Mathematical Physics Vol 1

Chapter 4. Field theory

156

c) In the time interval ∆ t the particle moved from position M to position N . The area "swept out" by the position vector, in that time interval, is approximately1 equal to half the area of the parallelogram with sides r and ∆ r , that is, r × ∆ r . The approximate area "swept out" by the position vector in a unit of time is thus equal to ∆ r ∆ t . Hence the instantaneous change in surface area over time is 1 2 1 2 r × where v is the current velocity of the particle. The value H = 1 2 r × d r d t = r × v is called sector velocity. As the sector velocity is constant, in the case of central forces (see (4.181)), we obtain 1 2 H = = c , where c is a vector constant. Given that r · H = r · = 0 (condition of coplanarity) it follows that the point moves in a plane. Namely, given that H is constant, it follows that this vector does not change its direction. However, the vector is perpendicular to the plane formed by the vectors r and v , so it can be concluded that the point moves in a plane with its normal in the direction of H . d) A planet (such as the Earth) is attracted by the Sun according to Newton’s law of gravity which reads: any two bodies of mass m and M are attracted to each other by a force equal to F = γ Mm r 2 r 0 , where r is the distance between the centers of the two bodies, and γ the universal gravitational constant. Let m and M be the masses of the planet and the Sun, respectively. Let us chose a coordinate system with its origin in O , placed in the center of the Sun. Then the equation of motion of the planet under the influence of the Sun is m d 2 r d t 2 = − γ mM r 2 r 0 , or d 2 r d t 2 = − γ M r 2 r 0 . According to c) the planet moves around the Sun so that its position vector sweeps out equal areas during equal intervals of time. This law is known in the literature as Kepler’s second law (law of equal areas). 1 2 r × d r d t 1 2 r × d r d t equal to lim ∆ t → 0 1 2 r × ∆ r ∆ t = 1 2 r × v ,

Exercise 96 Show that a planet moves around the Sun in an ellipse, where the Sun is in the focus of that ellipse.