In this paper we study the existence of sufficiently regular representations of Hamilton-Jacobi equations in optimal control theory with the compact control set. We introduce a new method to construct representations for a wide class of Hamiltonians, wider than was achieved before. Our result is proved by means of these conditions on the Hamiltonian that are necessary for the existence of a representation. In particular, we solve an open problem of Rampazzo [SIAM J. Control Optim., 44 (2005), pp. 867-884]. We apply the obtained results to reduce a variational problem to an optimal control problem.