We often say the Lagrangian is a function of some coordinates and only their first derivatives, $$ \mathcal{L}(q,\dot{q}). $$ Even in quantum field theory, the fields are only differentiated once, $$ \mathcal{L}(\phi,\partial_\mu\phi). $$ I heard Leonard Susskind mention in his lectures never to include second derivatives, and that it leads to trouble. I am just wondering why?
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1This is down to the so-called Ostrogradsky instability. Cf. https://physics.stackexchange.com/a/57912/297348 – kricheli Apr 08 '23 at 16:59
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1Possible duplicates: https://physics.stackexchange.com/q/4102/2451 , https://physics.stackexchange.com/q/18588/2451 and links therein. – Qmechanic Apr 08 '23 at 17:04