In this formulation , the solutions of the Hamilton -- Jacobi equations are the integral curves of Hamiltonian vector fields .
The study of the solutions of the Hamilton -- Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology .
In this formulation , the solutions of the Hamilton -- Jacobi equations are the integral solutions of Hamiltonian vector fields .
The study of the solutions of the Hamilton -- Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology .