The gravitational n-body problem

For the classical n-body problem in dynamics [2], the Hamiltonian is of potential or separable form:

$$H({\bf q},{\bf p})=T({\bf p}) + V({\bf q})$$

with

$$T({\bf p}) = {{\bf p}.{\bf p}\over 2} \quad , \quad V({\bf q}) = {1\over \sqrt{{\bf q}.{\bf q}}}$$

where ${\bf x}.{\bf x}$ denotes the scalar product of ${\bf x}$.For example, in 2 degrees of freedom,

$$H({\bf q},{\bf p})~= ({p_1}^2~+~{p_2}^2)/2 + 1/{\sqrt{({q_1}^2~+~{q_2}^2)}}$$

where subscripts denote vector components.