Eric Guiot
email: guiot.eric_1@yahoo.fr

Trajectory under harmonic potential and magnetic force

Keywords: Central force, magnetic field, harmonic oscillator

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Introduction

The study of the trajectories of particles which move under the influence of restoring forces is an important topic in Physics, considering the many physical applications in almost all the areas. This is particularly true when these forces are attractive, i.e. when the mechanical system is referred to a harmonic motion, and when the particle is simultaneously submitted to the influence of a magnetic field. Naturally, this kind of motion has already been studied in the history of classical Physics, especially when an atom was modelled as a simple harmonic oscillator, in order to explain the Zeeman effect ([1], [START_REF] Kox | The discovery of the electron : II. The Zeeman effect[END_REF]). This paper presents results which seems represent a significant progress in this topic. Indeed, studying in our turn this problem, we have obtained others and original solutions, which contain the solutions already known as limiting cases. In particular, we show that the trajectories of a charged bidimensional harmonic oscillator, under a constant and perpendicular magnetic field, is represented by a part of the centered trochoïde curves. It is well known that this important family of curves, which have been studied since several centuries by generations of mathematicians, are classically build from geometrical considerations, using the motion of two circles. Paper provides thus, that seems for the first time, physical significations to these trajectories. Complete mathematical demonstrations of this result are detailed, as several fundamental characteristics of the motion, such angular momentum and energy. Moreover, limiting cases are presented, naturally ellipse of Lissajous-Bowdith (if magnetic field is null), but also circular trajectories (if restoring fore is null) and rose trajectories (if initial harmonic oscillator is mono dimensional).

In its second part the paper presents another contribution to the topic. Indeed, we study the case where the restoring force is repulsive. Using the same approach, we show that trajectories are described by a

Attractive restoring force : Dynamics

Note : In all the paper the mass of the point particle is considered equal to 1, such forces and accelerations are confused.

We consider thus the motion of a charged point particle which is simultaneously submitted to an attractive restoring force and a magnetic force due to a constant magnetic field 𝐵 perpendicular to the plane on which a charged particle moves. The force at distance we propose to study can be written

𝐹 ⃗ = -𝑘𝑟 ⃗ + 𝑞𝑉 ⃗⃗ * 𝐵 ⃗⃗
(1) Where 𝑘 is the spring constant 𝑞 the charge of the particle 𝑉 its speed.

A limiting case is obtained when the magnetic field is null: this case is corresponding on the harmonic oscillator and the point particle describes an ellipse of Lissajous -Bowditch.

Solving

We begin introducing the cartesian system of coordinate (𝑂; 𝑋 ⃗ ; 𝑌 ⃗⃗ ; 𝑍 ⃗ ) where 𝑂 is the center of the restoring force. We can rewrite (1) as

𝐹 ⃗ = -𝑘(𝑋𝑥 ⃗ + 𝑌𝑦 ⃗) + +𝑞𝐵(𝑋 ̇𝑦 ⃗ -𝑌 ̇𝑥 ⃗)
For convenience for the calculation we introduce the pulsation of the initial harmonic oscillator and the Larmor frequency (per mass unity) [START_REF] Hand | Analytical Mechanics[END_REF]

{ 𝑤 0 = √𝑘 𝛼 = 𝑞𝐵 2 }
System becomes using the Newton law of motion

{ 𝑋 ̈= -𝑤 0 2 𝑋 -2𝛼𝑌 Ẏ̈= -𝑤 0 2 𝑌 + 2𝛼𝑋 ̇}
A classical method to solve this kind of system is to introduce the complex number (comparable solving can be found for example in references [1], [START_REF] Malherbe | L'observatoire de Paris, MOOC (Astro)Physique I : Électromagnétisme, Oscillateur harmonique en présence de champ magnétique et effet Zeeman MOOC[END_REF]) 𝑢 = 𝑋 + 𝑗𝑌 We obtain the differential equation

𝑑 2 𝑢 𝑑𝑡 2 + 𝑤 0 2 𝑢 -2𝑗𝛼 𝑑𝑢 𝑑𝑡 = 0
To solve it we consider a limiting case whose we know an exact solution, i.e. the ellipse of Lissajous-Bowditch we have evoked, corresponding on 𝛼 = 0. We name 𝑎 and 𝑏 respectivelly the semi major and minor axis of this conic. By choosing initial conditions such

{ 𝑡 = 0 𝑋 = 𝑎 𝑌 = 0 }
The solution is given by the relations { 𝑋 = 𝑎 cos 𝑤𝑡 𝑌 = ±𝑏 sin 𝑤𝑡 } (Dependent on the direction of rotation). In this case the complex number 𝑢 is 𝑢 = 𝑎 cos 𝑤𝑡 ± 𝑗𝑏 sin 𝑤𝑡 Or,

{ 𝑢 = 𝑎 + 𝑏 2 𝑒 𝑗𝑤𝑡 + 𝑎 -𝑏 2 𝑒 -𝑗𝑤𝑡 𝑢 = 𝑎 -𝑏 2 𝑒 𝑗𝑤𝑡 + 𝑎 + 𝑏 2 𝑒 -𝑗𝑤𝑡 }
We add now a magnetic fields perpendicular to the plane of this ellipse. We choose to investigate solutions of the form

𝑢 = 𝑎 0 𝑒 𝑗(𝑤+𝛼)𝑡
Introducing it in the differential equation we obtain

-(𝑤 + 𝛼) 2 + 𝑤 0 2 + 2𝛼(𝑤 + 𝛼) = 0 Consequently 𝑤 0 2 = 𝑤 2 -𝛼 2
Rewriting now our equation

𝑑 2 𝑢 𝑑𝑡 2 + (𝑤 2 -𝛼 2 )𝑢 -2𝑖𝛼 𝑑𝑢 𝑑𝑡 = 0
We notice that a second solution exists

𝑢 = 𝑏 0 𝑒 𝑗(-𝑤+𝛼)𝑡
Where 𝑏 0 is our second constant length. Whereas these results we investigate thus a general solution such 𝑢 = 𝑎 0 𝑒 𝑗(𝑤+𝛼)𝑡 + 𝑏 0 𝑒 𝑗(-𝑤+𝛼)𝑡 To determine the constants we study the initial conditions : considering the initial ellipse, i.e. for 𝛼 = 0 , and considering the origin of the time we obtain for 𝑤𝑡 = 0 𝑎 = 𝑎 0 + 𝑏 0

And for 𝑤𝑡 = 𝜋 2

, we obtain two possibilities depending on the direction of rotation

{ 𝑏 = 𝑎 0 -𝑏 0 -𝑏 = 𝑎 0 -𝑏 0 }
Where 𝑎 and 𝑏 are successively the semi major and minor axis of the conic. We deduce that

{ 𝑎 0 = 𝑎 + 𝑏 2 𝑏 0 = 𝑎 -𝑏 2 } Or { 𝑎 0 = 𝑎 -𝑏 2 𝑏 0 = 𝑎 + 𝑏 2 }
We obtain thus two general solutions

{ 𝑢 = 𝑎 + 𝑏 2 𝑒 𝑗(+𝑤+𝛼)𝑡 + 𝑎 -𝑏 2 𝑒 𝑗(-𝑤+𝛼)𝑡 𝑢 = 𝑎 -𝑏 2 𝑒 𝑗(+𝑤+𝛼)𝑡 + 𝑎 + 𝑏 2 𝑒 𝑗(-𝑤+𝛼)𝑡 }
We recognize the expressions of the centered trochoïde curves. as presented in reference [4] under the mathematical form

𝑢 = 𝑟 1 𝑒 𝑗𝑤 1 𝑡 + 𝑟 2 𝑒 𝑗𝑤 2 𝑡
And, by identifying the real and imaginary parts

{ 𝑋 = 𝑎 + 𝑏 2 cos(𝑤 + 𝛼)𝑡 + 𝑎 -𝑏 2 cos(𝑤 -𝛼)𝑡 𝑌 = 𝑎 + 𝑏 2 sin(𝑤 + 𝛼)𝑡 - 𝑎 -𝑏 2 sin(𝑤 -𝛼)𝑡 } (2.
𝑤 0 2 = 𝑘 = 𝑤 2 -𝛼 2 > 0
Or, introducing the ratio of pulsations

𝑛 = 𝛼 𝑤 If -1 < 𝑛 < 1

Verification

At this step we verify the solutions by direct checking: considering for example the first of them and the successive derivatives with respect to the time

{ 𝑢 1 = 𝑟 1 𝑒 𝑗𝑤 1 𝑡 + 𝑟 2 𝑒 𝑗𝑤 2 𝑡 𝑢 2 = 𝑟 2 𝑒 𝑗𝑤 1 𝑡 + 𝑟 1 𝑒 𝑗𝑤 2 𝑡 }
Geometrical properties of these centered trochoïde curves has already been studied by generations of Mathematicians and Physicians. It appears the construction of the curves is classicaly done using geometrical considerations : it is a roulette traced by a point attached to a circle rolling around a second fixed circle. We have provided a physical signification, valuable for -1 < 𝑛 < 1, consequently when

𝑤 2 < 0
This indicates that trajectories are corresponding on the hypotrochoïde curves [6]. The second part of the family (epitrochoïdes) are not corresponding on a real solution (𝑤 0 should be imaginary).

Several possible trajectories

To do it we, choose the solution (2.1). Using the ratio 𝑛 they are given by { 𝑋 = 𝑎 cos 𝑤𝑡 cos 𝑛𝑤𝑡 -𝑏 sin 𝑤𝑡 sin 𝑛𝑤𝑡 𝑌 = 𝑎 cos 𝑤𝑡 sin 𝑛𝑤𝑡 + 𝑏 sin 𝑤𝑡 cos 𝑛𝑤𝑡 }

These equations are corresponding on a precession of the initial ellipse of Lissajous-Bowdith with an angle of precession equal to 𝛼 . This precession is visible when 𝑛 is small (see for example Figure 1). Hypotrochoïdes curves can thus be obtained making a precession of Larmor on an ellipse of Hooke. It is illustrated in Figure 1 Corresponding now the cases where the ratio 𝑛 is proximate to 1. 𝑏 = 0.8 𝑛 = -0.9 𝑛 = 0.9

Trajectories becomes gradually circular, as we detail it later.

Energy and angular momentum

For the reason that magnetic force doesn't work Mechanical Energy 𝐸 (sum of the kinetic energy 𝑇and the harmonic potential 𝑉) is conserved on all the trajectories

𝐸 = 𝑇 + 𝑉 = 1 2 (𝑋 ̇2 + 𝑌 ̇2 + 𝑘(𝑋 2 + 𝑌 2 )) = 1 2 𝑤(𝑤𝑏 2 + 2𝑎𝑏𝛼 + 𝑎 2 𝑤)
Angular momentum is given at O by the relation

𝑝 ⃗= 𝑟 ⃗ * 𝑉 ⃗⃗

And can be written in our system of coordinate

𝑝 ⃗= (𝑋𝑥 ⃗ + 𝑌𝑦 ⃗) * (𝑋 ̇𝑥 ⃗ + 𝑌 ̇𝑦 ⃗) = (𝑋𝑌 ̇-𝑌𝑋 ̇)𝑧 ⃗

Using previously relations we obtain after simplifications

𝑝 ⃗ = (𝑎𝑏𝑤 + 𝛼𝑎 2 [1 -𝑒 2 𝑠𝑖𝑛 2 𝑤𝑡])𝑧 ⃗

Radius of curvature

Radius of curvature is naturally an important property of the curves. This one, for plane parametrized curves depending on the time is given by In fact it corresponds to a well-known situation. Indeed, in these cases the coefficient 𝑘 given by

𝑅(𝑡) = [𝑥̇2 + 𝑦̇2]
𝑘 = (𝑤 2 -𝛼 2 )
Becomes equal to zero : This indicates that the harmonic force is null and, consequently, point particle is only submitted to the constant magnetic field. Trajectories are thus circular and we re-obtain for each cases the classical relation

𝑉 = 𝑞𝐵𝑅

Moreover these two circles are the circles used to draw the curves.

Polar equation

It is often difficult to write a system of parametric equations in a polar system of coordinate (𝑂, 𝑒 𝑟 ⃗⃗⃗⃗, 𝑒 𝜃 ⃗⃗⃗⃗⃗). In our case we managed to do it only for certain limiting cases, writing the classical equations

{ 𝑋 = 𝑅 cos 𝜃 𝑌 = 𝑅 sin 𝜃 }
Where 𝑅 is given by

𝑅 = √𝑋 2 + 𝑌 2 = √𝑎 2 𝑐𝑜𝑠 2 𝑤𝑡 + 𝑏 2 𝑠𝑖𝑛 2 𝑤𝑡
Indeed, we can obtain the angular speed using the relation

𝜃 ̇= - ( 𝑋 𝑅 ) Ẏ 𝑅 Which leads to 𝜃 ̇= (𝑎 2 𝑐𝑜𝑠 2 𝑤𝑡 + 𝑏 2 𝑠𝑖𝑛 2 𝑤𝑡)𝛼 + 𝑤𝑎𝑏) 𝑎 2 𝑐𝑜𝑠 2 𝑤𝑡 + 𝑏 2 𝑠𝑖𝑛 2 𝑤𝑡
We have noted note that the angular speed is constant and given by 𝜃 ̇= 𝛼 if 𝑤 = 0, 𝑎 = 0 or 𝑏 = 0

Which allows to obtain in these cases the polar angle

𝜃 = 𝛼𝑡

Considering for example the case where 𝑏 = 0. The curves becomes

{ 𝑋 = 𝑎 cos 𝑤𝑡 cos 𝑛𝑤𝑡 𝑌 = 𝑎 cos 𝑤𝑡 sin 𝑛𝑤𝑡 }

And can be written in the polar system of coordinate

{ 𝑋 = 𝑎 cos 𝜃 𝑛 cos 𝜃 𝑌 = 𝑎 cos 𝜃 𝑛 sin 𝜃 }
Which is corresponding on a well known familly of curves (limiting cases of the centered trochoïde), called in the history of mathematics as « Rose Curves » since they have been studied by the mathematician Guido Grandi ( [START_REF] John | Mac Tutor History of mathematics archive[END_REF], [START_REF] Rosace | [END_REF]). To illustrate the paper we draw the case 𝑛 = 1/3 (« regular trifolium »)

𝑛 = 1/3
This indicates that when the initial oscillator is moved in a straight (i.e. when it is a one-dimensional oscillator) the point particle (after addition of the magnetic field) describes one of these curves, where the angular speed is constant and simply given by 𝜃 ̇= 𝛼. The speed of the point particle along these lines are thus

𝑉 = 𝑎𝛼 𝑛 √ 1 + (𝑛 2 -1)𝑐𝑜𝑠 2 𝜃 𝑛
We list the extrema in the following table:

𝜃 = 2𝑘𝑛𝜋 𝑉 𝑚𝑖𝑛 = 𝑎𝛼 𝜃 = 𝑛𝜋 ( 2𝑘 + 1 2 ) 𝑉 𝑚𝑎𝑥 = 𝑎𝛼 𝑛 Tab 2.

Speed on the rose curves

Speed is well at the most when the point particle reaches the origin of the restoring force.

Repulsive restoring force

We consider in this second part the case where the restoring force is repulsive. In this case the differential equation we have to solve is We investigate a form such

𝑢 = 𝑎 0 𝑒 (𝑤+𝑗𝛼)𝑡
Introducing it in the differential equation we obtain

(𝑤 + 𝑗𝛼) 2 -𝑘 -2𝑗𝛼(𝑤 + 𝑗𝛼) = 0 So 𝑤 2 -𝛼 2 + 2𝑗𝑤𝛼 -𝑘 -2𝑗𝑤𝛼 + 2𝛼 2 = 0
Consequently relation is this time 𝑘 = 𝑤 2 + 𝛼 2 Rewriting our initial equation

𝑑 2 𝑢 𝑑𝑡 2 -(𝑤 2 + 𝛼 2 ) -2𝑗𝛼 𝑑𝑢 𝑑𝑡 = 0
We note that a second solution is possible

𝑢 = 𝑏 0 𝑒 (-𝑤+𝑗𝛼)𝑡
Indeed we obtain well We see that the two expressions are in correct agreement as much the magnetic field is weak. Indeed classical explaination of the effect provides that "The circular motion of the oscillating electron in the xy-plane at angular frequencies ω0 + ΩL and ω0 -ΩL produces radiation at these frequencies » [1] (ΩL is the frequency of Larmor). We note we obtain well the same gap between the two frequencies. However we also note a difference about the central position, which isn't in our solution 𝑤 0 but 𝑤. This difference is thus

∆= 𝑤 0 -√𝑤 0 2 + 𝛼 2
Naturally this difference is small in the case of the Normal Zeeman effect, because the magnetic fields is weak. But, when it increases (case of the Paschen-Back effect) the difference could also increase, which constitutes a prediction for the model.

Central force is repulsive

The trajectories we have obtained seems this time entirely original (except the limiting case) and can be considered as a contribution to classical electrodynamics, but not also. Indeed, it appears the study of this kind of spirals trajectories present a modern interest in other fields of the modern Physics, for example to study the motion of spacecraft. For this reason, furthers works should be devoted to a mathematical study of them. Moreover, we attract the attention of the reader on the fact that the solution we have obtained shouldn't be confused with the Poinsot-Cotes spirals [START_REF]Spirale de la tige en rotation[END_REF].

Conclusion

We have presented two families of plane parametrized curves which are obtained by performing a precession around the center of an initial ellipse or hyperbola. We have showed that a part of these curves, known as hypotrochoïde curves, can describe the trajectory of a point-particle simultaneously submitted to a harmonic force and a force perpendicular to its speed. Besides their mathematical and physical interest, these results could have applications in some areas such, for example, the classical electrodynamic, or others. We suggest a prediction about the normal Zeeman and the Paschen-Back effects. Research perspectives could be to verify this prediction and to extend the study, for example to a three-dimensional harmonic oscillator. Another point should to consider the consequences of quantum effects.

Figure 1 :

 1 Figure 1 : 𝑛 = 0.05

𝑎𝑏𝑤 3 +Figure 5 :

 35 Figure 5 : 𝑛 = ±1

  𝐹 ⃗ = 𝑘𝑟 ⃗ + 𝑞𝑉 ⃗⃗ * 𝐵 ⃗⃗ As previously we rewrite the force 𝐹 ⃗ = 𝑘(𝑋𝑥 ⃗ + 𝑌𝑦 ⃗) + 2𝛼(𝑋 ̇𝑦 ⃗ -𝑌 ̇𝑥 ⃗) Finally we obtain the system of differential equations { 𝑋 ̈= 𝑘𝑋 -2𝛼𝑌 Ẏ̈= 𝑘𝑌 + 2𝛼𝑋 ̇} Introducing again the complex number 𝑢 = 𝑋 + 𝑗𝑌 We obtain this time the differential equation

𝑤 2 -

 2 𝛼 2 -2𝑗𝑤𝛼 -𝑘 + 2𝑗𝑤𝛼 + 2𝛼 2 = 0 Consequently we suggest a general solution on the form 𝑢 = ( 𝑎 + 𝑗𝑏 2 )𝑒 (𝑤+𝑗𝛼)𝑡 + ( 𝑎 -𝑗𝑏 2 )𝑒 (-𝑤+𝑗𝛼)𝑡 Such the initial condition (if 𝛼 = 0) is verified. This general solution, identifying real and imaginary parts, leads to the relations { 𝑋 = 𝑎 cosh 𝑤𝑡 cos 𝛼𝑡 -𝑏 sinh 𝑤𝑡 sin 𝛼𝑡 𝑌 = 𝑎 cosh 𝑤𝑡 sin 𝛼𝑡 + 𝑏 sinh 𝑤𝑡 cos 𝛼𝑡 } This family of curves seems this time original. We present the simulation, for values of 𝛼 variant de 𝛼 ∈ [0,2], and for 𝑎 = 1, 𝑏 = 0.8, 𝑘 = 1. Considering an initial trajectory of the point particle We add a magnetic field perpendicular to its osculating plane. When magnitude is weak we note an inflexion of the trajectory, depending on the direction of the magnetic field 𝑛 = -0.1 𝑛 = +0.1 When magnitude of magnetic field increases the spiral form of the curves appears { 𝑋 = 𝑎 cos(𝑤 0 -𝛼)𝑡 𝑌 = 𝑎 sin(𝑤 0 -𝛼)𝑡 }

  {𝑋 ̇= -𝑎(𝑤 sin 𝑤𝑡 cos 𝛼𝑡 + α cos 𝑤𝑡 sin 𝛼𝑡) -𝑏(𝑤 cos 𝑤𝑡 sin 𝛼𝑡 + 𝛼 sin 𝑤𝑡 cos 𝛼𝑡) 𝑌 ̇= -𝑎(𝑤 sin 𝑤𝑡 sin 𝛼𝑡 -𝛼 cos 𝑤𝑡 cos 𝛼𝑡) + 𝑏(𝑤 cos 𝑤𝑡 cos 𝛼𝑡 -𝛼 sin 𝑤𝑡 sin 𝛼𝑡) }

	And
	𝑋 ̈= 𝑎(-𝑤 2 cos 𝑤𝑡 cos 𝛼𝑡 + 2𝑤𝛼 sin 𝑤𝑡 sin 𝛼𝑡 -𝛼 2 cos 𝑤𝑡 cos 𝛼𝑡)
		+𝑏(𝑤 2 sin 𝑤𝑡 sin 𝛼𝑡 -2𝑤𝛼 cos 𝑤𝑡 cos 𝛼𝑡 + 𝛼 2 sin 𝑤𝑡 sin 𝛼𝑡)
	𝑌 ̈= -𝑎(𝑤 2 cos 𝑤𝑡 sin 𝛼𝑡 + 2𝑤𝛼 sin 𝑤𝑡 cos 𝛼𝑡 + 𝛼 2 cos 𝑤𝑡 sin 𝛼𝑡)
	{	-𝑏(𝑤 2 sin 𝑤𝑡 cos 𝛼𝑡 + 2𝑏𝑤𝛼 cos 𝑤𝑡 sin 𝛼𝑡 + 𝛼 2 sin 𝑤𝑡 cos 𝛼𝑡) }
	A check by direct proof leads well to
			{	𝑋 ̈= -(𝑤 2 -𝛼 2 )𝑋 -2𝛼𝑌 Ẏ̈= -(𝑤 2 -𝛼 2 )𝑌 + 2𝛼𝑋 ̇}
	Where
					𝑘 = 𝑤 2 -𝛼 2
	2. Presentation of the curves
	2.1. Definition
	Curves are thus definite with
		{ 𝑢 2 = 𝑢 1 =	𝑎 + 𝑏 2 𝑎 -𝑏 2	𝑒 𝑗(+𝑤+𝛼)𝑡 + 𝑒 𝑗(+𝑤+𝛼)𝑡 +	𝑎 -𝑏 2 2 𝑎 + 𝑏	𝑒 𝑗(-𝑤+𝛼)𝑡 𝑒 𝑗(-𝑤+𝛼)𝑡	}
	Rewritting them

𝑛 = 1 𝑛 = 2

-Polar equation and limiting cases : rotating rod spirals

Using the same method than in 2.5., with 𝑅 = √𝑋 2 + 𝑌 2 = √𝑎 2 𝑐𝑜𝑠ℎ 2 𝑤𝑡 -𝑏 2 (1 -𝑐𝑜𝑠ℎ 2 𝑤𝑡)

We obtain this time 𝜃 ̇= (𝑎 2 𝑐𝑜𝑠ℎ 2 𝑤𝑡 -𝑏 2 (1 -𝑐𝑜𝑠ℎ 2 𝑤𝑡)𝛼 + 𝑤𝑎𝑏) 𝑎 2 𝑐𝑜𝑠ℎ 2 𝑤𝑡 -𝑏 2 (1 -𝑐𝑜𝑠ℎ 2 𝑤𝑡)

We note that 𝜃 ̇ is constant and equal to 𝛼 for the three cases : if 𝑤 = 0, 𝑎 = 0 or 𝑏 = 0. Consequently it exists two possible trajectories which can be written in the polar system of coordinate

It appears these curves are already known under the name of « rotating rod spirals». Classicaly they describes the « trajectory of a massive ring sliding without friction on a horizontal rod in rotation around a center » [START_REF]Spirale de la tige en rotation[END_REF]. They constitute thus a limiting case of the solutions.

Discussion

Central force is attractive

The solutions of the problem, as centered trochoïde curves, seem haven't been noticed until today. Indeed, the solutions historically due to Lorentz, to explain the Normal Zeeman effect, are different, as we present it. Demonstrations of solving equation (1) can be found in reference [1].

It appears these solutions have been obtained considering that the angle between the direction of the constant magnetic field and the plane of the initial trajectory is random, which is not the case of our work. But fundamental difference is they have been obtained considering approximations, which isn't the case here, for example assuming that 𝛼 ≪ 𝑤 0 [1] or that the point particle describes initially a circular orbit around the center of harmonic force [START_REF] Malherbe | L'observatoire de Paris, MOOC (Astro)Physique I : Électromagnétisme, Oscillateur harmonique en présence de champ magnétique et effet Zeeman MOOC[END_REF].

The solutions presented in the paper are different because we have chosen different conditions:

-Initial trajectory of the point particle (before the addition of the magnetic field) is a conic and not necessary a circle. -Angle between the magnetic field and the osculating plane of the trajectory is right.

-We didn't do approximation on the ratio 𝛼/𝑤 0 Consequently, the family of curve is different but can be compared when initial trajectory is circular, i.e. when the two lengths 𝑎 and 𝑏 are equals, and when the orientation of the magnetic field is perpendicular. Indeed, our two possible trajectories are in this case given by