Healthy degenerate theories with higher derivatives

Understand

In the context of classical mechanics, we study the conditions under which higher-order derivative theories can evade the so-called Ostrogradsky instability.

• More precisely, we consider general Lagrangians with second order time derivatives, of the form $L(\ddot\phi^a,\dot\phi^a,\phi^a;\dot q^i,q^i)$ with $a = 1,\cdots, n$ and $i = 1,\cdots, m$.
• For $n=1$, assuming that the $q^i$'s form a nondegenerate subsystem, we confirm that the degeneracy of the kinetic matrix eliminates the Ostrogradsky instability.
• The degeneracy implies, in the Hamiltonian formulation of the theory, the existence of a primary constraint, which generates a secondary constraint, thus eliminating the Ostrogradsky ghost.