The evolution of the Milky Way disk, which contains most of the stars in the Galaxy, is affected by several phenomena. For example, the bar and the spiral arms of the Milky Way induce radial migration of stars(1) and can trap or scatter stars close to orbital resonances(2). External perturbations from satellite galaxies can also have a role, causing dynamical heating of the Galaxy(3), ring-like structures in the disk(4) and correlations between different components of the stellar velocity(5). These perturbations can also cause 'phase wrapping' signatures in the disk(6-9), such as arched velocity structures in the motions of stars in the Galactic plane. Some manifestations of these dynamical processes have already been detected, including kinematic substructure in samples of nearby stars(10-12), density asymmetries and velocities across the Galactic disk that differ from the axisymmetric and equilibrium expectations(13), especially in the vertical direction(11,14-16), and signatures of incomplete phase mixing in the disk(7,)(1)(2,)(1)(7,)(18). Here we report an analysis of the motions of six million stars in the Milky Way disk. We show that the phase-space distribution contains different substructures with various morphologies, such as snail shells and ridges, when spatial and velocity coordinates are combined. We infer that the disk must have been perturbed between 300 million and 900 million years ago, consistent with estimates of the previous pericentric passage of the Sagittarius dwarf galaxy. Our findings show that the Galactic disk is dynamically young and that modelling it as time-independent and axisymmetric is incorrect.