A note on duality of first order partial differential equations
The following generalized Legendre contact transformation of \(J^1 (\mathbb{R}^n, \mathbb{R})\) with the local coordinates \((x, u, p)= (x_1, \dots, x_n, y, p_1, \dots,p_n)\) and \(I\subset \{1, \dots, n\}\), \[ X_{I^c}= x_{I^c}, \quad X_I = p_I, \quad Y= p_I x_I- y, \quad P_{I^c}= p_{I^c}, \quad P_I= x_I \] is used to formulate and prove a duality principle for systems of first order PDE's with complete solutions. A stronger statement is proved for holonomic systems of first order PDE's. These results generalize the earlier work of Izumiya given in the references.
scientific article; zbMATH DE number 784565