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I am trying to understand geometrically the relation between a conservative classical system described by the hamiltonian $H$ for which the trajectories of particles are given by $$\dot{x} = \nabla_p H(x,p) \qquad \dot{p} = - \nabla_x H(x,p)$$ and that of the Hamilton-Jacobi equation $$u_t - H(x,\nabla u) = 0.$$

Namely: the trajectories of particles with hamiltonian $H$ are constrained to level sets of that hamiltonian. Are these level sets related to the characteristic curves given by the corresponding Hamilton-Jacobi equation? Is there a direct way to go between the level sets to the characteristic curves or vice versa?

You may restrict the problem to 1 space dimension if necessary.

Thank you!

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The function $u$ is known as Hamilton's principal function. Since OP's system is apparently autonomous, one should (in order to simplify) use Hamilton's characteristic function $W$ instead. The HJ equation then simplifies to $$H(x,\nabla W)~=~E. $$ The corresponding characteristic curves certainly preserves the energy level $E$.

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