qdot

qdot, typically written as q̇, denotes the time derivative of a quantity q: q̇ = dq/dt. The overdot is a compact notation used across physics and engineering to indicate a rate of change with respect to time. It is commonly applied to scalar or vector quantities and can be extended to multiple generalized coordinates in dynamical systems.

In classical mechanics, if q_i(t) are generalized coordinates, their time derivatives q̇_i = dq_i/dt are called generalized velocities. Many equations of motion are formulated in terms of q and q̇, particularly in Lagrangian and Hamiltonian formalisms. In electrical engineering and related fields, if q represents electric charge, then i = dq/dt, so q̇ corresponds to current. In robotics and control theory, q̇ often represents the velocities of joints or other actuated coordinates. In fluid dynamics and continuum mechanics, q could denote a time-dependent field quantity and q̇ its local time rate of change.

For higher-order derivatives, the notation extends to q̈ for the second derivative d^2q/dt^2, and so forth. When a system involves multiple coordinates, q̇ denotes the vector of all time derivatives [dq_1/dt, dq_2/dt, ...]. The dot notation is most commonly used for time derivatives; when differentiation with respect to other variables is required, other symbols such as dq/dt or ∂q/∂t may be used.

History and usage: the overdot convention has become a standard in physics and engineering to express time rates of change succinctly. It remains a fundamental tool in teaching and analyzing dynamical systems.