  We describe a technique for solving the combined collisionless Boltzmann and
Poisson equations in a discretised, or lattice, phase space. The time and the
positions and velocities of `particles' take on integer values, and the forces
are rounded to the nearest integer. The equations of motion are symplectic. In
the limit of high resolution, the lattice equations become the usual
integro-differential equations of stellar dynamics. The technique complements
other tools for solving those equations approximately, such as $N$-body
simulation, or techniques based on phase-space grids. Equilibria are found in a
variety of shapes and sizes. They are true equilibria in the sense that they do
not evolve with time, even slowly, unlike existing $N$-body approximations to
stellar systems, which are subject to two-body relaxation. They can also be
`tailor-made' in the sense that the mass distribution is constrained to be
close to some pre-specified function. Their principal limitation is the amount
of memory required to store the lattice, which in practice restricts the
technique to modeling systems with a high degree of symmetry. We also develop a
method for analysing the linear stability of collisionless systems, based on
lattice equilibria as an unperturbed model.
