A shadow of the repulsive Rutherford scattering and Hamilton vector

The fact that repulsive Rutherford scattering casts a paraboloidal shadow is rarely exploited in introductory mechanics textbooks. Another used construction such textbooks the Hamilton vector, cousin of more famous Laplace-Runge-Lenz vector. We will show how latter (Hamilton's vector) can be to explain and clarify former (paraboloidal shadow).