  For higher derivative theories, using the approach of Caratheodory's
equivalent Lagrangian, we show that there exist novel formulations of
Hamilton-Jacobi equations, which are different from the formulations derived
from Hamilton's canonical approach. The quantum mechanical correspondences of
these novel Hamilton-Jacobi equations lead to nonlinear quantum mechanics,
which seem being able to avoid the unbounded negative energy problem in the
quantum mechanics of higher derivative theories.
