Foucault Pendulum-like problems: A tensorial approach

Abstract

The paper offers a comprehensive study of the motion in a central force field with respect to a rotating noninertial reference frame. It is called Foucault Pendulum-like motion and it is a generalization of a classic Theoretical Mechanics problem. A closed form vectorial solution to this famous problem is presented. The vectorial time-explicit solution for the classic Foucault Pendulum problem is obtained as a particular case of the considerations made in the present aproach. Inedite conservation laws for the Foucault Pendulum-like motion are deduced by using simple differential and vectorial computations. They help to visualize the shape of the trajectories. Exact vectorial expressions for the law of motion and the velocity are also offered. The case of the driven Foucault Pendulum is also analyzed, and a closed form solution is deduced based on the general considerations. In the end, an inedite tensorial prime integral for the Foucault Pendulum problem is offered. It helps to reveal in a concise form, within a single entity, all the scalar and vectorial conservation laws for the Foucault Pendulum motion.Two important engineering applications to this approach are presented: the motion of a satellite with respect to a rotating reference frame and the Keplerian relative orbital motion. The latter has a great importance in modeling the problems concerning satellite formation flying, satellite constellations and space terminal rendezvous. The classic problem of the harmonic oscillator in an electromagnetic field is also solved by using the instruments presented in this paper.

Spacecraft Relative Orbital Motion

“…This geometric interpretation shows that the relative orbital motion is in fact a Foucault pendulum like motion (Condurache & Martinusi (2008a)). Excluding the situation h = 0, the case ζ ≤ 0 is equivalent with the Deputy elliptic inertial motion.…”

“…It involves the Lie group of proper orthogonal tensor functions and its associated Lie algebra of skew-symmetric tensor functions. Then, the solution was generalized to the problem of the relative motion in a central force field (Condurache & Martinusi (2007e;2008a;b)). An inedite solution to the Kepler problem by using the algebra of hypercomplex numbers was offered in (Condurache & Martinusi (2007d)).…”

On Six DOF Relative Orbital Motion of Satellites

“…The exact closed form, free of coordinate, solution of the translational motion can be found in [11,12,31,32,34].…”

A Quaternion Solution of the Motion in a Central Force Field Relative to a Rotating Reference Frame

“…Applying the quaternion operator F ω , the Equation (4.1), we will produce the below initial value problem: The solution of Equation (4.5) coresponds to a harmonic planar oscillation (with ω * being the pulsation of the pendulum) composed with a precession of −ω angular velocity of the oscillation plane [5]. In order to compute the closed form solutions of Equation (4.1), we must recall that we've assumed that the direction of the vector ω associated with the quaternion ω is considered to be fixed ( ) ( ) than the Equation (4.6) can be rewritten as following:…”