  We present a collective coordinate approach to describe coupled phase
oscillators. We apply the method to study synchronisation in a Kuramoto model.
In our approach an N-dimensional Kuramoto model is reduced to an n-dimensional
ordinary differential equation with n<<N, constituting an immense reduction in
complexity. The onset of both local and global synchronisation is reproduced to
good numerical accuracy, and we are able to describe both soft and hard
transitions. By introducing 2 collective coordinates the approach is able to
describe the interaction of two partially synchronised clusters in the case of
bimodally distributed native frequencies. Furthermore, our approach allows us
to accurately describe finite size scalings of the critical coupling strength.
We corroborate our analytical results by comparing with numerical simulations
of the Kuramoto model with all-to-all coupling networks for several
distributions of the native frequencies.
