Why Are So Many Equations of Motion Second Order?

Many familiar equations of motion are second order in time, but there is no universal rule that physical equations have order at most two. The order depends on the variables, the action, and whether we mean time or spatial derivatives.

This note revises my original Chinese answer.

Why ordinary mechanics is often second order

For an action built from a Lagrangian $L(q,\dot q,t)$ , stationarity gives

\frac{d}{dt}\frac{\partial L}{\partial\dot q^i} -\frac{\partial L}{\partial q^i}=0.

Expanding the time derivative introduces accelerations but no higher derivatives. If the velocity Hessian $\partial^2L/\partial\dot q^i\partial\dot q^j$ is invertible, these equations can locally be solved for $\ddot q$ . Initial positions and velocities are then the natural initial data, subject to the usual existence and uniqueness conditions.

The same dynamics can also be written as a first-order system in position and momentum. Differential order is partly a matter of representation.

What Ostrogradsky's theorem actually restricts

A Lagrangian depending nondegenerately on higher time derivatives generally produces additional canonical variables. Under the assumptions of Ostrogradsky's construction, its Hamiltonian is linear in some unconstrained momenta and is unbounded above and below. This creates a serious stability problem for such theories. See Woodard's review of the theorem.

The nondegeneracy and unconstrained-variable assumptions are essential. Degenerate higher-derivative theories can evade the construction through constraints. Higher-derivative terms also occur in effective theories treated perturbatively within a limited regime; treating a truncated effective equation as an exact theory can introduce spurious solutions.

Higher spatial derivatives are another matter. An ideal Euler–Bernoulli beam equation contains a fourth spatial derivative while remaining second order in time. The Schrödinger equation is first order in time. These examples already rule out the blanket claim in the original title.

The useful question is therefore why a particular model has its chosen variables and derivative order, and whether its initial-value problem and stability properties are appropriate to the physics it describes.