Quantity conservation and calculus

[C.E]

Hi, I was hoping someone could offer some guidance with the following, I don't even know how to start it.

The motion of a planet of mass m around the sun of mass M is governed by the following equation (note r is a vector):

d^2r = -k r (this should be the second derivative of vector r dt but I could not get it to underline)
dt^2 ||r||^3

where K=Gm (G is the gravitational constant) and r(t) is the position of the planet relative to the sun.

1. Show the following quantities are conserved.

a. Total energy (the sum of the potential and kinetic energies of the planet).

b. The angular momentum of the planet J.

c.The lenz- Runge vector L= dr x J - mk r
dt ||r|| (the dt should be under the dr and the ||r|| under the J - mkr )
Hint: (a x b) x c = (a.c)b- (b.c)a (note a.c is scalar product of a and c similarly b.c is scalar product of b and c).

2. (a) Interpret the constancy of J geometrically.
(b). Assume the planet moves in an ellipse with one focus at the sun, show by considering the point when the planet is furthest from the sun that L points in the direction of the major axis of the ellipse.

 

[LightHero]

For 1. I would start by taking the derivatives of both sides of the equation and showing that they are equal to zero. For 2. (a) I'm not sure how to interpret the constancy of J geometrically and for 2. (b) I'm not sure how to show that L points in the direction of the major axis of the ellipse. Any help would be greatly appreciated! Thanks!

 

[astrokreng]

Hello,

Conservation laws are fundamental principles in physics that describe the behavior of physical systems. In this case, we are dealing with the conservation of energy, angular momentum, and the Lenz-Runge vector for a planet orbiting the sun. These quantities are conserved because the gravitational force between the planet and the sun is a conservative force, meaning that it does not depend on the path taken by the planet.

1. To show that the total energy is conserved, we can use the equation for the total energy of the system:

E = K + U = 1/2 mv^2 - GmM/r

where K is the kinetic energy, U is the potential energy, and v is the velocity of the planet. Taking the derivative with respect to time, we get:

dE/dt = mv(dv/dt) - GmM(dr/dt)/r^2

Using the given equation for the motion of the planet, we can substitute dv/dt and dr/dt to get:

dE/dt = -kv^2/r^2 - kv^2/r^2 = 0

Since the derivative of the total energy is zero, we can conclude that the total energy is conserved.

To show that the angular momentum is conserved, we can use the definition of angular momentum:

J = r x p = r x mv

where p is the momentum of the planet. Taking the derivative with respect to time, we get:

dJ/dt = r x (mdv/dt) + (dr/dt) x mv

Using the given equation for the motion of the planet, we can substitute dv/dt and dr/dt to get:

dJ/dt = r x (-kr/r^3) + (dr/dt) x mv

= -kr x (r/r^3) + (dr/dt) x (dr/dt)

= -kr x (1/r^2) + (dr/dt) x (dr/dt)

= 0

Since the derivative of the angular momentum is zero, we can conclude that the angular momentum is conserved.

To show that the Lenz-Runge vector is conserved, we can use the definition:

L = dr x J - mk r/||r||

where k is the gravitational constant. Taking the derivative with respect to time, we get:

dL/dt = (d^2r