Trouble understanding angular momentum in relation to orbits

[Contadoren11]

Hi,

Two questions.

1) I'm having trouble understanding the stability of the stable Lagrangian points (L_4 and L_5); Wikipedia explains that if an object in the L_4 or L_5 of a planet is pushed closer towards the common center of gravity of the Sun and the planet, the increased speed that comes from a reduced distance to the center of gravity (due to conserved angular momentum) compensates for this (and the other way around (i.e. if it's pushed farther from the center of gravity)). I don't understand it intuitively nor can I explain it physically. Anyone care to have a go explaining it? Google seems to yield nothing.

2) I don't understand this formula that gives the eccentricity of an orbit expressed by the specific mechanical energy of the object (assuming orbit around the Sun, so virtually a one-body problem):

Intuitively, it makes sense to me that the eccentricity would vary depending on the mechanical energy, but I can't understand the presence of the h^2, nor how (or indeed if at all) the value of h^2 would change if, say, the speed (and thus mechanical energy) of an object in orbit were to suddenly increase, for instance. Again, anyone feel like having a go at explaining to me how the formula makes sense? (Also, am I correct in assuming that the lowest possible energy for an object in orbit would result in a circular orbit, while the eccentricity would gradually increase with mechanical energy?)

Oh, and the bottom paragraph of this site confuses me even more, as it states that the mechanical energy and e aren't connected (the way I understand it): http://courses.physics.northwestern.edu/Phyx125/Orbital%20Stuff.pdf

Any help is much appreciated (my lack of comprehension frustrates me greatly...)

 

[Mark Eaton]

I feel a good way to look at angular velocity is a mass moving in 360 degrees, or as mostly seen as rotating while still moving in one direction. Un like most force or momentum, released in all directions, as a light bulb does or a fire cracker does. Though these electrons, or photons also rotate, as all electro magnetic energy does on 360 degrees. This is a rotating circular moving mass, affected by varied amounts, of the force of gravity. Relating this to orbits. We know that a mass increases with increased velocity. This increase of mass due to increased rate of space displacement, or displacement of interactive momentum. This is seen as friction. Interactive Momentum causes friction. No interactive momentum = No friction =, zero Interactive Momentum.
With increased velocity or increased rate of space displacement, causing varied amounts of gravity, and mass, as the gravitational pull does from getting closer to the force.
Look at this way also. It takes something like a sine wave that rotates, much longer distance or time to move forward, then if you were moving in one straight line, or direction. Also this increased gravity affect this rotation, along with tit's orbit or angular velocity, and angular directional velocity.. the more gravity applied to this rotating moving mass, the more it affects this process on proportional levels. Thus affecting orbits, angular velocity, and gravity, all interrelated. I hope you agree.
I feel a good definition of Rotating Linear Directional Velocity would be 2pi r x frequency x directional velocity = rotating linear directional velocity. From UTOE Copyright M Eaton 2012, and Space displacement and Mass Intrusion Causing Gravity. Copyright M Eaton, 2004
I hope this helps you get the picture of this action in your mind to see, other than in a formula, though without the formula, it may not be useful ,for exacting calculations.

 

[A.T.]

1) I'm having trouble understanding the stability of the stable Lagrangian points (L_4 and L_5);

Are they stable?

 

[Contadoren11]

Are they stable?

They are, no? Hence the massive number of Trojans in Jupiter's orbit.

 

[A.T.]

They are, no?

They are maxima of the potential. Any deviation from their exact position will make an object move away from them.

 

[jbriggs444]

They are maxima of the potential. Any deviation from their exact position will make an object move away from them.

On the contrary. Any deviation from the exact position of L4 or L5 will tend to cause the object to drift back toward them.

 

[HomogenousCow]

They are maxima of the potential. Any deviation from their exact position will make an object move away from them.

L4 and L5 are minimas if I remember correctly.

 

[A.T.]

L4 and L5 are minimas if I remember correctly.

https://en.wikipedia.org/wiki/Lagrangian_point#Mathematical_details

 

[jbriggs444]

Some good discussion at http://physics.stackexchange.com/questions/36092/why-are-l4-and-l5-lagrangian-points-stable. As I read that thread, L4 and L5 are indeed potential maxima. But orbits about those points are nonetheless possible and stable. Coriolis acceleration apparently provides the required impetus toward the Lagrange points in spite of the potential gradient in the opposite direction.

 

[Contadoren11]

Some good discussion at http://physics.stackexchange.com/questions/36092/why-are-l4-and-l5-lagrangian-points-stable. As I read that thread, L4 and L5 are indeed potential maxima. But orbits about those points are nonetheless possible and stable. Coriolis acceleration apparently provides the required impetus toward the Lagrange points in spite of the potential gradient in the opposite direction.

Thanks so much!

 

[A.T.]

Coriolis acceleration apparently provides the required impetus toward the Lagrange points

It's not necessarily towards the Lagrange points, but perpendicular to velocity in the rotating frame. So I can see that when they drift away from the Lagrange points due to the potential gradient, Coriolis acceleration can make them go in a circles, preventing or delaying the escape from the potential plateau around the maximum.