Lagrangian submanifolds and the Hamilton-Jacobi equation
The authors present a generalized version of the geometric Hamilton-Jacobi theory in terms of certain Lagrangian submanifolds of a symplectic manifold. This allows them to give a unified presentation of Hamilton-Jacobi equations for systems with both holonomic and linear nonholonomic constraints. The formulation treats both the time-independent and the time-dependent case, where the authors need to use a time-dependent family of Lagrangian submanifolds. The application to the linear nonholonomic case gives more general results than those known in the literature. The most interesting feature of the approach in terms of Lagrangian submanifolds is that it gives a unified framework that also allows to deal with singular Lagrangian systems, as shown by \textit{W. M. Tulczyjew} (see, e.g. [C. R. Acad. Sci., Paris, Sér. A 283, 15--18 (1976; Zbl 0327.58007); ibid. A 283, 675--678 (1976; Zbl 0334.58008)]), and can be adapted to even more general settings.
scientific article; zbMATH DE number 6210441
scientific article