Conservative Systems and The Hamilton-Jacobi Equation

differential-geometry partial-differential-equations ordinary-differential-equations classical-mechanics dynamical-systems

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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