First integrals without integrating factors and symmetries

 My latest research has just been published with Springer Nature in Qualitative Theory of Dynamical Systems. Read here.



Finding conserved quantities—first integrals—is one of the most powerful ways to tame an ordinary differential equation (ODE). Traditionally, this requires either discovering a symmetry of the equation (à la Lie theory) or digging up an integrating factor. Both approaches are powerful, but they come with heavy baggage: symmetry detection can fail, and solving the partial differential equations for integrating factors is often impossible in practice.

In this paper, I propose a different route: a geometric method built on the complete integrability of the Pfaffian system naturally associated with an ODE. By moving the problem into the language of jet bundles and contact forms, the method bypasses both symmetries and integrating factors. Instead, it turns the hunt for a first integral into solving a system of PDEs derived from Frobenius’ theorem.

What makes this exciting is not that the method is “easier” (it can actually be more computationally involved), but that it works in situations where the standard toolkit breaks down. The paper demonstrates this on second-, third-, and fourth-order ODEs—including equations with no Lie symmetries—successfully extracting first integrals and, in some cases, families of solutions.

The result is a complementary tool for researchers tackling stubborn ODEs. Even better, its algorithmic nature makes it a natural candidate for computer algebra implementation, where systematic exploration of ansätze could extend its reach even further.


 

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