Abstract: In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimizes the integral of the Lorentz inner product of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.
Abstract: In this work, we successfully extended one-dimensional differential transform method (DTM), by presenting and proving some theorems, to solving nonlinear high-order multi-pantograph equations. This technique provides a sequence of functions which converges to the exact solution of the problem. Some examples are given to demonstrate the validity and applicability of the present method and a comparison is made with existing results.
Abstract: In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a timelike curve. This method uses the coordinate free Maximum Principle of Optimal control and results in the theory of integrable Hamiltonian systems. The presence of several conversed quantities inherent in these Hamiltonian systems aids in the explicit computation of the rigid body motions.