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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 ?

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Not enough reputation to comment, sorry. It should still be true that if there is no explicit $t$-dependence and the potential is a function of $q$, then the Lagrangian conserves energy.

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    $\begingroup$ What does it mean for a Lagrangian to conserve energy? Do you mean that energy is conserved along the solutions of the EOM associated to the Lagrangian? $\endgroup$ – NDewolf Mar 10 at 18:51
  • $\begingroup$ Yeah, that's definitely a better way to word it. $\endgroup$ – user3517167 Mar 10 at 18:56

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