Lagrangian stability and effective motion

Take a one-coordinate autonomous Lagrangian with positive effective mass and a smooth potential. At a rest equilibrium the potential derivative vanishes. Linearize about that point: the effective mass at equilibrium multiplies the displacement acceleration, and the potential curvature supplies the linear restoring coefficient. Positive curvature gives a positive squared oscillation frequency; negative curvature produces an exponentially growing direction. The derivative of a position-dependent mass does not contribute at linear order about rest because it multiplies products of small quantities.

In this one-coordinate conservative setting, a strict local potential minimum with a regular positive kinetic term gives Lyapunov stability through the conserved energy: sufficiently low nearby energy confines motion to a neighborhood. It is not asymptotic stability. A nearby nonzero-energy orbit generally keeps oscillating rather than approaching rest. Damping can create attraction, but that changes the equations and the energy balance. If curvature vanishes, the quadratic approximation is inconclusive and the first relevant higher-order term must be examined.

Let a vertical circular hoop of radius a rotate about its vertical diameter with angular speed Ω maintained by a motor. Measure the bead angle θ from the bottom. Its kinetic energy includes both motion along the hoop and azimuthal motion imposed by rotation. Eliminating the prescribed azimuthal coordinate gives an autonomous θ Lagrangian when Ω is constant. Its effective potential contains a negative centrifugal term. This is not fixed-angular-momentum Routh reduction: the motor supplies whatever torque is needed to maintain its commanded speed.

The effective θ energy is conserved for constant Ω in this ideal model, but the bead's laboratory mechanical energy need not be. The motor can do work through the moving hoop. At the bottom, the curvature is proportional to g−aΩ². Below the threshold the bottom is stable; above it the bottom is unstable and new off-bottom equilibria satisfy cos θ=g/(aΩ²). Their existence and stability should be checked in the effective problem rather than inferred by borrowing the positive centrifugal barrier from a different fixed-momentum model.

At the threshold Ω²=g/a, the quadratic term at the bottom vanishes. Expanding the stated effective potential gives a positive quartic leading term, so the ideal bottom is still a strict local minimum there, although the small-oscillation frequency is zero and the harmonic approximation fails. The period then depends on amplitude. This example shows why a zero eigenvalue should trigger a higher-order calculation rather than an automatic stable or unstable label. Real friction, finite bead size, and imperfect speed control require additional model terms.

The oscillator lab illustrates the distinction between conservative oscillations and damped attraction after a stable linearization. At damping ratio zero, predict persistent oscillation; at damping ratio one, predict a nonoscillatory return for its fixed initial displacement and zero velocity. Natural frequency is directly controlled, so the lab does not compute the hoop threshold, quartic potential, or unstable inverted dynamics. Its normalized amplitude must be separately scaled to remain a small physical perturbation. Use it to inspect a linear model only where that model is justified.