  Under conventional Legendre transformation, systems with a non-convex
Lagrangian will result in a multi-valued Hamiltonian as a function of conjugate
momentum. This causes problems such as non-unitary time evolution of quantum
state and non-determined motion of classical particles, and is physically
unacceptable. In this work, we propose a new construction of single-valued
Hamiltonian by applying Legendre-Fenchel transformation, which is a
mathematically rigorous generalization of conventional Legendre transformation,
valid for non-convex Lagrangian systems, but not yet widely known to the
physics community. With the new single-valued Hamiltonian, we study spontaneous
breaking of time translation symmetry and derive its vacuum state. Applications
to theories of cosmology and gravitation are discussed.
