Significance of the order of derivatives in an action

Question

What is the significance of having higher order derivatives in an action describing some system? For example, suppose I have the following two actions \begin{align} S_1&\propto \int \text{d}^4x \psi \partial \psi\\ S_2&\propto \int \text{d}^4x \psi \square \psi. \end{align} The field $\psi$ may or may not have multiple components (and describe a non-zero spin particle), I'm not sure if that is important to answer the question. What is the significance of $S_1$ being of first order in derivatives and $S_2$ being second order? To my knowledge the higher the order of derivatives contained in the Lagrangian, the more restrictions we place on the field when varying the action and ignoring whatever bounder terms may arise. I don't know why this is so, and this probably not the only significance of the order.

Answer 1

Note that you can rewrite $S_2$ in terms of first order derivatives in the following way:

$$\psi\square\psi=\psi\partial_\mu\partial^\mu\psi=\partial_\mu(\psi\partial^\mu\psi)-\partial_\mu\psi\partial^\mu\psi.$$

Throwing away the first term (total derivative implies boundary term), you are left with:

$$S_2\propto\int\mathrm{d}^4x\partial\psi\partial\psi.$$