dthetadt

dthetadt, often written as θ̇, is the time derivative of the angle θ in polar or cylindrical coordinates. It represents the instantaneous rate at which the angle changes with time and is a central quantity for describing angular motion in two-dimensional and rotational systems. In many contexts θ̇ is also called the angular speed (though angular speed can be defined more broadly as a non-signed magnitude).

In a planar motion with position (x(t), y(t)), θ is the angle that the position vector makes with the x-axis, commonly defined by θ = arctan2(y, x). Differentiating gives θ̇ = (x dy/dt − y dx/dt) / (x^2 + y^2). This shows that the angular rate depends on both the instantaneous velocity components and the distance from the origin. The tangential component of velocity is v_t = r θ̇, where r = sqrt(x^2 + y^2) is the radial distance. The total speed satisfies v^2 = (dr/dt)^2 + (r θ̇)^2.

In cylindrical or spherical coordinates, θ̇ is the rate of change of the azimuthal angle around the primary axis (often the z-axis). For pure rotation about a fixed axis with constant radius, θ̇ equals the scalar angular velocity ω about that axis. More generally, the angular velocity vector in three dimensions has magnitude |ω| = θ̇ when rotation is about a fixed axis, but the full description of orientation change requires a vector form.

Notes: θ̇ is undefined at the origin (where r = 0) since the angle is not defined there, and numerical evaluation can be sensitive to angle wrap-around near 2π.