Metastable Balancing Oscillators

The problem of an oscillator performing oscillations at the top of an oscillating potential barrier is solved exactly. The conditions under which the motion is finite, quasi-finite (metastable), or infinite are determined. Examples are presented from mechanics (a Kapitsa pendulum on a solid support that oscillates in the horizontal direction), plasma physics (Bernstein–Green–Kruskal waves), and microwave electronics (metastable electrons near an oscillating virtual cathode).