On the dynamics of a particle on a cone

► We study the classical and quantum free particle and harmonic oscillator on a double cone. ► We analyze the solutions of classical equations and show that the motion in a generator is unstable. ► We identify the observables and Hilbert space for the quantum particle on a double cone. ► The case of the quantization on the double cone is compared with a single-nappe case. ► We solve the Schrödinger equations and discuss them in the context of the classical instabilities.