Completing Born's finite-precision predictability limit for classical mechanics: arbitrary conjugate-spread products

Max Born argued — first in 1955, and again in a 1959 companion paper in Physikalische Blätter — that classical mechanics is not deterministic in any physically testable sense: with an initial Gaussian phase-space distribution of finite spreads, the spread in the angle variables grows without bound, so prediction becomes impossible after a finite critical time for every motion except harmonic oscillators. His multi-degree-of-freedom derivation in action-angle variables, however, rests on the simplifying assumption that the products of the spreads in each conjugate pair are equal ($\sigma_k\tau_k=\mathrm{const}$), which makes the relevant matrix symmetric and allows decoupling, and it is confined to conservative integrable systems. The objective is to complete Born's predictability-limit analysis by removing the equal-products assumption: treat general initial spreads in multi-degree-of-freedom integrable systems, reducing to Born's critical times in the special cases. The physics of predictability beyond integrable systems — non-integrable, chaotic and dissipative dynamics — is treated as background and is not required here.

Background

Whether classical mechanics is deterministic may look like a purely mathematical question, but Max Born argued in 1955 that it has an empirical side: initial conditions are never measured exactly, so determinism can only mean that predictions made from finitely accurate initial data remain usable for long times. In 'Ist die klassische Mechanik tatsächlich deterministisch?' (Physikalische Blätter 11(2), 49–54, 1955), republished with a 1959 companion paper 'Voraussagbarkeit in der klassischen Mechanik' (Physikalische Blätter 15, 342–349, 1959) and appearing in English as 'Is Classical Mechanics in Fact Deterministic?' in Physics in My Generation (Springer 1969, pp. 78–83), Born gave this idea a quantitative form. The imperfectly known initial state is modeled by a probability distribution in phase space, taken to be Gaussian with spreads $\sigma$ in position and $\tau$ in velocity, and evolved by Liouville's theorem, which states that Hamiltonian flow preserves phase-space volume. A motion is called stable if, for every finite initial phase-space spread, the spread at any later time stays below a time-independent bound that vanishes as the initial spread goes to zero; otherwise it is unstable. For free motion on a circle of circumference $l$ with velocity spread $\tau$, the position spread grows linearly in time and position prediction becomes impossible once the spread reaches the period, giving the critical time $t_c=l/\tau$. Born also motivated his doubt from the kinetic theory of gases: for a molecule rebounding elastically among many others, a very small change in the initial velocity direction eventually changes the path qualitatively — an instability of non-integrable collision dynamics that his quantitative treatment, based on action-angle variables, does not itself cover.

For conservative integrable systems with several degrees of freedom, Born used action-angle variables: canonical coordinates $(\varphi_k,J_k)$, available whenever the system is integrable (possesses as many independent constants of motion as degrees of freedom), in which the actions $J_k$ are conserved and the angles $\varphi_k$ advance linearly, $\varphi_k(t)=\varphi_k(0)+v_k(J)t$, with frequencies $v_k(J)=\partial E/\partial J_k$ depending on the actions through the Hamiltonian $E(J)$. Harmonic oscillators are exceptional because their frequencies do not depend on the actions. For every other integrable system the frequencies change with the actions, so neighboring trajectories shear apart in the angle directions: an initial Gaussian with spreads $\sigma_k$ in $\varphi_k$ and $\tau_k$ in $J_k$ is stretched linearly in the angle directions. To obtain a closed formula, Born assumed that the products $\sigma_k\tau_k$ are the same for every degree of freedom $k$; under this assumption the matrix $M_{kl}=\mu_{kl}\tau_l/\sigma_k$, built from the frequency gradients $\mu_{kl}=\partial^2E/\partial J_k\partial J_l$, becomes symmetric, and an orthogonal transformation decouples the quadratic form in the exponent of the Gaussian into independent one-dimensional problems, yielding the critical time $t_c=1/(m\sigma)$, where $\sigma^2=\sum_k\sigma_k^2$ and $m$ is the smallest eigenvalue of $M$. For the simple pendulum (a non-harmonic oscillator) with amplitude $\vartheta_m$ and relative amplitude accuracy $\gamma$, Born obtained $t_c/T=8/(\vartheta_m^2\gamma)$, where $T$ is the period of the pendulum's oscillation — about 40 hours for typical laboratory values — and concluded that, except for harmonic oscillators, all mechanical motions are unstable in his sense, so classical mechanics is not deterministic in any physically testable sense. The quantum uncertainty relation $\sigma_k\tau_k\geq\hbar/2$ gives the products of conjugate spreads a physical floor and motivated the equal-products idealization.

The 1955/1959 analysis has two restrictions that later work did not remove within Born's own framework. First, the equal-products assumption is a technical convenience rather than a statement about real measurement errors; the case of arbitrary, unequal products $\sigma_k\tau_k$ was not worked out, and within the paper's simultaneous-diagonalization technique it is not obvious how the critical time depends on the individual spreads. This restriction is what the Problem Statement below asks to remove. Second, action-angle variables exist only for conservative integrable systems, so the quantitative analysis says nothing about non-integrable (chaotic) dynamics or dissipative forces. That physics was subsequently answered in a different framework rather than within Born's: the chaos tradition since Lorenz quantifies error growth by Lyapunov exponents (the asymptotic exponential rates at which neighboring trajectories separate), leading to predictability horizons of order one Lyapunov time (the inverse of the leading Lyapunov exponent) multiplied by the logarithm of the ratio of tolerated to initial error; philosophy of physics (Earman's primer on determinism, Svozil, Ben-Yami, Gisin's finite-information program) continues the conceptual debate; and stellar dynamics has computed the linear spreading of general Gaussian error ellipsoids in specific orbital systems. None of these completes Born's finite-precision critical-time analysis itself; the extension of the analysis to non-integrable or dissipative systems is out of scope here.

Problem Statement

The problem is to complete Born's finite-precision predictability-limit analysis of classical mechanics by removing its equal-products restriction.

General conjugate spreads in the integrable multi-dimensional case. For a conservative integrable system with $n\geq2$ degrees of freedom in action-angle variables $(\varphi_k,J_k)$ with Hamiltonian $E(J)$, take an initial Gaussian distribution with spreads $\sigma_k$ in the angles and $\tau_k$ in the actions with arbitrary, unequal products $\sigma_k\tau_k$. Determine the evolution of this distribution under the Hamiltonian flow, in the same linear approximation of the frequency map used by Born ($v_k(J)=\bar{v}_k+\sum_l\mu_{kl}(J_l-\bar{J}_l)$ with $\mu_{kl}=\partial^2E/\partial J_k\partial J_l$), and derive the analogue of Born's critical time: the time after which prediction of the angle variables becomes impossible, i.e. the critical time in the sense of Born's stability definition quoted in the Background — the motion is unstable when, for finite initial spread, the angle spread is not bounded by any time-independent bound that vanishes as the initial spread goes to zero, so the angle value ceases to be usefully predicted — expressed in terms of the individual initial spreads and the frequency gradients.

The general critical time must reduce to Born's known special cases: the one-dimensional results $t_c=l/\tau$ for free motion on a circle of circumference $l$ with velocity spread $\tau$ and $t_c/T=8/(\vartheta_m^2\gamma)$ for the pendulum with amplitude $\vartheta_m$ and relative amplitude accuracy $\gamma$, and the equal-products multi-dimensional result $t_c=1/(m\sigma)$.

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