Siacci's theorem

In kinematics, the acceleration of a particle moving along a curve in space is the time derivative of its velocity. In most applications, the acceleration vector is expressed as the sum of its normal and tangential components, which are orthogonal to each other. Siacci’s theorem, formulated by the Italian mathematician Francesco Siacci (1839–1907), is the kinematical decomposition of the acceleration vector into its radial and tangential components. In general, the radial and tangential components are not orthogonal to each other. Siacci’s theorem is particularly useful in motions where the angular momentum is constant. In kinematics, the acceleration of a particle moving along a curve in space is the time derivative of its velocity. In most applications, the acceleration vector is expressed as the sum of its normal and tangential components, which are orthogonal to each other. Siacci’s theorem, formulated by the Italian mathematician Francesco Siacci (1839–1907), is the kinematical decomposition of the acceleration vector into its radial and tangential components. In general, the radial and tangential components are not orthogonal to each other. Siacci’s theorem is particularly useful in motions where the angular momentum is constant. Let a particle P of mass m  move in a two-dimensional Euclidean space (planar motion). Suppose that C is the curve traced out by P and s is the arc length of C corresponding to time t. Let O be an arbitrary origin in the plane and {i,j} be a fixed orthonormal basis. The position vector of the particle is The unit vector er is the radial basis vector of a polar coordinate system in the plane. The velocity vector of the particle is where et is the unit tangent vector to C. Define the angular momentum of P as where k = i x j. Assume that h ≠ 0. The position vector r may then be expressed as in the Serret-Frenet Basis {et, en, eb}. The magnitude of the angular momentum is h = mpv, where p is the perpendicular from the origin to the tangent line ZP. According to Siacci’s theorem, the acceleration a of P can be expressed as where the prime denotes differentiation with respect to the arc length s, and κ is the curvature function of the curve C. In general, Sr and St are not equal to the orthogonal projections of a onto er and et. Suppose that the angular momentum of the particle P is a nonzero constant and that Sr is a function of r. Then