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The paper presents a quaternion approach of giving a closed form solution of the motion in a central force field relative to a rotating reference frame. This new method involves two quaternion operators: the first one transforms the motion from a non-inertial reference frame to a inertial one with a very significant consequence of vanishing all the non-inertial terms (Coriolis and centripetal forces); the second quaternion operator provides the solution of the motion in the noninertial reference frame by applying it to the solution in the inertial reference frame. This process will govern the inverse transformation of the motion and is proved on two particular cases, the Foucault Pendulum and Keplerian motions problems relative to rotating reference frames.

The present paper presents a quaternion solution of the motion in a central force field relative to a rotating ref- erence frame. It starts from the main Cauchy problem stated below:

where

The quaternion method which will be presented in this paper involves two quaternion operators from which the first one transforms the non-linear with variable coefficients initial value problem (1.1) in another one without the coefficients and the second quaternion operator, applied to the solution of the last problem, will provide the time-explicit closed form solutions for two specific cases, Foucault Pendulum and Keplerian motion problem when

The structure of this paper consists of the following four parts. Section 2 starts with a brief presentation of the quaternion algebra and continues with the presentation of Darboux problem in quaternion form in order to prepare the defining of the quaternion operators.

The next section represents the core of the paper because there the quaternion operators are defined, but not before the transformation in the quaternion form of the Equation (1.1) to be done.

Section 4 proves the accuracy of the method of using quaternion operators for computing the time-explicit closed form solutions for two particular cases, the Foucault Pendulum and Keplerian motions problems in rotating reference frame.

The quaternions were invented by William Rowan Hamilton in 1843 [

where

For the quaternion

where

The set of quaternions is denoted by

with

We already know that an algebra is a vector space where the product may be defined as an additional internal operation. Also, the dimension of an algebra is the algebraic dimension of the vector space. We will define a division algebra as an algebra where the division operation is possible. So, for any

We will denote with

when

We can describe the motion of a particle on a sphere with a constant radius with the help of time-depending quaternions such as:

where

where

It is well known that in rigid body kinematics, we need to describe the instantaneous rotation when we know the angular velocity [

If R is the rotation matrix, the rotation with angular velocity

If a vector

and if the matrix

the instantaneous angular velocity vector

The rotation matrix that models the rotation with a given instantaneous angular velocity

where

The rotation matrix

Consider

From Equation (2.14), it results that:

and using vector quaternions property (2.7) we will rewrite (2.18) as

Due to the fact that

where

Using (2.15) and the expression of

In order to find the solutions of the equations specific to the motions in a central force field relative to a rotating reference frame, two reciprocal transformations will be done: first, the motion in the non-inertial reference frame will be transformed in a inertial one through the quaternion operator

In this section, a quaternion operator

Knowing that

and further,

Now, the following quaternion operator

where

If

1. For any quaternions

2. For any quaternions

3. For any quaternion

4. If

5. If

6.

where

Theorem 3.1.

The solution of the Cauchy problem:

will be obtained by applying the quaternion operator

Proof. If we apply

Using the Equation (3.10), it results that:

Replacing

Consequently, by using the quaternion operator

In the next sections will be studied two particular cases of motions in central force field: the Foucault Pendulum and the Kepler’s motions relative to a rotating reference frame problems.

This section presents the methods adequate to the very known two topics: the Foucault Pendulum and Keplerian motion problems relative to a rotating reference frame problems. In order to achieve the goal of this paper, the motion in central force field Equation (1.1) will be particularized for these two specific cases giving for each of them the characteristic eqaution of

The Foucault Pendulum motion is described by the below initial value problem which is a particular form of the Equation (1.1) that coresponds to a spatial harmonic oscillator relative to a rotating reference frame, with

where

Applying the quaternion operator

The Equation (4.2) models the spatial harmonic oscillator and it’s solution is:

Due to the Theorem 3.1., the solution of the initial value problem

results from applying the the quaternion operator

The solution of Equation (4.5) coresponds to a harmonic planar oscillation (with

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

with

If we’ll note:

than the Equation (4.6) can be rewritten as following:

In conclusion, when the direction of the vector

The Keplerian motion in a rotating reference frame that rotates with the angular velocity

following linear initial value problem which is a particular form of the Equation (3.1) with

where

It was proved in the second section that the solution to the Cauchy problem is obtained by applying the quaternion operator

The Equation (4.10) describes a typical Keplerian motionunder certain conditions.

In the particular case of negative specific energy, the solution of (4.11) is: [

where

In the Equation (4.11), the coefficients

where the specific energy is noted with

and the specific angular momentum of the inertial trajectory is noted with

The eccentricity of the trajectory is given by:

with

and the mean motion is:

where

The function

with

Now, in order to find the solution to the Cuchy problem ( 4.21), the quaternion operator

Using the properties of the quaternion operator

Again, the direction of the vector

with

Consequenly, similar to the Foucault pendulum case, the Keplerian motion relative to a rotating reference frame consists of two motions: a Keplerian elliptical motion described by the Equation (4.11) and a rotation with the angular velocity

The quaternion method described in this work presents a new perspective to the clasical problem of motion in central force field relative to the rotating reference frames and provides us a very powerfull tool to solve the similar problems. Throughout the paper, two quaternion operators are defined in order to reveal the closed form solution to the two particular problems of the Foucault Pendulum and Keplerian motions in rotating reference frame.