# Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators

## Abstract

**:**

_{N+1}and SU

_{N}, and this does not depend on the functional form of the Hamiltonian. In particular, all classical spherically symmetric potentials have algebraic symmetries, namely O

_{4}and SU

_{3}; they possess an additional vector integral of the motion, while the quantal counterpart-operator does not exist. This offers possibilities that are absent in quantum mechanics.

## 1. Introduction

## 2. Overview of the General Formalism and of Its Application to Hydrogenic Atoms

_{n}and velocities v

_{n}= dq

_{n}/dt, where n = 1, 2, ..., N. From the Lagrangian of the system

_{n}, v

_{n})

_{n}= ∂L/∂v

_{n}

_{n}, v

_{n}, p

_{n}can be varied by small amounts δq

_{n}, δv

_{n}, δp

_{n}, respectively. The latter small quantities are of the order of ε and the variation should be worked to the accuracy of ε. As a result of the variation, the set of Equation (2) would not be satisfied any more. This is because their right side would differ from the corresponding left side by a quantity of the order of ε.

_{n}are not independent functions of velocities, it is possible to exclude velocities v

_{n}from the set of Equation (2) and obtain one or several weak equations

_{m}(q, p) ≅ 0, (m = 1, 2, ..., M)

_{m}(q, p) from Equation (4) with coefficients depending on q and p.

_{g}= H(q, p) + u

_{m}φ

_{m}(q, p)

_{g}and the conventional Hamiltonian H (q, p). Quantities u

_{m}are coefficients to be determined.”

_{g}≅ H(q, p) would be only a weak equation - in distinction to Equation (5).

_{m}[f, φ

_{m}]

_{m’}in Equation (6) instead of f and taking into account the set of Equation (4), one obtains:

_{m’}, H] + u

_{m}[φ

_{m’}, φ

_{m}] ≅ 0. (m’ = 1, 2, ..., M)

_{m}.”

_{m}(q, p), m = 1, 2, ..., M. They wrote the generalized Hamiltonian in the form (see Equations (4) and (5)):

_{g}= H(q, p) + u

_{m}{A

_{m}(q, p) − A

_{0m}}, A

_{0m}= const.

_{0m}is the value of A

_{m}(q, p) in a particular state of the motion, so that in this state

_{m}(q, p) − A

_{0m}≅ 0.

_{m}(q, p) are integrals of the motion, their Poisson bracket with H(q, p) vanishes and the consistency condition (7) reduces to the form

_{m}[A

_{m’}, A

_{m}] ≅ 0. (m’ = 1, 2, ..., M).

_{m}.

**L**=

**r**□

**p**and the Runge-Lenz vector (see, e.g., [16])

**A**(

**r**,

**p**) = {

**r**p

^{2}−

**p**(

**r·p**)}/(μZe

^{2}) −

**r**/r, where μ is the reduced mass. Therefore, Oks and Uzer [8] presented the generalized Hamiltonian in the form:

_{g}= p

^{2}/(2μ) − Ze

^{2}/r +

**u×**(

**r**□

**p − L**

_{0}) +

**w×**(

**A**(

**r**,

**p**) −

**A**

_{0}).

**L**

_{0},

**A**

_{0}, and the energy H

_{0}are connected by the well-known relation [16]:

_{0}

^{2}= μZ

^{2}e

^{4}(A

_{0}

^{2}− 1)/(2H

_{0}).

**r**□

**p**, H

_{g}] ≅ 0, [

**A**(

**r**,

**p**), H

_{g}] ≅ 0 resulted into the following equations for the unknown vector-coefficients

**u**and

**w**:

**u**□

**L**

_{0}+

**w**□

**A**

_{0}≅ 0,

**u**□

**A**

_{0}− 2

**w**□

**A**

_{0}H

_{0}/(μZ

^{2}e

^{4}) ≅ 0.

_{0}in the particular state of the atom. Oks and Uzer [8] showed that in terms of B(H

_{0}), the generalized Hamiltonian and the equations of the motion take the following form:

_{g}= p

^{2}/(2μ) − Ze

^{2}/r + 2B(H

_{0})H

_{0}{

**M**

_{0}·(

**r**□

**p**)/M

_{0}

^{2}− (1 −

**A**

_{0}

**·A**(

**r**,

**p**))/(1 − A

_{0}

^{2})}

**r**/dt = {1 + B(H

_{0})}

**p**/μ

**p**/dt = − {1 + B(H

_{0})}Ze

^{2}

**r**/r

^{3}

_{0}, A

_{0})}. Therefore, the transformation of the time

_{0})}t

_{g}and the generalized frequency ω

_{g}differ from their conventional values T

_{0}and ω

_{0}as follows:

_{g}= T

_{0}/|1 + B(H

_{0})|

_{g}= ω

_{0}|1 + B(H

_{0})| = |1 + B(H

_{0})||2H

_{0}|

^{3/2}/D

^{1/2}, D ≅ μZ

^{2}e

^{4}

_{0}has been used).”

_{g}= 0 despite H

_{0}≠ 0 (the conventional formalism allows to be ω

_{0}= |2H

_{0}|

^{3/2}/D

^{1/2}= 0 only for H

_{0}= 0). This is a state (or states) where B(H

_{0}) = −1. Therefore, such state or states would not emit the electromagnetic radiation, would not lose energy for the radiation, and would thus constitute stable states of the classical atom.”

**r**/dt = d

**p**/dt = 0, so that

**r**(t) =

**r**

_{0}and

**p**(t) =

**p**

_{0}, where

**r**

_{0}and

**p**

_{0}are some vector constants. Thus, the electron is motionless, but its momentum differs from zero. This is not surprising: the momentum

**p**is a more complex physical quantity than the velocity

**v**≡ d

**r**/dt. For example, for a charge in an electromagnetic field characterized by a vector-potential

**A**, it is also possible to have

**v**= [

**p**− e

**A**/(mc)]/m = 0 while

**p**= e

**A**/(mc) ≠ 0 [17].

_{g}in Equation (18) vanishes and so does the radiation.”

## 3. New Results

**M**, the conservation of the latter following from the geometrical (spherical) symmetry of this system. It is well-known that the SHO also possesses another set of conserved quantities, whose conservation is the consequence of the higher-than-geometric (algebraic) symmetry:

_{mn}= p

_{m}p

_{n}/μ + kx

_{m}x

_{n}, m = 1, 2, 3, n = 1, 2, 3

_{m}and x

_{m}are the Cartesian components of the momentum

**p**and of the radius-vector

**r**, respectively; μ is the mass of the SHO. Obviously, I

_{nm}= I

_{mn}, so that there are generally only six independent conserved quantities I

_{mn}. The unperturbed Hamiltonian H can be actually expressed via some of the conserved quantities from Equation (19) as follows:

_{11}+ I

_{22}+ I

_{33})/2

_{3}-axis (the z-axis) perpendicular to the orbital plane. Then, the dynamical variables are x

_{1}, p

_{1}, x

_{2}, p

_{2}.

_{g}, which differs from H only by the addition of the constraints corresponding to the conserved quantities responsible for the algebraic symmetry, i.e., the conserved quantities from Equation (19), but only those of them that are relevant to the motion in the orbital plane:

_{g}= (I

_{11}+ I

_{22})/2 + B

_{11}(E) (I

_{11}− I

_{11,0}) + B

_{22}(E) (I

_{22}− I

_{22,0}) + B

_{12}(E) (I

_{12}− I

_{12,0})

_{mn,0}are the values of these conserved quantities in the particular state of the system; E is the energy of the system in a particular state of the motion.

_{mn}“commute” with each other: the Poisson bracket of any two of them vanishes. Therefore, the consistency conditions from Equation (10) in this case reduce to equating to zero the Poisson brackets of the components of the angular momentum

**M**with the second term in the right side of Equation (21):

_{i}, a

_{mn}I

_{mn}] = [e

_{ijq}x

_{j}p

_{q}, B

_{mn}(p

_{m}p

_{n}/μ + kx

_{m}x

_{n})] = 0

_{ijq}is the Levi-Civita symbol.

_{mn}by I

_{mn,0}(as required by the GHD), lead to the following equations:

_{22}I

_{12,0}= B

_{12}I

_{22,0}

_{12}I

_{11,0}= B

_{11}I

_{12,0}

_{11}and I

_{22}are non-negatively defined.

_{11,0}differs from zero, i.e., I

_{11,0}> 0. Then, from Equations (23) and (24), it is easy to obtain

_{12}= B

_{11}I

_{12,0}/I

_{11,0}, B

_{22}= B

_{11}I

_{22,0}/I

_{11,0}

_{g}from three to one, so that H

_{g}can be represented in the form:

_{g}= (I

_{11}+ I

_{22})/2 + B

_{11}(E) {(I

_{11}− I

_{11,0}) + I

_{22,0}/I

_{11,0}(I

_{22}− I

_{22,0}) + I

_{12,0}/I

_{11,0}(I

_{12}− I

_{12,0})}

_{g}from Equation (26) and using dx

_{i}/dt = ∂H

_{g}/∂p

_{i}, dp

_{i}/dt = −∂H

_{g}/∂x

_{i}, we find the following equations of motion:

_{1}/dt = {(1+2B

_{11})p

_{1}+ (B

_{11}I

_{12,0}/I

_{11,0})p

_{2}, dx

_{2}/dt = (1+2 B

_{11}I

_{22,0}/I

_{11,0})p

_{2}+ (B

_{11}I

_{12,0}/I

_{11,0})p

_{1}}/μ

_{1}/dt = −k{(1+2B

_{11})x

_{1}+ (B

_{11}I

_{12,0}/I

_{11,0})x

_{2}, dp

_{2}/dt = (1+2 B

_{11}I

_{22,0}/I

_{11,0})p

_{2}+ (B

_{11}I

_{12,0}/I

_{11,0})x

_{1}}

^{2}x

_{1}/dt

^{2}= −ω

_{0}

^{2}{[(1+2B

_{11})

^{2}+ (B

_{11}I

_{12,0}/I

_{11,0})

^{2}]x

_{1}+ 2(B

_{11}I

_{12,0}/I

_{11,0})[1 + B

_{11}(1+ I

_{22,0}/I

_{11,0})]x

_{2}},

^{2}x

_{2}/dt

^{2}= −ω

_{0}

^{2}{[(1+2B

_{11}I

_{22,0}/I

_{11,0})

^{2}+ (B

_{11}I

_{12,0}/I

_{11,0})

^{2}]x

_{2}+ 2(B

_{11}I

_{12,0}/I

_{11,0})[1 + B

_{11}(1+I

_{22,0}/I

_{11,0})]x

_{2}}

_{0}= (k/μ)

^{1/2}

_{1}= exp(iω

_{g}t), x

_{2}= α exp(iω

_{g}t), α = const

_{g}is the (yet unknown) generalized frequency of the oscillator.

_{1}and x

_{2}from Equation (31) into the first equation in Formula (29), we obtain:

^{2}/ω

_{0}

^{2}= (1+2B

_{11})

^{2}+ (B

_{11}I

_{12,0}/I

_{11,0})

^{2}+ 2α(B

_{11}I

_{12,0}/I

_{11,0})[1 + 2B

_{11}(1 + I

_{22,0}/I

_{11,0})]

_{1}and x

_{2}from Equation (31) into the second equation in Formula (29), we obtain:

_{g}

^{2}/ω

_{0}

^{2}= (1+2B

_{11}I

_{22,0}/I

_{11,0})

^{2}+ (B

_{11}I

_{12,0}/I

_{11,0})

^{2}+ (2/α)(B

_{11}I

_{12,0}/I

_{11,0})[1 + 2B

_{11}(1+ I

_{22,0}/I

_{11,0})]

^{2}− 2γα − 1 = 0, γ = (I

_{22,0}− I

_{11,0})/I

_{12,0}

_{±}= γ ± (γ

^{2}+ 1)

^{1/2}

_{+}> 0 while α

_{−}< 0. Physically, these two solutions correspond to the two opposite directions of the revolution along the orbit (see Equation (31)).

_{g}

^{2}/ω

_{0}

^{2}= (4 + 2α

_{±}εδ +δ

^{2})B

_{11}

^{2}+ 2(2 + α

_{±}δ)B

_{11}+ 1

_{22,0}/I

_{11,0}), δ = I

_{12,0}/I

_{11,0}

_{±}εδ +δ

^{2}= (2 + α

_{±}δ)

^{2}

_{g}

^{2}/ω

_{0}

^{2}= [(2 + α

_{±}δ)B

_{11}+ 1]

^{2}

_{g}/ω

_{0}= |(2 + α

_{±}δ)B

_{11}+ 1|

_{g}/ω

_{0}= |{1 + I

_{22,0}/I

_{11,0}± [(I

_{22,0}/I

_{11,0}− 1)

^{2}+ I

_{12,0}

^{2}/I

_{11,0}

^{2}]

^{1/2}}B

_{11}(E) + 1|

_{11}(E). It is seen that, for each direction of the revolution of the charged particle in the orbital plane, there is a value of B

_{11}(E), for which the generalized frequency is ω

_{g}vanishes and so is the radiation. These non-radiating (stationary) states correspond explicitly to the following values, B

_{11+}(E

_{st}) and B

_{11–}(E

_{st}) of B

_{11}(E), where the subscript “st” stands for “stationary”:

_{11+}(E

_{st}) = −1/{1 + I

_{22,0}/I

_{11,0}+ [(I

_{22,0}/I

_{11,0}− 1)

^{2}+ I

_{12,0}

^{2}/I

_{11,0}

^{2}]

^{1/2}}

_{+}and

_{11–}(E

_{st}) = −1/{1 + I

_{22,0}/I

_{11,0}− [(I

_{22,0}/I

_{11,0}− 1)

^{2}+ I

_{12,0}

^{2}/I

_{11,0}

^{2}]

^{1/2}}

_{–}. Remember that α

_{±}(γ) is given by Equation (35), where γ = (I

_{22,0}− I

_{11,0})/I

_{12,0}.

_{11+}(denoted in the plot for brevity as B

_{+}) versus I

_{22,0}/I

_{11,0}(denoted in the plot as C) and I

_{12,0}/I

_{11,0}(denoted in the plot as D).

_{11–}(denoted in the plot for brevity as B

_{–}) versus I

_{22,0}/I

_{11,0}(denoted in the plot as C) and I

_{12,0}/I

_{11,0}(denoted in the plot as D).

_{22,0}/I

_{11,0}± [(I

_{22,0}/I

_{11,0}− 1)

^{2}+ I

_{12,0}

^{2}/I

_{11,0}

^{2}]

^{1/2}}B

_{11}(E) + 1|t

_{g}in Equation (41) vanishes—consequently, the radiation also vanishes.

_{22,0}= I

_{11,0}, corresponding to the circular orbits, the above formulas can be simplified as follows. In this case, from Equation (34) follows γ = 0, so that from Equation (35) we get α

_{+}= 1 and α

_{–}= −1, as it should be for the circular orbits (see Equation (31)). Then Equation (36) simplifies to

_{g}/ω

_{0}= |(2 ± |I

_{12,0}|/I

_{11,0})B

_{11}(E) + 1|

_{g}vanishes (and so does the radiation) at the following values of B

_{11}(E):

_{11+}(E

_{st}) = −1/(2 + |I

_{12,0}|/I

_{11,0}) for α = 1

_{11−}(E

_{st}) = −1/(2 − |I

_{12,0}|/I

_{11,0}) for α = −1

_{12,0}|/I

_{11,0}= 2. In the exceptional case, Equation (45) yields a trivial result: ω

_{g}= ω

_{0}.

_{11}(E) = B

_{11+}(E

_{st}) for α = 1 or B

_{11}(E) = B

_{11−}(E

_{st}) for α = −1.

_{11}(E)—given by Equations (42) and (43) in the general case of the elliptical orbits or by Equations (46) and (47) for the particular case of the circular orbits—for which the radiation vanishes. The fact that, for each direction of the revolution, there is only one value of B

_{11}(E), does not mean that there is only one classical stationary state. Indeed, if the dependence of B

_{11}on the energy E is oscillatory (with the amplitude greater than or equal to the absolute value of the right side of Equation (42) for α = α

_{+}, or with the amplitude greater than or equal to the absolute value of the right side of Equation (43) for α = α

_{–}), then there would be an infinite number of the energies of the classical stationary states E

_{st}—just as in the corresponding quantum solution.

_{11+}on the energy E:

_{11+}(E) = −|cos[π(E − C)/(E

_{st,0}− C)]|/(2 + |I

_{12,0}|/I

_{11,0})

_{st,0}is the energy of the lowest non-radiating state (the ground state) and both E and E

_{st,0}are measured in units of ħω

_{0}. In Equation (48), C is a constant, which is an analog of the Maslov index [18], which, for spherically symmetric potentials, is equal to 1/2 (see, e.g., the textbook [19]). With C = 1/2, Equation (48) takes the form

_{11+}(E) = −|cos[π(E − 1/2)/(E

_{st,0}− 1/2)]|/(2 + |I

_{12,0}|/I

_{11,0})

_{st,n}, where

_{st,n}− 1/2 = (n + 1)(E

_{st,0}− 1/2), n = 0, 1, 2, …,

_{11+}satisfies Equation (46), so that the sequence of values E

_{st,n}from Equation (50) is the sequence of the energies of the classical non-radiating stationary states. More explicitly,

_{st,n}= (n + 1)E

_{st,0}− n/2

_{st,0}= 3/2, then the sequence of values E

_{st,n}from Equation (51) would coincide with the corresponding quantum results.

## 4. Conclusions

_{N+1}and SU

_{N}, and this does not depend on the functional form of the Hamiltonian. In particular, all classical spherically symmetric potentials have algebraic symmetries, namely tO

_{4}and SU

_{3}; they possess an additional vector integral of the motion, while the quantal counterpart-operator does not exist [20,21,22]. (This fact was employed in paper [9], where the authors successfully applied the GHD to a modified Coulomb potential.) This offers possibilities that are absent in quantum mechanics, as noted in paper [8].

## Funding

## Conflicts of Interest

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**Figure 1.**Three-dimensional plot of B

_{11+}(denoted in the plot for brevity as B

_{+}) from Equation (42) versus I

_{22,0}/I

_{11,0}(denoted in the plot as C) and I

_{12,0}/I

_{11,0}(denoted in the plot as D).

**Figure 2.**Three-dimensional plot of B

_{11–}(denoted in the plot for brevity as B

_{–}) from Equation (43) versus I

_{22,0}/I

_{11,0}(denoted in the plot as C) and I

_{12,0}/I

_{11,0}(denoted in the plot as D).

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**MDPI and ACS Style**

Oks, E.
Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators. *Symmetry* **2020**, *12*, 1130.
https://doi.org/10.3390/sym12071130

**AMA Style**

Oks E.
Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators. *Symmetry*. 2020; 12(7):1130.
https://doi.org/10.3390/sym12071130

**Chicago/Turabian Style**

Oks, Eugene.
2020. "Application of the Generalized Hamiltonian Dynamics to Spherical Harmonic Oscillators" *Symmetry* 12, no. 7: 1130.
https://doi.org/10.3390/sym12071130