Siacci's theorem

Motion of a particle P in a plane.

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

${\displaystyle \mathbf {r} =r\mathbf {e} _{r}.}$

The unit vector e_r is the radial basis vector of a polar coordinate system in the plane. The velocity vector of the particle is

${\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}={\dot {s}}\mathbf {e} _{t}=v\mathbf {e} _{t},}$

where e_t is the unit tangent vector to C. Define the angular momentum of P as

${\displaystyle \mathbf {h} =\mathbf {r} \times m\mathbf {v} =h\mathbf {k} ,}$

where k = i x j. Assume that h ≠ 0. The position vector r may then be expressed as

${\displaystyle \mathbf {r} =q\mathbf {e} _{t}-p\mathbf {e} _{n}}$

in the Serret-Frenet Basis {e_t, e_n, e_b}. 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

${\displaystyle \mathbf {a} =-{\frac {\kappa v^{2}r}{p}}\mathbf {e} _{r}+{\frac {(h^{2})'}{2p^{2}}}\mathbf {e} _{t}=S_{r}\mathbf {e} _{r}+S_{t}\mathbf {e} _{t}.}$

where the prime denotes differentiation with respect to the arc length s, and κ is the curvature function of the curve C. In general, S_r and S_t are not equal to the orthogonal projections of a onto e_r and e_t.

Siacci's theorem can be extended to three-dimensional motions. 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. Then the acceleration vector of P can be expressed as

${\displaystyle \mathbf {a} =-{\frac {\kappa v^{2}r}{p}}\mathbf {e} _{r}+\left(v{\frac {dv}{ds}}+{\frac {\kappa v^{2}q}{p}}\right)\mathbf {e} _{t}.}$

The tangential component is tangent to the curve C. The radial component is directed from the point P to the point where the perpendicular from an arbitrary fixed origin meets the osculating plane. Other expressions for a can be found in,[1] where a new proof of Siacci's theorem is given.

• Acceleration
• Areal velocity
• Central force
• Serret-Frenet equations

• F. Siacci. Moto per una linea plana. Atti della Reale Accademia della Scienze di Torino, XIV, 750–760, 1879.
• F. Siacci. Moto per una linea gobba. Atti della Reale Accademia della Scienze di Torino, XIV, 946–951, 1879.
• E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th edition, Cambridge University Press, Cambridge. Reprinted by Dover Publications, Inc., New York (1944).
• Nathaniel Grossman. The sheer joy of celestial mechanics. Birkhäuser, Basel, 1996.