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An equilibrium is stable if small perturbations cause restoring forces (like a ball at the bottom of a bowl); unstable if perturbations grow (ball on top of a sphere); or neutral if energy is unchanged (ball on a flat surface). For conservative systems, stability is determined by whether the potential energy is at a minimum, maximum, or inflection point—a criterion crucial for analyzing mechanical systems.
From equilibrium analysis you know how to find *where* a system is in equilibrium — the conditions under which net force and net moment are zero. But finding equilibrium tells you nothing about whether that equilibrium is physically realizable. A pencil balanced on its tip is in equilibrium, yet it immediately falls under any perturbation. A pencil lying flat is also in equilibrium, and it stays there indefinitely. The classification of stability asks: what happens after a small push?
The key insight uses potential energy from your prerequisite. For conservative systems, where all active forces derive from a potential V(q), equilibrium occurs wherever dV/dq = 0 — the potential is stationary. But a stationary function can have three distinct local behaviors: a local minimum, a local maximum, or a flat inflection point. These correspond exactly to stable, unstable, and neutral equilibrium. At a potential minimum, any displacement increases V, which creates a restoring force F = −dV/dq pointing back toward equilibrium — the system returns. At a potential maximum, any displacement decreases V, creating a force that drives the system further away. At a flat point (neutral), V is locally constant and no restoring or destabilizing force acts.
The quantitative criterion uses the second derivative d²V/dq² evaluated at the equilibrium point. If d²V/dq² > 0, the potential is concave up (a bowl) — stable. If d²V/dq² < 0, concave down (an inverted bowl) — unstable. If d²V/dq² = 0, you must examine higher derivatives; in simple mechanical problems this usually indicates neutral equilibrium, but a full analysis requires checking d⁴V/dq⁴ and the sign of higher even-order terms. The second derivative also encodes the stiffness of the restoring force: a large positive d²V/dq² means a steep bowl, a strong restoring force, and a high natural frequency of oscillation. A small positive value means a shallow bowl, weak restoring force, and low natural frequency.
The engineering consequence of approaching instability is that the natural frequency approaches zero. As a column is loaded axially, bending stiffness is reduced by the compressive load. At the buckling load, effective stiffness reaches zero — the natural frequency of the bending mode drops to zero, marking the transition from stable to unstable equilibrium. This is why the buckling analysis of columns reduces to finding where d²V/dq² = 0: the critical load is the stability boundary, not the yield strength. For multi-degree-of-freedom systems, the equivalent condition is that the Hessian matrix of second partial derivatives of V transitions from positive definite to indefinite — a generalization of the scalar criterion.
An important subtlety: these criteria apply to conservative systems where energy is conserved. Dissipative systems (with damping) behave differently — a system can be at a potential maximum yet be stabilized by sufficiently strong damping. Conversely, systems with gyroscopic forces or follower forces may have potential minima that are dynamically unstable. The energy criterion is exact for conservative systems and a reliable first approximation for weakly damped ones, which covers most structural and mechanical equilibrium problems you will encounter.