qddot

qddot, written as q̈ in mathematical notation, denotes the second time derivative of a generalized coordinate q(t). In plain-text environments it is commonly written as qddot. It represents the acceleration associated with the coordinate, i.e., how fast q̇ changes over time; q̇ is the first derivative, the velocity along the generalized coordinate.

In classical mechanics, a system described by generalized coordinates q(t) has velocities q̇ and accelerations q̈. For example, a particle constrained to move along a straight line with coordinate x(t) has ẍ as its linear acceleration. For a robotic arm with joint angles θ(t), θ̈ gives the angular acceleration of each joint. In multi-dimensional systems, q is a vector of coordinates and q̈ is the vector of accelerations.

In Lagrangian mechanics, q̈ arises in the Euler-Lagrange equations; the general form involves d/dt(∂L/∂q̇) which is related to q̈ when L depends on q̇. In dynamics computations, q̈ is the quantity solved for in equations of motion, e.g., M(q) q̈ + C(q, q̇) q̇ + g(q) = τ, where M is inertia, C is Coriolis/centripetal terms, g gravity, and τ input torques.

In broader contexts, q̈ is used in numerical simulation to integrate motion and in control theory as part of second-order dynamics. The concept generalizes to non-Cartesian coordinates where q̈ represents the second derivative of a generalized coordinate and may involve curvature terms when coordinates describe motion on curved spaces.