  A brane-world universe consists of a 4-dimensional brane embedded into a
5-dimensional space-time (bulk). We apply the Arnowitt-Deser-Misner
decomposition to the brane-world, which results in a 3+1+1 break-up of the
bulk. We present the canonical theory of brane cosmology based on this
decomposition. The Hamiltonian equations allow for the study of any physical
phenomena in brane gravity. This method gives new prospects for studying the
initial value problem, stability analysis, brane black holes, cosmological
perturbation theory and canonical quantization in brane-worlds.
