Siacci's theorem

In kinematics, the acceleration of a particle moving along a curve in space is the time derivative of its velocity.

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.

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 Because the curvature at a point in an orbit is given by the function f can be conveniently written as a first order ODE The energy conservation equation for the particle is then obtained if f(r) is integrable.

Thus, let C be a space curve traced out by P and s is the arc length of C corresponding to time t. Also, suppose that the binormal component of the angular momentum does not vanish.