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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Does this answer your question? Lagrangian and conservation of energy – user3517167 Mar 10 '21 at 16:18
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@user3517167 My lagrangian contains $\ddot{q_i}$ – seVenVo1d Mar 10 '21 at 16:20
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There's this paper https://arxiv.org/abs/astro-ph/0601672 in which at section 2 there's a detailed discussion on the Ostrogradsky instability. Hope it helps. – Noone Mar 10 '21 at 16:31
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Possible duplicates: https://physics.stackexchange.com/q/610562/2451 , https://physics.stackexchange.com/q/489969/2451 and links therein. – Qmechanic Mar 10 '21 at 16:58
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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.
user3517167
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2What 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? – NDewolf Mar 10 '21 at 18:51
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