of system will then be ordinarily stable, but secularly unstable.

Newton’s worry consisted in having no argument against this latter contingency in the case of our solar system. Laplace’s triumph consisted in his recognition of the form of the problem. To prove the planetary system stable, then, he had to assume c_rsx_rx_s ≥ 0. And he had to postulate but very little friction in the system. If either of these suppositions are challenged, he must then infer that the stability of the solar system is not the result of gyroscopic terms merely – that forces other than gyroscopic determine the stable cohesion of the system.

This kind of argument is methodologically fascinating. In a way Laplace, to the degree that he reasoned thus, has not really proved anything about our planetary system as it exists de facto. Rather, he has deductively unpacked the semantical content of the suppositions: “Grant that the planetary system is quasi-gyroscope, in balanced ‘vortical’ motion around its own freely moving (but physically unoccupied) center of mass; and grant that very little friction is operative in the configuration; and grant that in its trigonometric description c_rsx_rx_s ≥ 0”. From this springboard one easily leaps to the conclusion that all oscillations will gradually be damped down, even those initiated by the sudden introduction into the system of quite violent and dislocating energy spasms from ‘outside’. Any oscillation that does not thus dampen down was too large to begin with!

The reasoning is magnificent: if our system were as supposed, then it would be secularly stable!

But for what reasons ought we to grant such suppositions about our planetary system? If we simply argue that by doing so a proof of the system’s stability follows, the reasoning is not only circular, but viciously so. This would give us no reason for being convinced of systematic stability other than a merely hypothetical kind – and no reason for making our ‘gyroscopic’ assumptions other than that the consequences are reassuring. Newton did as much by leaving it all to God!

No, we need some independent argument either in favor of the system’s quasi-gyroscopic nature, or else in favor of the system’s stability – from either of which we can then infer to the other! But we cannot at once assume both what is to be proved and also that from which the proof is to proceed.

What other reasons are there, therefore, for supposing the initial suppositions about our planetary system to be fulfilled? The answer is clear: we must find independent theoretical data from which it will appear that our planetary system’s gross motion is comparable to something quasi-gyroscopic, e.g., something like 6000 epicyclically-mounted roulette wheels on a Ferris wheel, itself mounted on a railway turntable, which in its turn … etc. But to show the latter really is gyroscopically stable (if it is) one must undertake an immensely complex computation, determining the force components of each roulette wheel, and the nature of the many dynamical systems it forms with other roulette wheels to which it is mechanically linked, and the relationships obtaining between the mass-centers of every such roulette-couple, and then the compositional effects of all these on the gross gyroscopic balance of the Ferris wheel, which in its turn must be … etc. Clearly this is the multi-body problem in gyroscopic-engineering terms. But this is precisely the kind of thing for which Laplace must find a general analytical solution in gravitational-dynamical terms if he is ever to proceed with his further deductions.

To show the planetary system stable in such terms, then, one must first address the multi-body problem. Since a general analytical solution to this problem is not yet indubitably secured (however probable) the ‘gyroscopic’ springboard assumed in any practical demonstration must be pieced together by successive approximation technique – the ultimate result of which can never be a quod erat demonstrandum of the Euclidean type.

Ergo, the proof without which Newton’s synthesis must be adjudged seriously incomplete may not yet have been finally given. One’s convictions as to the stability of our solar system indeed, may not actually be founded on any formal demonstration. Ultimately it may rest on an immense accumulation of inductive considerations. These may be quite sufficient for planetary theory. But we must remark to what degree the logic of the situation, and our expectations withal, have changed.