We obtain the dynamics in number and phase difference, for Bose condensates that tunnel between two wells of a double-well atomic trap, using the (nonlinear) Gross-Pitaevskii equation. The dynamical equations are of the canonical form for the two conjugate variables, and the Hamiltonian corresponds to that of a momentum-shortened pendulum, supporting a richer set of tunneling oscillation modes than for a superconductor Josephson junction, that has a fixed-length pendulum as a mechanical model. Novel modes include ‘inverted pendulum’ oscillations with an average angle of π; and oscillations about a self-maintained population imbalance that we term ‘macroscopic quantum self-trapping’. Other systems with this phase-number nonlinear dynamics include two-component (interconverting) condensates in a single harmonic trap, and He³B superfluids in two containers connected by micropores.