Order Equations Of Motion Research Articles

Effect of Limited Wordlength on Satellite Prediction Accuracy

Round-off error accumulation in numerical integration of the equations of motion in an inverse square field has been studied for approximately circular orbits. The second order equations of motion in Cowell formulation were assumed to be integrated with a one-step method of arbitrary order after reduction to a first order system. Analytical expressions for the first two statistical moments of the accumulated error in one position component have been derived via the adjoint method for particular classes of fixed and floating point arithmetics. The error mean increases linearily or quadratically with time dependent on the mechanization of the ADD-instruction regardless of number system (sign-magnitude or two’s complement) and method of scaling (fixed or floating point). The error variance increases with third power of time.

THE GROWTH OF CORRUGATIONS ON ACCELERATED FLUID SHEETS

The growth of waves or corrugations on sheets of infinitely conducting fluid accelerated by electromagnetic forces is studied. Two cases are considered: circular cylindrical sheets accelerated by constant axial currents and planar sheets accelerated by currents that are constant or increase linearly with time. Only corrugations that are parallel to the magnetic field lines are considered. For the cylindrical model the theory is valid only provided axial motion is very small. For the planar case initial corrugations of any amplitude can be considered, since it is not necessary to linearize the equations of motion in order to obtain solutions in a closed form. It is shown that the validity of the theory is limited by the axial motion of the fluid sheet and that, for sinusoidal corrugations of very small initial amplitude, the theory remains valid until this amplitude increases to λ/2π, where λ is the wavelength of the corrugation.

Equations of motion in the theory of gravitational perturbations

In the two lowest orders of the fast motion approximation for gravitational perturbations the equations of motion for a system of point particles are derived from the (slightly modified) equations of motion for a perfect compressible fluid by a transition from small drops to pointlike particles. In the first order equations the singular terms arising in the limit l → 0 ( l is the diameter of a drop), which do not cancel out because of the stability condition, are removed by a mass renormalization. Radiation damping terms occur like those in electrodynamics. In the second approximation there are needed two subtractions: one for the mass and one for the coupling constant. The “damping terms” in the second order equation of motion are symmetric under time reversal.

CXXIV. A note on the quantum dynamical principle

Abstract It is shown that commutation relations are derived consistently from the quantum dynamical principle, for systems obeying first order equations of motion.

XVII. Commutation relations in Lagrangian quantum mechanics

Abstract Schwinger's method for deriving commutation relations is shown to fail in the case of first order equations of motion. A new method, adapted from some recent work of Peierls is discussed in connection with two examples of first order equations in point mechanics.

Relativistic commutation rules in the quantum theory of fields (II)

Summary The analysis of the commutation rules between field variables at different times, begun in the first part of this paper 1) , is extended to more general cases, in connection with the Heisenberg-Pauli quantum theory of wave fields 2) . This theory is discussed in connection with the wave equations of the classical field. The theory involves difficulties in the case of first order equations of motion. It is shown that the commutation rules between field quantities at different times can be determined by means of the commutation rules at the same time and the equations of motion, when these equations are linear. The case of second order wave equations is explicitely treated, the commutation rules for the same time being known from the Heisenberg-Pauli theory. The study of the functions involved in the relativistic rules shows that the knowledge of these functions is equivalent to the complete solution of the wave equations, any solution being calculable by integrations involving the initial values.