We develop the formalism for canonical reduction of (1 + 1)-dimensional gravity coupled with a set of point particles by eliminating constraints and imposing coordinate conditions. The formalism itself is quite analogous to the (3 + 1)-dimensional case; however in (1 + 1) dimensions an auxiliary scalar field is shown to have an important role. The reduced Hamiltonian is expressed as a form of spatial integral of the second derivative of the scalar field. Since in (1 + 1) dimensions there exists no dynamical degree of freedom of the gravitational field (i.e. the transverse-traceless part of the metric tensor is zero), the reduced Hamiltonian is completely determined in terms of the particles' canonical variables (coordinates and momenta). The explicit form of the Hamiltonian is calculated both in post-linear and post-Newtonian approximations.

We consider the problem of three body motion for a relativistic one-dimensional selfgravitating system. After describing the canonical decomposition of the action, we find an exact expression for the 3-body Hamiltonian, implicitly determined in terms of the four coordinate and momentum degrees of freedom in the system. Non-relativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtaining orbits in both the hexagonal and 3-body representations of the system, and plot the Poincare sections as a function of the relativistic energy parameter η. We find two broad categories of periodic and quasi-periodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase-space between these two types. Despite the high degree of non-linearity in the relativistic system, we find that the the global structure of its phase space remains qualitatively the same as its non-relativisitic counterpart for all values of η that we could study. However 1 email: fburnell@physics.ubc.ca 2 email: jjmaleck@uwaterloo.ca 3 email: mann@avatar.uwaterloo.ca 4 email: t-oo1@ipc.miyakyo-u.ac.jp the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing η. For the post-Newtonian system we find that it experiences a KAM breakdown for η ≃ 0.26: above which the near integrable regions degenerate into chaos.

We present the exact solution of two-body motion in (1+1) dimensional dilaton gravity by solving the constraint equations in the canonical formalism. The determining equation of the Hamiltonian is derived in a transcendental form and the Hamiltonian is expressed for the system of two identical particles in terms of the Lambert W function. The W function has two real branches which join smoothly onto each other and the Hamiltonian on the principal branch reduces to the Newtonian limit for small coupling constant. On the other branch the Hamiltonian yields a new set of motions which can not be understood as relativistically correcting the Newtonian motion. The explicit trajectory in the phase space (r, p) is illustrated for various values of the energy. The analysis is extended to the case of unequal masses. The full expression of metric tensor is given and the consistency between the solution of the metric and the equations of motion is rigorously proved.1

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