The Beam That Behaves Like a Molecule

Two Hamiltonian systems that share almost nothing in their physical origins turn out to behave identically once their local structure near a 2:1 internal resonance is examined. One system models a pair of flexural modes in a micromechanical resonator through coupled Duffing oscillators. The other models the stretch–bend Fermi resonance of a molecule such as carbon dioxide through a Morse oscillator coupled to a harmonic bend. Both possess an elliptic equilibrium, equal Krein signatures, a frequency ratio close to 2:1, and a cubic resonant coupling. These shared features place them in the same resonant normal-form type, a difference resonance that confines energy exchange rather than driving escape. On the conserved polyad fixed by an initial excitation that places nearly all the energy in the higher-frequency mode, the truncated normal form of any such system collapses to a one-degree-of-freedom problem that depends on only two numbers: a scaled detuning and a scaled anharmonic curvature. Consequently the largest fraction of energy that can be transferred to the lower mode is given by a universal curve whose edges are independent of the curvature. All of the necessary invariants are computed directly from each Hamiltonian before any trajectory is integrated. Numerical integration then confirms the predictions. When the curvature is small the systems follow the one-parameter curve; when it is large they follow the two-parameter curve to high accuracy. A beam whose quartic coefficient is deliberately softened until its curvature matches that of the molecular model reproduces the molecular exchange curve directly, even though the remaining normal-form coefficients differ. An ensemble of sixty randomly drawn members of the same type likewise collapses onto the appropriate universal curve. The residual discrepancy grows roughly linearly with excitation amplitude, as expected from the truncation of the normal form. Two controls mark the boundaries of the result. Reversing one Krein signature converts the same frequency window from bounded exchange into unbounded escape. Weak viscous damping reduces the transferred fraction in the manner predicted by a correspondingly damped reduced model. A quantum treatment of the truncated normal form introduces the polyad number as a third invariant; the lowest Fermi dyad therefore follows a distinct quantum curve rather than the classical one. The findings remain local, confined to two-mode models, and classical except where the quantum analogue is stated explicitly. Within those limits they demonstrate that a dimensionless exchange relation established for one physical system can be transferred to an unrelated system once both have been rescaled by invariants computed from their own Hamiltonians.