Reviewing the geometric Hamilton–Jacobi theory concerning Jacobi and Leibniz identities

Abstract In this survey, we review the classical Hamilton–Jacobi theory from a geometric point of view in different backgrounds. We propose equation for structures attending to one particular characterization: whether they fulfill Jacobi and Leibniz identities simultaneously, or if at least satisfy them. This property reviews work present authors such way that it is presented according hierarchy nested brackets. new outlook which do not only themselves, but their hierarchical relations how affect dynamic through equation. regard, case time-dependent ( t -dependent sequel) dissipative physical systems as identity Leibnitz identity, instances cosymplectic contact mechanical problems. Let us remark here means dissipation provided by parameter included Hamiltonian function, instance, entropy some thermodynamical systems. Furthermore, contact-evolution split off regular geometry, actually satisfies rule instead Jacobi. include novel result, conformal vector fields generalization well-known on symplectic manifold, retrieved zero factor. The interest primordial observation field X H can be projected into configuration manifold one-form d W , then integral curves X H d W transformed solution Geometrically, plays role Lagrangian submanifold certain bundle. Exploiting these features scenarios multiple depending fundamental satisfies. Different examples are pictured reflect results provided, being all them new, except reassessment previously considered example. Finally, let mention an important issue study separability, relevant finding conserved quantities. However, will deal with subject although refer reader papers [16, 17, 136].