Dirac’s Generalized Hamiltonian Dynamics (GHD) is a purely classical formalism for systems having constraints: it incorporates the constraints into the Hamiltonian. Dirac designed the GHD specifically for applications to quantum field theory. In one of our previous papers, we redesigned Dirac’s GHD for its applications to atomic and molecular physics by choosing integrals of the motion as the constraints. In that paper, after a general description of our formalism, we considered hydrogenic atoms as an example. We showed that this formalism leads to the existence of classical non-radiating (stationary) states and that there is an infinite number of such states—just as in the corresponding quantum solution. In the present paper, we extend the applications of the GHD to a charged Spherical Harmonic Oscillator (SHO). We demonstrate that, by using the higher-than-geometrical symmetry (i.e., the algebraic symmetry) of the SHO and the corresponding additional conserved quantities, it is possible to obtain the classical non-radiating (stationary) states of the SHO and that, generally speaking, there is an infinite number of such states of the SHO. Both the existence of the classical stationary states of the SHO and the infinite number of such states are consistent with the corresponding quantum results. We obtain these new results from first principles. Physically, the existence of the classical stationary states is the manifestation of a non-Einsteinian time dilation. Time dilates more and more as the energy of the system becomes closer and closer to the energy of the classical non-radiating state. We emphasize that the SHO and hydrogenic atoms are not the only microscopic systems that can be successfully treated by the GHD. All classical systems of N degrees of freedom have the algebraic symmetries ON+1 and SUN, and this does not depend on the functional form of the Hamiltonian. In particular, all classical spherically symmetric potentials have algebraic symmetries, namely O4 and SU3; 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.

中文翻译：

---

广义哈密顿动力学在球谐振荡器中的应用

Dirac 的广义哈密顿动力学 (GHD) 是具有约束的系统的纯经典形式主义：它将约束合并到哈密顿量中。狄拉克专门为量子场论的应用设计了 GHD。在我们之前的一篇论文中，我们通过选择运动的积分作为约束，重新设计了 Dirac 的 GHD 以将其应用于原子和分子物理学。在那篇论文中，在对我们的形式主义进行了一般性描述之后，我们以氢原子为例。我们证明了这种形式主义导致经典非辐射（静止）状态的存在，并且存在无限多个这样的状态——就像在相应的量子解中一样。在本文中，我们将 GHD 的应用扩展到带电球面谐波振荡器 (SHO)。我们证明，通过使用 SHO 的高于几何对称性（即代数对称性）和相应的附加守恒量，可以获得 SHO 的经典非辐射（静止）状态，一般来说，有是无限数量的 SHO 状态。SHO 的经典稳态的存在和无限数量的这种状态都与相应的量子结果一致。我们从第一性原理中获得了这些新结果。在物理上，经典静止状态的存在是非爱因斯坦时间膨胀的表现。随着系统的能量越来越接近经典非辐射状态的能量，时间会越来越膨胀。我们强调，SHO 和氢原子并不是 GHD 可以成功处理的唯一微观系统。所有 N 自由度的经典系统都具有 ON+1 和 SUN 的代数对称性，这与哈密顿量的函数形式无关。特别是，所有经典球对称势都具有代数对称性，即 O4 和 SU3；它们拥有额外的运动矢量积分，而量子对应算子不存在。这提供了量子力学中不存在的可能性。所有经典球对称势都具有代数对称性，即 O4 和 SU3；它们拥有额外的运动矢量积分，而量子对应算子不存在。这提供了量子力学中不存在的可能性。所有经典球对称势都具有代数对称性，即 O4 和 SU3；它们拥有额外的运动矢量积分，而量子对应算子不存在。这提供了量子力学中不存在的可能性。