Tadayuki Ohta

A dynamical formalism of singular Lagrangian system with higher derivatives

The singular Lagrangian system with higher derivatives is analyzed with the aid of the Ostrogradski transformation and the Dirac formalism. The formulation of canonical theory is developed so that the equivalence between the Lagrange formalism and the Hamilton one is maintained. As a practical example, the acceleration‐dependent potentials appearing in the Lagrangian of two‐point particles interacting gravitationally are dealt with and the equivalence between the two Hamiltonians that follow from the two Lagrangians which are related by the coordinate transformations is shown. It is also shown, when the constraints are all first class, that a consistent generator of gauge transformation is constructed. Typical examples are given.

Exact Relativistic Two-Body Motion in Lineal Gravity

We consider the N-body problem in (1+1) dimensional lineal gravity. For 2 point masses (N=2) we obtain an exact solution for the relativistic motion. In the equal mass case we obtain an explicit expression for their proper separation as a function of their mutual proper time. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant.

Chaos in a relativistic 3-body self-gravitating system.

We consider the 3-body problem in relativistic lineal [i.e., (1+1)-dimensional] gravity and obtain an exact expression for its Hamiltonian and equations of motion. While general-relativistic effects yield more tightly bound orbits of higher frequency compared to their nonrelativistic counterparts, as energy increases we find in the equal-mass case no evidence for either global chaos or a breakdown from regular to chaotic motion, despite the high degree of nonlinearity in the system. We find numerical evidence for mild chaos and a countably infinite class of nonchaotic orbits, yielding a fractal structure in the outer regions of the Poincaré plot.

Exact solution for relativistic two-body motion in dilaton gravity

We present an exact solution to the problem of the relativistic motion of two point masses in (1 + 1)-dimensional dilaton gravity. The motion of the bodies is governed entirely by their mutual gravitational influence and the spacetime metric is likewise fully determined by their stress - energy. A Newtonian limit exists, and there is a static gravitational potential. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant.

Chaos in an exact relativistic three-body self-gravitating system

We consider the problem of three-body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the three-body Hamiltonian, implicitly determined in terms of the four coordinates and momentum degrees of freedom in the system. Nonrelativistically 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 obtain orbits in both the hexagonal and three-body representations of the system, and plot the Poincaré sections as a function of the relativistic energy parameter eta. We find two broad categories of periodic and quasiperiodic 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 nonlinearity in the relativistic system, we find that the global structure of its phase space remains qualitatively the same as its nonrelativistic counterpart for all values of eta that we could study. However, the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing eta. For the post-Newtonian system we find that it experiences a chaotic transition in the interval 0.21

Exact charged two-body motion and the static balance condition in lineal gravity

We find an exact solution to the charged two-body problem in (1+1)-dimensional lineal gravity which provides the first example of a relativistic system that generalizes the Majumdar-Papapetrou condition for static balance.

Exact solutions of relativistic two-body motion in lineal gravity

We develop the canonical formalism for a system of

Derivation of many‐body potential among charged particles in the S‐matrix method

N

Canonical formalism of action‐at‐a‐distance electrodynamics and many‐particle potential among charged particles

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.

Addendum to a dynamical formalism of singular Lagrangian system with higher derivatives

A general method of deriving a classical potential from the S‐matrix element of particle scattering in the theory of quantized fields is applied to electrodynamics to the post‐post‐Coulombian approximation. To obtain the many‐body potential, a consistent prescription is implemented in subtracting the contributions of the repetition of lower‐order potential from the S‐matrix elements of the higher‐order diagrams. The result shows that the four‐body potential between charged particles has a characteristic feature at a large distance and the two‐body potential is identical with that given in the reduced Hamiltonian of Wheeler–Feynman electrodynamics. The advantage of the S‐matrix method over the canonical formalism is to give the potential directly, without complicated treatment of the interaction with higher derivatives by a method of constrained dynamics.