  We develop the canonical formalism for a system of $N$ bodies in lineal
gravity and obtain exact solutions to the equations of motion for N=2. The
determining equation of the Hamiltonian is derived in the form of a
transcendental equation, which leads to the exact Hamiltonian to infinite order
of the gravitational coupling constant. In the equal mass case explicit
expressions of the trajectories of the particles are given as the functions of
the proper time, which show characteristic features of the motion depending on
the strength of gravity (mass) and the magnitude and sign of the cosmological
constant. As expected, we find that a positive cosmological constant has a
repulsive effect on the motion, while a negative one has an attractive effect.
However, some surprising features emerge that are absent for vanishing
cosmological constant. For a certain range of the negative cosmological
constant the motion shows a double maximum behavior as a combined result of an
induced momentum-dependent cosmological potential and the gravitational
attraction between the particles. For a positive cosmological constant, not
only bounded motions but also unbounded ones are realized. The change of the
metric along the movement of the particles is also exactly derived.
