secondorder

Secondorder, often written as second-order or second order, is a term used across mathematics, physics, and engineering to denote concepts that involve degree two. It serves as a general descriptor for second derivatives, second-degree equations, or second-order systems rather than a single object.

In mathematics, second-order differential equations involve the second derivative, such as y''(t) or x''(t). They occur in both ordinary differential equations and partial differential equations, with canonical forms like y'' + p(t) y' + q(t) y = g(t).

In dynamics and control, a second-order system is described by an equation with inertia and damping, typified by a mass-spring-damper model. Key characteristics include a natural frequency and a damping ratio; the associated transfer function often has the form H(s) = ω_n^2 / (s^2 + 2 ζ ω_n s + ω_n^2).

In statistics and signal processing, second-order statistics refer to variance, covariance, and autocovariance, capturing the second moments of random processes and enabling measurements of spread and correlation over time.

In logic and computation, second-order logic extends first-order logic by allowing quantification over predicates and relations, enabling greater expressive power. In optimization, second-order methods such as Newton's method use curvature information from the Hessian to improve convergence.