A Lagrange with Higher Derivatives (Ostrogradsky instability)

[Arman777]

In class our teacher told us that, if a Lagrangian contain $\ddot{q_i}$ (i.e., $L(q_i, \dot{q_i}, \ddot{q_i}, t)$) the energy will be unbounded from below and it can take any lower values (in other words be unstable). In this type of systems can we show that the energy is conserved ? Or in such system does energy conservation is applicable ?

 

[wrobel]

there must be some conservation law which follows from the time shift invariance of such systems; I would try to derive it first.

If I did not make an error the "energy" integral is as follows
$$H(q,\dot q,\ddot q,\dddot q)=-L+2\frac{\partial L}{\partial \ddot q}\ddot q+\frac{\partial L}{\partial \dot q}\dot q-\frac{d}{dt}\Big(\frac{\partial L}{\partial \ddot q}\dot q\Big)$$
This is under the assumption that $L$ does not depend on t surely.

Instability arises due to the last term I guess. This term is linear in $\dddot q$

 

[vanhees71]

Indeed, the most famous example is the Abraham-Lorentz-Dirac equation for a point particle moving in an external electromagnetic field including the radiation reaction, i.e., the backreaction on the motion of the particle by its own (radiation) field. It leads to serious problems with causality and "run-away solutions".

 

[Arman777]

So the energy conservation is applicable.