Comment on Clebsch’s 1857 and 1859 papers on using Hamiltonian methods in hydrodynamics

Abstract

The present paper is a companion of two translated articles by Alfred Clebsch, titled "On a general transformation of the hydrodynamical equations" and "On the integration of the hydrodynamical equations". The originals were published in the Journal für die reine and angewandte Mathematik" (1857 and 1859). Here we provide a detailed critical reading of these articles, which analyzes methods, and results of Clebsch. In the first place, we try to elucidate the algebraic calculus used by Clebsch in several parts of the two articles that we believe to be the most significant ones. We also provide some proofs that Clebsch did not find necessary to explain, in particular concerning the variational principles stated in his two articles and the use of the method of Jacobi's Last Multiplier. When possible, we reformulate the original expressions by Clebsch in the language of vector analysis, which should be more familiar to the reader. The connections of the results and methods by Clebsch with his scientific context, in particular with the works of Carl Jacobi, are briefly discussed. We emphasize how the representations of the velocity vector field conceived by Clebsch in his two articles, allow for a variational formulation of hydrodynamics equations in the steady and unsteady case. In particular, we stress that what is nowadays known as the "Clebsch variables", permit to give a canonical Hamiltonian formulation of the equations of fluid mechanics. We also list a number of further developments of the theory initiated by Clebsch, which had an impact on presently active areas of research, within such fields as hydrodynamics and plasma physics.

Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

“…which, recalling the pressure representation (33b), becomes again p = p. The left-hand side of Equation ( 52) is closely related to the Lagrangian density proposed by Clebsch [37] for the variational formulation of Euler equations using Clebsch potentials; see, also, [57]. What differs, obviously, is the presence of additional terms, which is due to the additional features of compressibility, conservative body forces, and applied pressure on the free surface.…”

“…The right-hand side of the latter equation, where u = u (k, A i , a i ), is closely related (although more general) to the Lagrangian density provided in works relevant to the approach of Clebsch [37]; see, also, [57]. Further, Equations (33) are essential for the establishment of the connection between Equation ( 7) and the two alternative forms of the dynamic free-surface condition that occur variationally in Section 5; see Equations ( 51) and (52).…”