Chaos and Analytic Non-Integrability of the Classical Double Pendulum: A Computer-Assisted Proof

The planar double pendulum with two equal point masses on two equal massless rods is the standard example of chaos in classical mechanics, but, as far as we could find, neither its chaos nor its non-integrability has been proved at these parameters: the known proofs need a small parameter (a weak coupling, a small mass ratio, a special link geometry), and the one variational criterion that needs none has been applied only under a parameter condition that, in the text available to us, the equal case does not satisfy. We give a computer-assisted proof. At each of the energies E = -1/2, 0 and 1/2, in units in which the lower rest state has E = -3 and E = 0 is the energy of releasing both arms from rest in the horizontal position, the flow on the energy level has a hyperbolic periodic orbit with a transversal homoclinic orbit, and the same holds simultaneously for every energy in [-10^(-10), 10^(-10)]. Consequently every real-analytic function on such a level that is invariant under the flow is constant, and the double pendulum has no real-analytic first integral functionally independent of the energy on any connected open set that contains the level E = 0. At each of these three energies, explicit covering relations between h-sets give a topological horseshoe (a compact invariant set semiconjugate to a subshift of finite type) with explicit bounds; at E = 0: the Poincaré return map has a compact invariant set on which its topological entropy exceeds 0.1016, and the topological entropy of the flow on the level exceeds 0.0138 per unit time. The proofs combine interval arithmetic (the CAPD library), the time-reversal symmetry of the pendulum, cone conditions for the local unstable manifold and covering relations; the programs and their output accompany the paper. Meromorphic non-integrability in the sense of Morales-Ruiz and Ramis remains open.

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