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” (https://doi.org/10.1140/epjh/s13129-021-00015-8, https://doi.org/10.1140/epjh/s13129-021-00016-7). 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.
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Change history
10 August 2021
The article has been updated as of the addition of“(PEMAT)” in the affiliation of Gérard Grimberg
Notes
Kummer (1847).
Jacobi (1844): 251.
Lagrange (1760–1761): 462, where he derives Euler equations for the case of a barotropic fluid.
Truesdell (1954a): 27.
Pfaff (1814).
Clebsch uses the index notation of Jacobi, \(a,\,a',\,a^{(2)},\ldots a^{(n-1)}\). Here we deviate slightly from the original notation by Clebsch and prefer to let the values of the index of the variables go from 1 to n.
Jacobi (1844): 203.
Basset (1888): 34–38.
Euler (1757): 347.
Hesse (1855): 248.
Jacobi (1866): 77.
Lin (1963).
Kuznetsov and Mikhailov (1980).
Sahraoui et al. (2003).
Morrison (1998).
Yoshida (2009).
Scholle et al. (2020).
References
Balkovsky, E. 1994. Some Notes on the Clebsch Representation for Incompressible Fluids. Physics Letters A 186 (1–2): 135–136. https://www.sciencedirect.com/science/article/abs/pii/0375960194909342
Basset, A.B. 1888. A Treatise on Hydrodynamics, 1961. London: Reprinted by Dover Publications. https://ia600900.us.archive.org/4/items/atreatiseonhydr02bassgoog/atreatiseonhydr02bassgoog.pdf
Bateman, H. 1929. Notes on a Differential Equation Which Occurs in the Two-Dimensional Motion of a Compressible Fluid and the Associated Variational Problems. Proceedings of the Royal Society 125 (799): 598–618. https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1929.0189
Beltrami, E. 1871. Sui Principi Fondamentali Della Idrodinamica. Memoirs Academy of Sciences Bologna 1 (1871): 431–476.
Beltrami, E. 1872. Sui Principi Fondamentali Della Idrodinamica. Memoirs Academy of Sciences Bologna 2: 381–437.
Beltrami, E. 1873. Sui Principi Fondamentali Della Idrodinamica. Memoirs Academy of Sciences Bologna 3: 349–407.
Beltrami, E. 1874. Sui Principi Fondamentali Della Idrodinamica. Memoirs Academy of Sciences Bologna 5: 443–484.
Beltrami, E. 1904. Richerche Sulla Cinematica Dei Fluidi, Opere Matematiche Tome 2 Milano. 202–379. http://gallica.bnf.fr/ark:/12148/bpt6k99434d/f6.image
Bretherton, F. 1970. A Note on Hamilton’s Principle for Perfect Fluids. Journal of Fluid Mechanics 44 (1): 19–31. https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/note-on-hamiltons-principle-for-perfect-fluids/6094DF5E50773252481F4558144447DA
Burau, W. 1970–1980. Clebsch, Rudolf Friedrich Alfred. In Dictionary of scientific biography, vol. 3. 313–315, New York: Charles Scribner Sons.
Calkin, M.G. 1963. An Action Principle for Magnetohydrodynamics. Canadian Journal of Physics 41 (1): 2241–2251. https://www.nrcresearchpress.com/doi/pdf/10.1139/p63-216
Cartan, E. 1899. Sur certaines expressions différentielles et le problème de Pfaff. Annales scientifiques de L’E.N.S., 3ème Ser. 16(1): 239–332. http://www.numdam.org/article/ASENS_1899_3_16__239_0.pdf
Cauchy, A-L. 1815/1827. Théorie de la propagation des ondes à la surface d’un fluide pesant d’une profondeur indéfinie - Prix d’analyse mathématique remporté par M. Augustin-Louis Cauchy, ingénieur des Ponts et Chaussées. (Concours de 1815). Mémoires présentés par divers savans à l’Académie royale des sciences de l’Institut de France et imprimés par son ordre. Sciences mathématiques et physiques. Tome I, imprimé par autorisation du Roi à l’Imprimerie royale: 5–318. http://gallica.bnf.fr/ark:/12148/bpt6k90181x/f14.image.r=Oeuvres%20completes%20d%27Augustin%20Cauchy.langFR
Cayley, A. 1845. Chapters in the analytical geometry of n dimensions. Cambridge Mathematical Journal IV, 119–127. https://doi.org/10.1017/CBO9780511703676.012
Cayley, A. 1847. Recherches sur L’élimination, et sur la Théorie des Courbes. Journal für die Reine und Angewandte Mathematik 34: 34–49. https://www.digizeitschriften.de/download/PPN243919689_0034/PPN243919689_0034___log5.pdf
Cayley, A. 1851. Note sur la théorie des Hyperdéterminants. Journal für die Reine und Angewandte Mathematik 42: 368–371. https://www.digizeitschriften.de/download/PPN243919689_0042/PPN243919689_0042___log42.pdf
Cayley, A. 1857. Mémoire sur la Forme Canonique des Fonctions Binaires. Journal für die Reine und Angewandte Mathematik 54 (48–58): 292. https://www.digizeitschriften.de/download/PPN243919689_0054/PPN243919689_0054___log7.pdf
Cendra, H., and J.E. Marsden. 1987. Lin Constraints, Clebsch Potentials and Variational Principles. Physica D 27: 63–89.
Clebsch, A. 1854. De motu ellipsoidis in fluido incompressibili viribus quibuslibet impulsi. Dissertatio Inaugurali Physico-Matematica, P.P: O F. E. Neumann, Regiomonti PR, impressit Ernestus Julius Daldowski. http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10054016-1
Clebsch, A. 1856. Über die Bewegung eines Ellipsoids in einer tropfbaren Flüssigkeit. Journal für die Reine und Angewandte Mathematik 52: 103–132. https://www.digizeitschriften.de/download/PPN243919689_0052/PPN243919689_0052___log12.pdf
Clebsch, A. 1857a. Über die Bewegung eines Ellipsoids in einer tropfbaren Flüssigkeit, Note zu der Abhandlung im Band LII dieses Journals. Journal für die Reine und Angewandte Mathematik 53: 293–297. https://www.digizeitschriften.de/download/PPN243919689_0053/PPN243919689_0053___log26.pdf
Clebsch, A. 1857b. Über eine Allgemeine Transformation der Hydrodynamischen Gleichungen. Journal für die Reine und Angewandte Mathematik 54: 293–312. http://www.digizeitschriften.de/download/PPN243919689_0054/PPN243919689_0054___log30.pdf
Clebsch, A. 1858a. Über die Reduction der zweiten Variation auf ihre Einfachste Form. Journal für die Reine und Angewandte Mathematik 55: 254–273. https://www.digizeitschriften.de/download/PPN243919689_0055/PPN243919689_0055___log17.pdf
Clebsch, A. 1858b. Über Diejenigen Probleme der Variationsrechnung, Welche nur eine Unabhaängige Variable Enthalten. Journal für die Reine und Angewandte Mathematik 55: 335–355. https://www.digizeitschriften.de/download/PPN243919689_0055/PPN243919689_0055___log24.pdf
Clebsch, A. 1858c. Über die Criterien des Maximums und des Minimums in der Variationsrechnung. Monatsberichte der Königlichen Preuss. Akademie der Wissenschaften zu Berlin Aus dem Jahre 1857. Königlichen Akademie der Wissenschaften 1858: 618–621. https://catalog.hathitrust.org/Record/100321563
Clebsch, A. 1859a. Über die Integration der Hydrodynamischen Gleichungen. Journal für die Reine und Angewandte Mathematik 56: 1–10. http://www.digizeitschriften.de/download/PPN243919689_0056/PPN243919689_0056___log4.pdf
Clebsch, A. 1859b. Üeber die Zweite Variation Vielfacher Integrale. Journal für die Reine und Angewandte Mathematik 56: 122–48. https://www.digizeitschriften.de/download/PPN243919689_0056/PPN243919689_0056___log15.pdf
Clebsch, A. 1861. Über Jacobis Methode, die Partiellen Differentialgleichungen erster Ordnung zu integriren und ihre Ausdehnung auf das Pfaffsche Problem, Auszug aus einem Schreiben an den Herausgeber. Journal für die Reine und Angewandte Mathematik 59: 190–192. https://www.digizeitschriften.de/download/PPN243919689_0059/PPN243919689_0059___log15.pdf
Clebsch, A. 1862. Über das Pfaffsche Problem. Journal für die Reine und Angewandte Mathematik 60: 193–251. https://www.digizeitschriften.de/download/PPN243919689_0060/PPN243919689_0060___log7.pdf
Clebsch, A. 1863. Über das Pfaffsche Problem. Zweite Abhandlung. Journal für die Reine und Angewandte Mathematik 61: 146–179. https://www.digizeitschriften.de/download/PPN243919689_0061/PPN243919689_0061___log9.pdf
Deser, S., R. Jackiw, and A.P. Polychronakos. 2001. Clebsch (String) Decomposition in d = 3 Field Theory. Physics Letters A 279: 151–153. https://www.sciencedirect.com/science/article/abs/pii/S0375960100008513
Dictionary of scientific biography 1970–1980, 16 vol, ed. Charles Coulston Gillipsie, Charles Scribner Sons, New York.
Duhem P. 1901. Sur les équations de l’hydrodynamique. Commentaire à un mémoire de Clebsch. Annales de la faculté des sciences de Toulouse 2e série, tome 3(2): 253–279. http://www.numdam.org/article/AFST_1901_2_3_2_253_0.pdf
Euler, 1757. Continuation des recherches sur la théorie du mouvement des fluides. Mémoires de l’académie des sciences de Berlin, Volume 11: 316–361. In Euler, Opera omnia, ser. 2, 12 (Lausanne), pp. 92 –132. https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?article=1226&context=euler-works
Falkovich, G., and L’vov V.S. . 1995. Isotropic and Anisotropic Turbulence in Clebsch Variables. Chaos, Solitons and Fractals 5: 1855–1869.
Frisch, U., and B. Villone. 2014. Cauchy’s Almost Forgotten Lagrangian Formulation of the Euler Equation for 3D Incompressible Flow. The European Physical Journal H 39: 325–351. https://arxiv.org/pdf/1402.4957.pdf
Frisch, U., G. Grimberg, and B. Villone. 2017. A Contemporary Look at Hermann Hankel’s 1861 Pioneering Work on Lagrangian Fluid Dynamics. The European Physical Journal H. 42: 537–546. https://arxiv.org/pdf/1707.01882.pdf
Gallavotti, G. 2010. Foundations of Fluid Dynamics. Berlin: Springer.
Ghosh, S. 2002. ‘Gauging’ the fluid. Journal of Physics A: Mathematical General 35: 10747. https://iopscience.iop.org/article/10.1088/0305-4470/35/50/306/pdf
Grad, H., and H. Rubin. 1958. Hydromagnetic equilibria and force-free fields, Proceedings of the Second United Nation Conference on Peaceful Uses of Atomic Energy, Geneva 31: 190.
Grassmann, H.G. 1844. Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik. Leipzig: Wiegand. https://ia802205.us.archive.org/2/items/dielinealeausde00grasgoog/dielinealeausde00grasgoog.pdf
Hamilton, W.R. 1834. On a General Method in Dynamics. Philosophical Transactions of the Royal Society Part 2: 247–308. http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/GenMeth.pdf
Hamilton, W.R. 1835. Second Essay on a General Method in Dynamics. Philosophical Transactions of the Royal Society Part 1: 95–144. http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/SecEssay.pdf
Hankel, H. 1861. Zur allgemeinen Theorie der Bewegung der Flüssigkeiten. Eine von der philosophischen Facultät der Georgia Augusta am 4. Juni 1861 gekrönte Preisschrift, Göttingen. http://babel.hathitrust.org/cgi/pt?id=mdp.39015035826760;view=1up;seq=5
Hawkins, G. 2005. Frobenius, Cartan, and the Problem of Pfaff. Archive for History of Exact Sciences 59: 381–436. https://www.researchgate.net/publication/225915292_Frobenius_Cartan_and_the_Problem_of_Pfaff
Hawkins, G. 2013. The Problem of Pfaff. In The Mathematics of Frobenius in Context, 155–204. New York: Springer.
Helmholtz, H. 1858. Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. Journal für die Reine und Angewandte Mathematik 55: 25–55. https://www.digizeitschriften.de/download/PPN243919689_0055/PPN243919689_0055___log6.pdf
Herivel, J.W. 1954. A General Variational Principle for Dissipative Systems: II. Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 56 (1953/1954), 56(1953/1954): 67–75. https://www.jstor.org/stable/20488563?origin=JSTOR-pdf
Herivel, J.W. 1955. The derivation of the equations of motion of an ideal fluid by Hamilton’s principle. Mathematical Proceedings of the Cambridge Philosophical Society, 51, 2(799): 344–349. https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/derivation-of-the-equations-of-motion-of-an-ideal-fluid-by-hamiltons-principle/97C179B0BFF6DE2938D0BEE01D3E5531
Hesse, O. 1855. Über Determinanten und ihre Anwendung in der Geometrie, insbesondere auf Curven vierter Ordnung. Journal für die Reine und Angewandte Mathematik 49: 243–264. https://www.digizeitschriften.de/download/PPN243919689_0049/PPN243919689_0049___log18.pdf
Hicks, W. M. 1882. Report on Recent Progress in Hydrodynamics, Part 1. Report of the Fifty-First Meeting of the Britisch association for the advancement of Science Held at York in August and September 1881 London, 1882: 57–88. https://ia802807.us.archive.org/19/items/reportofbritisha82brit/reportofbritisha82brit.pdf
Holm, D.D., and B.A. Kupershmidt. 1983. Poisson Brackets and Clebsch Representations for Magnetohydrodynamics, Multifluid Plasmas, and Elasticity. Physica D 6: 347–363. https://www.sciencedirect.com/science/article/abs/pii/0167278983900179
Holm, D.D., J.E. Marsden, T. Ratiu, and A. Weinstein. 1985. Nonlinear Stability of Fluid and Plasma Equilibria. Physics Reports 123: 1–116. https://www.sciencedirect.com/science/article/abs/pii/0370157385900286
Jackiw, R., V.P. Nair, and So-Young, P. 2000. Chern–Simons Reduction and Non-Abelian Fluid Mechanics. Physical Review D 62: 085018. https://arxiv.org/abs/hep-th/0004084
Jackiw, R., and A.P. Polychronakos. 2000. Supersymmetric Fluid Mechanics. Physical Review D 62: 085019. https://arxiv.org/abs/hep-th/0004083
Jacobi, C.G.J. 1837. Über die Reduction der Integration der partiellen Differentialgleichungen erster Ordnung zwischen irgend einer Zahl Variabeln auf die Integration eines einzigen Systems gewöhnlicher Differentialgleichungen. Journal für die Reine und Angewandte Mathematik 17: 1–189. http://gdz.sub.uni-goettingen.de/pdfcache/PPN243919689_0017/PPN243919689_0017___LOG_0012.pdf
Jacobi, C.G.J. 1844. Theoria Novi Multiplicatoris Systemati Aequationum Differentialium Vulgarium Applicandi. Journal für die Reine und Angewandte Mathematik 27: 199–268. https://www.digizeitschriften.de/download/PPN243919689_0027/PPN243919689_0027___log23.pdf
Jacobi, C.G.J. 1845. Theoria Novi Multiplicatoris Systemati Aequationum Differentialium Vulgarium Applicandi. Journal für die Reine und Angewandte Mathematik 29: 213–279. https://www.digizeitschriften.de/download/PPN243919689_0029/PPN243919689_0029___log16.pdf
Jacobi, C.G.J. 1846. Mathematische Werke, Band I. Berlin. https://ia802606.us.archive.org/12/items/cgjjacobimathem00jacogoog/cgjjacobimathem00jacogoog.pdf
Jacobi, C.G.J. 1851. Mathematische Werke, Band II. Berlin. https://ia802606.us.archive.org/12/items/cgjjacobimathem00jacogoog/cgjjacobimathem00jacogoog.pdf
Jacobi, C.G.J. 1862. Nova Methodus, Aequationes Differentiales Partiales Primi Ordinis Inter Numerum Variabilium Quemcunque Propositas Integrandi. Journal für die Reine und Angewandte Mathematik 60: 1–181. https://www.digizeitschriften.de/download/PPN243919689_0060/PPN243919689_0060___log4.pdf
Jacobi, C.G.J. 1866. Vorlesungen über Dynamik. Clebsch: Herausgegeben von A. https://books.googleusercontent.com/books/content?req=AKW5Qafv_LfzV2dahwaN6k1qn1Pee3SK8n9d3i8XY4q8EE-ODivjEQ84KaUkr0fp8EPmFsXTcNlSqWOCaVfDXypyIVSVXjccMppclSBeubOn5Vf2JOAdw79bWbsMEGYFyR3opNt6DKIneXRYB_2jj-0_YWokzyo3H-quKgh9dJVaXjrP4UAgwEyIMWyEflhZYKEHQtxGfhYrkxuwATrhptHxSqekd0QLN7Kq6HxyEladMlMVPIhQIcFhU39tP0dOS6nYlfNYmJNXJslvwiF_Uh6RwHHmncPESQ
Jacobi, C.G.J. 1890. Gesammelte Werke Band 5 Berlin. https://ia800204.us.archive.org/9/items/gesammeltewerke05cgjj/gesammeltewerke05cgjj.pdf
Kummer, E.E. 1847. Zur Theorie der Complexen Zahlen. Journal für die Reine und Angewandte Mathematik 35: 319–326. https://www.digizeitschriften.de/download/PPN243919689_0035/PPN243919689_0035___log25.pdf
Kuznetsov, E.A., and A.V. Mikhailov. 1980. On the Topological Meaning of Canonical Clebsch Variables. Physics Letters A 5: 3986–3989. https://www.sciencedirect.com/science/article/abs/pii/0375960180906271
Lagrange, J.L. 1760–1761. Application de la méthode exposé dans le Mémoire précédent à la solution de différents problémes de dynamique, Miscellanea Taurinensia, Oeuvres. 1: 365–468. http://gallica.bnf.fr/ark:/12148/bpt6k2155691/f415
Lamb, H. 1895. Hydrodynamics, 1st ed. Cambridge: Cambridge University Press. https://ia902908.us.archive.org/35/items/hydrodynamics00horarich/hydrodynamics00horarich.pdf
Lanczos, C. 1970. The Variational Principles of Mechanics. New York: Dover Publications.
Landau, L.D. and E.M. Lifshitz 1976 Mechanics, Course of Theoretical Physics, 1, Third Ed. Elsevier.
Lin, C.C. 1963. Liquid Helium. In Proceedings of International School of Physics, Course XXI. New York: Academic Press.
Marsden, J., and A. Weinstein. 1983. Coadjoint Orbits, Vortices, and Clebsch Variables for Incompressible Fluids. Physica D: Nonlinear Phenomena 7 (1–3): 305–323. https://www.sciencedirect.com/science/article/pii/0167278983901343
Marsden, J., T. Ratiu, and A. Weinstein. 1984. Semidirect Products and reduction in Mechanics. Transactions of the American Mathematical Society 281 (1): 147–177. https://www.researchgate.net/publication/46175465_Semidirect_Products_and_Reduction_in_Mechanics
Mendes, A.C.R., C. Neves, W. Oliveira, and F.I. Takakura. 2005. Hidden Symmetries in (Relativistic) Hydrodynamics. Journal of Physics A: Mathematical and General 38 (1): 8747–8762. https://iopscience.iop.org/article/10.1088/0305-4470/38/40/016
Merzbach, U. 2018. Dirichlet: A Mathematical Biography. Birkhauser.
Morrison, P.J., and J.M. Greene. 1980. Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics. Physical Review Letters 45: 790–794. https://pdfs.semanticscholar.org/6239/4c7804d194ff80c9990e1e32fce9da6e7c89.pdf
Morrison, P.J. 1982. Poisson Brackets for Fluids ans Plasmas in Mathematical Methods, Hydrodynamics and Integrability in Related Dynamical Systems, AIP Conference Proceedings No. 88 edited by M. Tabor and Y.Treve, 13–46. New York: AIP. https://www.researchgate.net/publication/236117225_Poisson_Brackets_for_Fluids_and_Plasmas
Morrison, P.J., and R.J. Hazeltine. 1984. Hamiltonian Formulation of Reduced Magnetohydrodynamics. Physics of Fluids 27: 886–897. https://pdfs.semanticscholar.org/8c90/6fee8dca334f832cc1eb934b619addc785d2.pdf
Morrison P.J., Caldas I.L. and H. Tasso. 1984. Hamiltonian formulation of two-dimensional gyroviscous MHD. Zeitschrift für Naturforschung 39a: 1023–1027. https://web2.ph.utexas.edu/~morrison/84ZNF_morrison.pdf
Morrison, P.J. 1998. Hamiltonian Description of the Ideal Fluid. Reviews of Modern Physics 70: 467–521. https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.70.467
Nakane, M., and C.G. Fraser. 2002. The Early History of Hamilton-Jacobi Dynamics 1834–1837. Centaurus 44: 161–227. https://europepmc.org/article/med/17357243
Pfaff, J.F. 1814. Methodus Generalis, Aequationes Differentiarum Partialium nec non Aequationes Differentiales Vulgates, Ultrasque Primi Ordinis, Inter Quotcunque Variables, Complete Integrandi. Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin, 76–136. https://ia600405.us.archive.org/4/items/bub_gb_KoY_AAAAcAAJ/bub_gb_KoY_AAAAcAAJ.pdf
Riemann, B. 1854. Über die Hypothesen, welche der Geometrie zugrunde liegen., Abh. Kgl. Ges. Wiss., Göttingen 1868. https://www.emis.de//classics/Riemann/Geom.pdf
Rund, H. 1977a. Clebsch Potentials in the Theory of Electromagnetic Fields Admitting Electric and Magnetic Charge Distributions. Journal of Mathematical Physics 18: 84. https://doi.org/10.1063/1.523121
Rund, H. 1977b. Clebsch Potentials and Variational Principles in the Theory of Dynamical Systems. Archive for Rational Mechanics and Analysis 65: 305–334. https://link.springer.com/article/10.1007/BF00250431
Sahraoui, F., G. Belmont, and L. Rezeau. 2003. Hamiltonian Canonical Formulation of Hall-Magnetohydrodynamics: Toward an Application to Weak Turbulence Theory. Physics of Plasmas 10: 1325. https://doi.org/10.1063/1.1564086
Scholle, M., F. Marner, and P.H. Gaskell. 2020. Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances. Water 12: 1241. https://doi.org/10.3390/w12051241
Seliger, R.L., and G.B. Whitham. 1968. Variational Principles in Continuum Mechanics. Proceedings of the Royal Society of London 305: 1–25. https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1968.0103
Truesdell, C. 1954a. The Kinematics of Vorticity, Indiana University Science Series no. 19. Indiana University Press (Bloomington).
Truesdell, C. 1954b. ‘Rational fluid mechanics, 1657–1765.’ In Euler, Opera omnia, ser. 2, 12 (Lausanne), IX–CXXV.
Truesdell, C., Toupin, R.A., 1960 Classical Fields Theory, in Encyclopedia of Physics, ed. S. Flugge, Vol. III/1, Principles of Classical Mechanics and Field Theory, pp. 226–793. Springer.
Various Authors. 1873a. Zum Andenken an Rudolf Friedrich Alfred Clebsch. Mathematische Annalen 6: 197–202. http://gdz.sub.uni-goettingen.de/pdfcache/PPN235181684_0006/PPN235181684_0006___LOG_0021.pdf
Various Authors. 1873b. Versuch einer Darlegung und Würdigung seiner wissenschaftlichen Leistungen von einigen seiner Freunde: “R.F. Alfred Clebsch’s Mathematische Arbeiten”, Mathematische Annalen 7: 1–50. http://gdz.sub.uni-goettingen.de/pdfcache/PPN235181684_0007/PPN235181684_0007___LOG_0007.pdf
Yoshida, Z. 2009. Clebsch Parametrization: Basic Properties and Remarks on its Applications. Journal of Mathematical Physics 50: 113101. https://doi.org/10.1063/1.3256125
Zakharov, V.E. 1989. The Algebra of Integrals of Motion of Two-dimensional Hydrodynamics in Clebsch Variables. Functional Analysis and its Applications 23 (3): 189–196. https://link.springer.com/article/10.1007/BF01079524
Zakharov, V.E., S.L. Musher, and A.M. Rubenchik. 1985. Hamiltonian Approach to the Description of Non-linear Plasma Phenomena. Physics Reports 129 (5): 285–366.
Zakharov, V.E., and E.A. Kuznetsov. 1997. Hamiltonian Formalism for Nonlinear Waves. Physics Uspekhi 40 (11): 1087. https://iopscience.iop.org/article/10.1070/PU1997v040n11ABEH000304
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GG and ET are grateful to Uriel Frisch, without whom the realization of this article would not have been possible. GG and ET are also thankful to the two Reviewers, who provided useful and constructive comments which helped improving the paper. GG wishes to thank the Observatoire de la Côte d’Azur and the Laboratoire J.-L. Lagrange for their hospitality and financial support.
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Grimberg, G., Tassi, E. Comment on Clebsch’s 1857 and 1859 papers on using Hamiltonian methods in hydrodynamics. EPJ H 46, 17 (2021). https://doi.org/10.1140/epjh/s13129-021-00014-9
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DOI: https://doi.org/10.1140/epjh/s13129-021-00014-9