The Hooke-Newton transmutation system of magnetic force

Recently, it was reported that the Hooke-Newton transmutation of magnetic force can be generated by a conformal mapping in a uniform magnetic field. We perform the classical analysis of the transmutation system in this paper. First, the action variables are calculated and the energy is expressed in terms of the variables. The quantum spectrum of the system is manifested by the condition of the angular variables for the closed trajectories. Second, the trajectory equations are presented for the charge in the transmuted Coulomb field whose characteristics, attractive or repulsive, are determined by the signatures of both the angular momentum and the charge. It is shown that, given the released energy E and angular momentum pφ of the charged particle, the types of trajectories can be classified by the critical energy associated with the magnetic field strength E C = ∣ ω c / p φ ∣ ℏ 2 / 4 . The feature makes the system only trap the particles with energy E < EC and the definite signatures of angular momentum and charge.


Introduction
It was pointed out by Issac Newton that there exists duality between Hooke's linear force law and the inverse square force law (Cor. III, Prop, VII, [1]). Newton revealed this duality by showing a transmutation between these two kinds of forces when given a planet orbit, and, amazingly, proved it only with the skills of elementary geometry. In modern terminology, the transmutation is established by a quadric conformal mapping (see, e.g., appendix I in [2,3], and section 6 in [4]). Recently, it was shown that there exists a magnetic force version of the Hooke-Newton (H-N) transmutation for a charged quantum particle in a uniform magnetic field [5]. This is more or less unexpected since the magnetic force is quite different from the central force field, e.g., doing no work, and unlike the fields of the Newtonian gravitational attraction and of electrostatic interaction generated by a massive object and a charge, which are the sources of the inverse square force fields, there is no magnetic monopole discovered so far. Although the magnetic version of the H-N transmutation is due to a conformal image, it is possible to realize the system by reinterpreting the conformal structure as a physical effect by transformation design means [6][7][8]. In reference [5], the quantum mechanics of the magnetic transmutation system have been discussed. The purpose of this article is to carry out the classical analysis for the system.
The paper is arranged as follows: in section 2, the physical realization of the mechanical conformal image is formulated. In section 3, the H-N transmutation of a charge in the uniform magnetic field is investigated by way of the action and angular variables. The energy of the closed trajectory is expressed in terms of the action variables. The quantum spectrum of the system is then identified after the Bohr's quantization rules have been adopted for the action variables. It follows from the action representation of energy that the angular variables are calculated, from which the relation between the allowed quantum bound states and the conditions of the angular variables for the closed trajectories are established. Section 4 is used to obtain the trajectory equations of a charge in the transmutation system. It is shown that the types of trajectories can be controlled by the strength of the magnetic field. Finally, the conclusion is provided in section 5.

Form-invariant Hamilton-Jacobi equation and mechanical conformal mapping
The purpose of this section is to show that a form-invariant Hamilton-Jacobi (H-J) equation can be generated by a kind of conformal mapping, and one can endow the conformal factor in the equation with the meaning of a potential field such that it is possible to carry out the transformed system by physical means. The evolution of a charge moving in a 2D space is governed by the solution of the H-J equation where S=S(q) is the reduced action (p. 149, [9]). It may be regarded as a function of the coordinates. The partial derivatives of the action with respect to the coordinates are equal to the corresponding generalized momenta: where the new coordinate variables =¯( ) q q q i i i are only the function of q i , and the metric coefficientsḡ ii are defined by obtained by replacing the variable q i withq i for the inverse of the covariant metric components g ii (q) calculated by the standard definition Here, let us introduce the metric coefficients for the new coordinate system {¯} q i to figure out the transformation leading to the invariant H-J equation. It is easy to find the relation between the metric coefficients g ii (q) for the systems { } q i and (¯) This makes (7) turn into where the determinants = | | G G ii and = |¯| g g ii . The simultaneous equations determine the definite form of the transformation. Now equation (10) becomes The invariant squared distance between two neighboring points of the transformed space is depicted by One interesting outcome which could emerge from the H-J equation is that the motion of the particle in the conformal space can be obtained with a physical effect by moving the factorḡ G to the right hand side of the equation, which yields where E is the released energy of the particle. This reinterpretation turns the geometric effect of the transformation into a real mechanical effect. It allows us to reveal and construct novel systems through conformal mapping. Equation (14) is effective for higher dimensional space as long as the conformal conditions =Ḡ g G g ii ii are solvable. In the coming sections, the formulation is applied to investigate the H-N transmutation in the uniform magnetic field.

Action variables and the energy of the Hooke-Newton transmutation system in the uniform magnetic field
The action variables of the transmutation system are calculated in this section so as to obtain the system energy.
To generate the transmutation, let us evaluate the transformation functions with respect to the cylindrical coordinate system (ρ, j). According to the definitions ofḡ ii and G ii in (9), we have r r , and diag 1 , Substituting the transformation into the components in (16) and a short calculation give the conformal factor rr j r So we obtain the invariant distance of transformed space from (13), and the transformed H-J equation from (14). It turns out that the geometric effect is now achieved physically by the potential U. It is not difficult to generate the potential field since it corresponds to a repulsive linear force which can be produced by the electric field E=F/q=−∇U/q. Now let us turn our attention to consider the influence of a uniform magnetic field on the conformal image of a charge's motion. For the magnetic field directed along the Z-axis, =Be B z , the H-J equation is obtained by replacing p j /ρ with the minimal coupling r - qA since the magnetic field is calculated through = B A , and one can choose the vector potential The replacement guarantees the conservation of the total momentum of the charge and field. The motion of the charge q is now governed by the H-J equation with A j =Bρ /2. The angular variable j is obviously a cyclic coordinate. So the generalized momentum This is the angular momentum of the system. The corresponding action variable to the momentum is It follows from (23) that the momentum p ρ can be evaluated by We note that this is an action variable for an inverse square force generated by the Coulomb field lw r j ( )p 8 c . Therefore, the mapping in (18) does indeed generate the H-N transmutation of the magnetic force in the uniform magnetic field. It is remarkable to note that the force which is attractive or repulsive is determined by both the signatures of charge q in ω c and the angular momentum p j . Given a kind of charged particle in the Coulomb field, a right or left turn will determine the particles with closed or unclosed trajectories. To perform the integration of I ρ , let us put .
Since  r 0, it implies b0. So the signature of q has to be the same as that of the angular momentum p j , and energy satisfies The conditions of the signature and the inequality actually stand for the attractive force and closed trajectory conditions as we shall see later. With the integral formula (p. 102, [10]), The radial action variable is found to be Solve the equation for the energy. It is expressed in terms of action variables by Action variables are adiabatic invariant. Let us choose the Bohr's quantization conditions for the actions Here we remember that = ¹ j j I p 0. Otherwise, the attractive force no longer exists. The energy turns into the discrete spectrum This is exactly the quantum spectrum of the transmutation system (See equation (46) in [5]). Equations (34) and (37) show that the existence of ω c , and thus the magnetic field, is the crucial point of the closed trajectories and bound states. With the representation in (34 ), the angular variables can be calculated, yielding where C is the prefactor in formula (34), and It is a rational number, and the condition of the closed trajectories. This exhibits the corresponding relation between the bound states in quantum mechanics and the closed trajectories in classical mechanics. Before finishing the discussion of the section, let us determine the constant λ. The best choice is the characteristic length of the system 1 . The expression is particularly useful when we would like to compare the interaction strengths of the magnetic H-N system and the electric Coulomb system, as we shall see in the notes of the conclusion. Furthermore, the choice also makes the wave function normalized to unity [5]. To obtain the expression, it is noted that the characteristic length of the hydrogen atom system with the electric Coulomb interaction r q e . This completes the proof of (42). Let us estimate the order of λ when the magnetic field B=1 Tesla. According to quantum mechanics, the angular momentum is always quantized by The substitution of the cyclotron frequency w = »qB M 1.

Trajectory equations for the Hooke-Newton transmutation system of magnetic force
The trajectory is defined by the equation in the H-J formulation of mechanics (p. 149, [9]) Equation (23) shows that the differential form of the H-J equation is Since S=S(ρ ) is a function of ρ, and r r l r = ( )dS d dS d 2 , solving the above equation for the factor r dS d gives r l w l w The reduced action is then given by ò l w l w Using the representation, we get ò ò for short. To go further, let us define where we have used (42) to get to the second equality. The action (50) may be for the attractive or repulsive force depending on the coupling constant (λω c p j )/8>0 or <0. We shall first consider the attractive case. = E 0, and 3. > E 0, according to the strength of the magnetic field when the released energy and angular momentum of the charge are given.

Trajectory equation for <
E 0. In this case the strength of the magnetic field is strong enough such that The first integral in (51) can be manipulated to yield where the minus (plus) sign is for the negative (positive) charged particle. To reflect the cyclic characteristic of j for a general value of e, let us take the cosine operation with respect to both sides of (65 ), yielding where The top sign is for the positive charge. It is possible to obtain an alternative representation of the trajectory equation without resorting to the inverse function. In order to prove the statement, we need the known equalities (p. 58, [10]) and  -= -+ -- It is easy to verify that the trajectory equation can be expressed in terms of The top sign is for the positive charge. Figure 1 shows the patterns of the trajectories with the value β=2, i.e. = e 3 2, where L=1 was chosen. The left pattern is for the negative charge corresponding to the minus sign in (66) or the plus sign in (71). The patterns can be obtained either through (66) or by (71). Figure 2 shows the trajectories for several different values of e. They exhibit simpler designs when the charge is moving in the stronger magnetic fields, corresponding to the smaller values of e. Each of them only spans a smaller region of the trajectory plane. When the charge is moving in the weaker magnetic fields, the larger values of e, the trajectories  It is easy to see that the trajectories of the equation are elliptic with two different tilting angles for a given n.
The condition is equivalent to e=1. As calculated above, the first integral in (51) can be performed, and gives  Accordingly, the equation of the trajectory is The plus is for the negative charged particle. The patterns in the first row of figure 4 exhibit the trajectory, where L=1 was chosen. It is a kind of uniform spiral. The patterns in the second row are for the positive charged particle.
The equivalent condition to this is e>1. The first integral in (51) in this case is As above, the plus is for the negative charge. The patterns in the first (second) row of figure 5 show two trajectories of the negative (positive) charged particle produced from the equation. They are a non-uniform spiral.  With l »´-| | m 8.12 10 2 μm from (46 ), the electric field which generates the corresponding effective potential is given by The corresponding electric field for the potential U is » -´r ( ) ( ) e E 3 10 N C . 93 6 The strength is about 10 times the electric field in a photocopier and is easy to achieve. Equations (90) and (92) show that the electric and magnetic fields of the magnetic transmutation system can select and trap electrons with extremely low energy and a definite signature of angular momentum. This function is obviously effective for arbitrary charged particles. It thus offers a novel means for the choosing and trapping of the low energy charged particles in the uniform magnetic field.

Repulsive force
In accordance with equation (50), the force generated by the Coulomb field of the transmutation system would be repulsive when the charge q and momentum p j have different signatures, i.e., a lw = < The top (bottom) pattern in figure 6 is a trajectory of the negative (positive) charged particle generated from the equation with the plus (minus) sign. It is a non-uniform spiral like the previous cases of > E 0 and α>0. However, it escapes more quickly from the region of the electromagnetic field.

Conclusion
This article performs the classical analysis for the Hooke-Newton transmutation in the uniform magnetic field. The first part is devoted to discussing the physical realization of mechanical conformal mapping of the orthogonal systems based on the form-invariant Hamilton-Jacobi equation. Then, it is applied to calculate the action and angular variables of the transmutation system since they are the easiest way to manifest the coherence of energy between classical and quantum mechanics. The second part of the article is used to evaluate the different kinds of trajectory equations for the motion of a charge in the system. It is shown that the types of trajectories for the attractive field are classified by the magnetic field parameter  w j | | p 4 c 2 . Although the transmutation system is created by the quadratic conformal mapping, it is possible to actualize the system in physics by reinterpreting the conformal structure as an action of the force field. Several notes are worth making as follows: (i) The interaction strength of the transmuted Coulomb field is weak, but adjustable. To appreciate