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The paper studies the motion of the Foucault Pendulum in a rotating non-inertial reference frame and provides a closed form vector solution determined by vector and matrix calculus. The solution is determined through vector and matrix calculus in both cases, for both forms of the law of motion (for the Foucault Pendulum Problem and its “Reduced Form”). A complex vector which transforms the motion equation in a first order differential equation with constant coefficients is used. Also, a novel kinematic interpretation of the Foucault Pendulum motion is given.

Swinging with elegance across the meridian of Paris inside the grand hall of the observatory, the pendulum built by Bernard Léon Foucault (1819-1868) proved the rotation of the Earth for the first time by terrestrial methods. It was a true kick for both mathematicians and physicists because none of them could write the equations or imagine this simple experiment. As we now know, Cauchy never thought that is possible that a pendulum can change the oscillation plan and Poisson said in 1827 that a pendulum cannot move such way.

The “non-mathematician” Foucault, as the members of the French Academy named him, wrote the first equation which computes the period of the whole rotation of the oscillation plan depending of the latitude of the place of oscillation. The as-known “Foucault formula” or “The law of sinus” is

The famous experience done by Léon Foucault in 1851 emphasized the movement of the Earth around the poles, without the need for astronomical observations. The problem is very important out of the theoretical point of view. Modeling this experiment involves the study of a harmonic oscillator with respect to a non-inertial frame of reference with uniform rotation.

But finding the equation of the movement of the pendulum proved to be for mathematicians a really “hard nut” due to the non-inertial character of the reference frame. Long time, the solution had been obtained after many approximations which had to simplify the differential equations.

The type of motion that will be named “Foucault Pendulum-like motion” is described by the non-linear initial value problem [

where

The motion which is described by the below linear initial value problem will be named “Foucault Pendulum motion”:

In this case, the function

Many times is used the simplified form of (1.2) written below, when the inertial centripetal force is ignored (see for example [

where

The present paper presents a closed form vector solution which exploits the benefits of the dualism of vector calculus and matrix calculus with extension to tensors. It is structured in five sections described below.

In the second section, two theorems which put the basis of the correspondence between vector operations and their matrix representation are stated. Two symbolic representations are defined which creates the two ways of the cross-representations of equations in vector and matrix forms.

The third section presents the vector solution of the Foucault pendulum problem (1.2) using the two symbolic representations. Here a workaround is used through a complex vector which transforms (1.2) in a first order differential equation with constant coefficients.

Section 4 prepares the next one because it presents the tensor method of representation of vector functions which will be very useful when we will find the vector solution of the Reduced Foucault Pendulum Problem (1.3). Therefore, the transformation

Finally, in Section 5, we will compute the solution of (1.3) and we will be able to extract the surprising conclusion that the solution of the Reduced Foucault Pendulum Problem is less simple than the solution of the whole Cauchy problem (1.2).

Many times, the solution to the Cauchy problem (1.3) is given only for the planar case, using polar coordinates [

Consider the vector space

Consider

A function

is an isomorphism of vectors spaces.

If

is an endomorphism of

If

is an endomorphism on

We want to find the link between the vector

So, if

then:

Using (2.6) and (2.7) we have:

Therefore, with the notation:

from (2.8) results the relation:

with

Theorem 2.1. The function

So, it is an exact symbolic representation, with respect to the endomorphism (2.4) of vector space

Note: The matrix

The characteristic polynomial of the skew-symmetric matrix (2.9) is:

Solutions (roots) of the equation

Using one of the known proceedings for determination of an exponential matrix, it follows that:

Theorem 2.2. If

where

Note: due to the Cayley-Hamilton theorem, any square matrix verifies her characteristic equation; consequently, from Equation (2.12), it follows that:

If we denote by

If we denote by

The mathematical model of this experiment is given by the Cauchy problem:

In Equation (3.1),

Using the symbolic representation, we will find a vector exact solution for the problem (3.1). Applying to the problem (3.1) the correspondence

We will consider now the column matrix with complex functions elements given by:

First, we will differentiate this column matrix:

Replacing

After developing, (3.5) becomes

Grouping the terms, (3.6) becomes:

and this means that

We will note

It results that the function

and the solution of the problem (3.10) is:

Using Equation (3.3), and knowing that _{1} and z_{2} complex numbers, it follows that the solution of the problem (3.2) is the real part of Equation (3.11):

Using Theorem 2.2 and the definition of a matrix exponential, it follows:

with (3.3) and (3.13), Equation (3.12) becomes:

After restructuring, Equation (3.14) looks like:

with the notation:

Equation (3.15) becomes:

Applying to Equation (3.17), the correspondence

where:

After elementary transformation, the solution (3.18) of the Cauchy problem (3.1) it will be written:

Note:

1) The function (3.19) is the solution of the Cauchy problem:

The differential equation of the problem (3.21) can be found from the differential equation of the Cauchy problem (3.1) for

2) The solution (3.20) is the vector form of an equation matrix (3.17). This has special significance.

Let

The function

Because from Equation (3.22), we have

The angular velocity corresponding to this rotation is:

The transformation (3.17) is therefore an own rotation with angular velocity

with the above observation, we can obtain the next theorem:

Theorem 3.1. The solution of the Cauchy problem

will be obtained applying the tensor of the rotation operator with the angular velocity

to the solution of the next Cauchy problem:

Note: The hodograph of the solution of the problem (3.21)

The solution of the problem (3.1) can be viewed by the rotation of the plane of the ellipse, with the angular velocity

The tensor relation (3.25) suggests a direction to approach the symbolic representation of a vector function of real variable which will be developed in the next paragraph.

This section describes the tensor method of representation the vector functions which will be used in the next chapter when we will give the solution to the Reduced Foucault Pendulum Problem.

We will denote by

Let

Let

The problem (4.1) has a unique solution

Indeed, be a tensor function of a real variable:

Using Equation (4.1), it follows that

The solution of the problem (4.3) is unique and because the identity tensor

Let

The unique solution of Equation (4.1) will be further named as “the rotation tensor corresponding to the angular velocity

If

where

If the vector function

where

In this condition, the solution of the Cauchy problem (4.1) will be written in explicit form:

Using one of the known procedures to determine the exponential matrix, we will have:

where:

and

or:

Using the relation

where

If

Theorem 4.1.

If

1)

2)

3)

4)

5)

6)

7)

Proof:

1)

2)

Equation (4.15) can be obtained directly from Equation (4.14).

The proper tensor

From (4.16) using (4.15) it follows:

The matrix form of the tensor relation (4.17) is:

From Equation (4.18) it follows that the matrix

3) We will prove the matrix form of Equation (3). Let

Using Equation (4.1), Equation (4.20) will be written:

Using Equation (4.18) we will have:

The corresponding tensor of Equation (4.22) is:

4) We will apply twice Equation (4.23):

5) The transformed (4.5) being a proper rotation is also an isometry, so:

6) In matrix notation we have

Equation (4.26) can be also written:

To compute

7) Being an orthogonal transformation,

From (4.1), by transposition, it follows:

Knowing that

From Equation (4.19), by transposition and considering

From Equations (4.30) and (4.31) we have:

Therefore

Note: The transformation

This transformation “gives an algebraic form” to a class of vector differential equations that model the motion of mechanical systems in non-inertial frames, whom are in the motion of non-uniform rotation, on fixed direction, also the motion with respect to the inertial frames in the fields of gyroscopic forces.

The motion of the Foucault Pendulum is described by the following non-linear initial value problem:

If the force field is elastic, the type

We will use the present method in order to resolve the reduced form of the problem (5.2):

The mathematical model of the Foucault pendulum is presented of the type (5.3) in the theoretical mechanics [

1) If

The solution of the problem (5.4):

verifies the initial conditions of (5.3) and has the property:

2) If

Assuming

Therefore:

Applying the transformation

The solutions of (5.9) with the property

The solution of the problem (5.10) is:

In the hypothesis

Considering the fact that

Now, let be the Cauchy problems:

In accordance with those shown in the points a) and b) the solutions of the problems (5.14) and (5.15) are:

where:

respectively:

where:

with

The relations

Theorem 5.1:

The solution of Cauchy problem:

is given by the vector function:

where

Note: Also, the problem (5.21) shapes the movement of the vibration for a class of gyroscopic instruments. Even in the case of planar motion, the literature shows only the approximate solutions assuming

The work presents the closed form vector solution for the well-known Foucault pendulum problem. Both forms of the Foucault problems (the whole form and the as known “reduced form” when the centripetal force is neglected) are considered. The last one models the movement of the harmonic oscillator in uniform magnetic field, also. Therefore, a specific isomorphism between the free vectors map and the column matrix map is used. The short solution of the Foucault pendulum problem is obtained using vectors as column matrix of complex numbers adequate defined. With this method, the second order Cauchy vector problem which describes the spatial movement of the Foucault pendulum becomes a first order differential matrix equation with constant coefficients. The closed form vector solution obtained in this way allows a suggestive kinematic representation of the spatial movement of the Foucault pendulum. The closed form vector solution for Foucault pendulum problem is obtained by means of a time dependent tensor operator which reduces this problem to only two classic problems very easy to be solved. The tensor operator as introduced can extend the study of all Foucault type movements in the case of non-inertial reference frame with time dependent angular velocity.