Note: This post is an English adaptation of my original Chinese article (URL). Some parts have been modified for clarity, cultural relevance, or to better fit the English-speaking audience.
This is a compelling question, and the explanation is rooted in a fundamental principle of physical systems and a key mathematical theorem. The essential reason is as follows:
The energy of any non-degenerate physical system must possess a lower bound. Let us designate this as postulate $\Psi$. This principle is regarded as an empirical conclusion derived from consistent experimental observation.
There exists a mathematical theorem known as Ostrogradsky’s Instability. It states that if the Lagrangian ($L$) of any non-degenerate system depends on time derivatives of the generalized coordinates of an order higher than one (e.g., $\ddot{q}$), then the Hamiltonian of the system is necessarily unbounded from below.
By applying the contrapositive of Ostrogradsky’s theorem in conjunction with postulate $\Psi$, we can deduce a critical constraint: The Lagrangian of any stable, non-degenerate physical system must not depend on second or higher-order time derivatives of its generalized coordinates. The Lagrangian must be a function of the form $L(q, \dot{q}, t)$.
Furthermore, the conventional definitions of potential energy and kinetic energy in physics also conform to the constraint that their derivative to generalized coordinate and generalized velocity is 0th-order respectively.
Therefore, the Euler-Lagrange equations of motion for any such non-degenerate physical system will be second-order (at most) differential equations.
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