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The paper reviews some parts of classical potential theory with applications to two-dimensional fluid dynamics, in particular vortex motion. Energy and forces within a system of point vortices are similar to those for point charges when the vortices are kept fixed, but the dynamics is different in the case of free vortices. Starting from Euler’s and Bernoulli’s equations we derive these laws. Letting the number of vortices tend to infinity leads in the limit to considerations of equilibrium distributions, capacity, harmonic measure and many other notions in potential theory. In particular various kinds of Green functions have a central role in the paper, and we make a distinction between electrostatic and hydrodynamic Green functions. We also consider the corresponding concepts in the case of closed Riemann surfaces provided with a metric. From a canonically defined monopole Green function we rederive much of the classical theory of harmonic and analytic forms on Riemann surfaces. In the final section of the paper we return to the planar case, which then reappears in the form of a symmetric Riemann surface, the Schottky double. Bergman kernels, electrostatic and hydrodynamic, come up naturally as well as other kernels, and associated to the Green function there is a certain Robin function which is important for vortex motion, and which also relates to capacity functions in classical potential theory.

The essential point of Bohr–Mottelson theory is to assume an irrotational flow. As was already suggested by Marumori and Watanabe, the internal rotational motion, i.e., the vortex motion, however, may exist also in nuclei. So, we must take the vortex motion into consideration. In classical fluid dynamics, there are various ways to treat the internal rotational velocity. The Clebsch representation, $v(x)=-\nabla \varphi (x)+\lambda (x)\nabla \psi (x)(\varphi ;\text{velocity potential},\lambda \text{and}\psi :\text{Clebsch parameters})$ is very powerful and allows for the derivation of the equations of fluid motion from a Lagrangian. Making the best use of this advantage, Kronig–Thellung, Ziman and Ito obtained a Hamiltonian including the internal rotational motion, the vortex motion, through the term $\lambda (x)\nabla \psi (x)$. Going to quantum fluid dynamics, Ziman and Thellung finally derived the roton spectrum of liquid Helium II postulated by Landau. Is it possible to follow a similar procedure in the description of the collective vortex motion in nuclei? The description of such a collective motion has not been considered in the context of the Bohr–Mottelson model (BMM) for a long time. In this paper, we will investigate the possibility of describing the vortex motion in nuclei on the basis of the theories of Ziman and Ito together with Marumori’s work.