Proving the Runge-Lenz vector is zero for circular orbits

[samjohnny]

Homework Statement

Please find it attached.

Homework Equations

The Attempt at a Solution

For the first part of the question I managed to show that the vector is a constant of the motion by differentiating it with respect to time and showing that it yields a value of zero.

As for the second part where I need to calculate the magnitude and direction of the vector then I'm not sure how to go about doing that explicitly. I know that the way vector is defined is such that it is equivalent to the eccentricity vector and so for the circle part it would necessarily be equal to zero. But to show that explicitly I'm not to sure. Any ideas?

 

[Kiarash]

Homework Statement

Please find it attached.

Homework Equations

The Attempt at a Solution

For the first part of the question I managed to show that the vector is a constant of the motion by differentiating it with respect to time and showing that it yields a value of zero.

As for the second part where I need to calculate the magnitude and direction of the vector then I'm not sure how to go about doing that explicitly. I know that the way vector is defined is such that it is equivalent to the eccentricity vector and so for the circle part it would necessarily be equal to zero. But to show that explicitly I'm not to sure. Any ideas?

Hi,

First of all, notice that the Rung-Lenz Vector could be written by velocity of the particle, because momentum divided my mass is equal to velocity, and K is a constant. Obviously, it is necessary to know initial conditions of the particle, so assume that we know the initial distance and initial velocity, and therefore, the Runge-Lenz Vector (I show this with A). Let me show the particle position vector's with R. Now, calculate Dot Product of A and R in two different ways; in the first path, consider the A has its initial value, but in the second way, put its definition expression. Hence, by some calculation, you can find the R in terms of angle (from a specific axis.)

 

[samjohnny]

Hi,

First of all, notice that the Rung-Lenz Vector could be written by velocity of the particle, because momentum divided my mass is equal to velocity, and K is a constant. Obviously, it is necessary to know initial conditions of the particle, so assume that we know the initial distance and initial velocity, and therefore, the Runge-Lenz Vector (I show this with A). Let me show the particle position vector's with R. Now, calculate Dot Product of A and R in two different ways; in the first path, consider the A has its initial value, but in the second way, put its definition expression. Hence, by some calculation, you can find the R in terms of angle (from a specific axis.)

Thank you for the reply. Would it not be the case that, by dotting the Runge Lenz vector with R, the resultant expression would describe motion of the object? I'm not clear on what relation this would have to finding the magnitude/direction of the vector in the case of circular, elliptical etc orbits.

 

[Kiarash]

Thank you for the reply. Would it not be the case that, by dotting the Runge Lenz vector with R, the resultant expression would describe motion of the object? I'm not clear on what relation this would have to finding the magnitude/direction of the vector in the case of circular, elliptical etc orbits.