# History of Pfaff's problem and integrability conditions

I'm curious about Pfaff's problem and all the various mathematicians who have worked on it. From Wikipedia articles I was able to gather the following timeline of events.

In the modern language:

Pfaff's problem: Given a collection of $$1$$-forms on a manifold, find a submanifold where they vanish.

Frobenius theorem: A distribution on a manifold is integrable if and only if it is involutive.

Darboux's theorem: Given a $$1$$-form $$\theta$$ with $$d\theta$$ having constant rank, locally $$\theta$$ has a nice canonical form.

There is a similar canonical form for a collection of involutive vector fields and this is used in modern proofs of the Frobenius theorem.

Timeline: (from Wikipedia)

1814-1815: Pfaff asks about the integrability of a system of $$1$$-forms, presumably just a system of ODEs in the modern language.

1840: Deahna provides a sufficient condition for the Frobenius theorem.

1866: Clebsch provides a necessary condition for the Frobenius theorem.

1887: Frobenius applies "Frobenius" theorem to the Pfaffian problem.

1882: Darboux proves his theorem, solving Pfaff's problem because the solution can be obtained, at least locally, by setting some coordinates (in a specific parametrization) to zero.

Questions:

Today, differential equations are either treated as just mathematical objects that describe surfaces et c., or in relation to solving some physical system or finding trajectories of objects and so on. I suspect Pfaff was concerned with the mechanics and dynamics of physical systems, but I'm not sure. What prompted Pfaff to pose his problem and did he work on it?

Secondly, in modern terms one of the implications of Frobenius theorem is fairly trivial using the notion of Lie brackets, pushforwards et c. So, what exactly were Deahna and Clebsch concerned with? Were they simply answering a question of "a system of ODEs has a solution if..." and "if a solution exists, then the coefficients of the ODES should satisfy..."? Of course, without the modern language, I'd say that both implications would look like a mess of symbols and far from trivial.

Lastly, I'm completely in the dark as to what Frobenius' contribution is. Wikipedia simply states that he "applied" Frobenius theorems to the problem. Did Frobenius give an answer to Pfaff's problem or give a hint on tackling Pfaff's problem? Was Darboux aware of Frobenius' work or was it a different kind of solution answering a differently stated problem?

References: (from Wikipedia)

Pfaff, Johann Friedrich (1814–1815). "Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi" [A general method to completely integrate partial differential equations, as well as ordinary differential equations, of order higher than one, with any number of variables]. Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin (in Latin): 76–136

Deahna, F. (1840). "Über die Bedingungen der Integrabilitat ..." J. Reine Angew. Math. 20: 340–350. doi:10.1515/crll.1840.20.340. S2CID 120057555.

Clebsch, A. (1866). "Ueber die simultane Integration linearer partieller Differentialgleichungen". J. Reine. Angew. Math. (Crelle). 1866 (65): 257–268. doi:10.1515/crll.1866.65.257. S2CID 122439486.

Frobenius, G. (1877). "Über das Pfaffsche Problem". J. Reine Angew. Math. 1877 (82): 230–315. doi:10.1515/crll.1877.82.230. S2CID 119848431.

Darboux, Gaston (1882). "Sur le problème de Pfaff" [On the Pfaff's problem]. Bull. Sci. Math. (in French). 6: 14–36, 49–68.

• Scans from the Göttingen Digitization Center: A. Clebsch, "Ueber das Pfaffsche Problem." Journal für die reine und angewandte Mathematik, Vol. 60, 1862, pp. 193-261. ### A. Clebsch, "Ueber die simultane Integration linearer partieller Differentialgleichungen." Journal für die reine und angewandte Mathematik, Vol. 65, 1866, pp. 257-268. Commented Dec 4, 2023 at 6:41
• More scans from GDZ: A. Clebsch, "Ueber das Pfaffsche Problem." Journal für die reine und angewandte Mathematik, Vol. 61, 1863, pp. 146-179. ### Frobenius, "Ueber das Pfaffsche Problem." Journal für die reine und angewandte Mathematik, Vol. 82, 1877, pp. 230-315. Commented Dec 4, 2023 at 6:47
• In addition to various publications by Jacobi, Frobenius also cites: L. Natani, "Ueber totale und partielle Differentialgleichungen." Journal für die reine und angewandte Mathematik, Vol. 58, 1861, pp. 301-328. Commented Dec 4, 2023 at 6:53
• Darboux mentions Frobenius's 1877 publication ("un beau memoire") in the footnotes on the second page of his paper, so clearly he was aware of this work. Commented Dec 4, 2023 at 6:59
• Unfortunately, while I do know German I know nothing about the subject matter plus these are some pretty voluminous publications, so I cannot summarize them. I added the references with links to the scans for the convenience of some expert that has the necessary skills (not everybody knows about GDZ). Commented Dec 4, 2023 at 8:08