Drawing movements as an outcome of the principle of least action (Q1598971)

The authors discuss the kinematics of drawing a picture. They note that it has been established empirically that the angular velocity of drawing a segment is proportional to the two-thirds power of the curvative of the trajectory. They show this implies that the product of tangential velocity and centripetal acceleration is constant over a segment, and that this implies that in rectangular Cartesian coordinates \({x^3d^2y\over dx^2}\) is constant over the segment. This implies that \(x(t)\) and \(y(t)\), \(t\) being the time, minimise the integral \[ \int^{T_1}_{T_0} {x^3 d^2y\over dx^2}\;dt, \] the so-called action of the system. Further, if \(\underline r\) is the position vector, the equation of the path is of the form \({d^2\underline r\over dt^2}+ q(t)\underline r=\underline 0\) (apart from a constant velocity which can be ignored). It is pointed out that the solution of this equation does not include Lissajous figures except as a special case which is treated in detail.