Lagrangian Mechanics Made Simple Part 5 The Stable Lagrange Points and the Coriolis Force

# Lagrangian Mechanics Made Simple Part 5 The Stable Lagrange Points and the Coriolis Force

Last month we discussed the motion of a small body in the gravitational field of two larger bodies. We showed that the Lagrangian was

 (1)

Here, is the mass of the smaller body. is the angular velocity of the two larger bodies in their orbits around each other. and are the distances from each of the larger bodies to their common center of rotation, given in vector form by

 (2)

Finally, is the distance between the two bodies.

We described the five solutions to the equations of motion (known as the Lagrange points) where is a constant. Two of these solutions,

 (3)

are known to be stable in certain cases. Our goal this month is to show this.

# Perturbations About the Stability Points

In the last column, we briefly discussed the stability of the solutions by perturbing them a small constant distance . To do a full stability analysis, we examine the dynamics of these perturbations by allowing to be a function of the time . That is, we define

 (4)

In terms of , the Lagrangian is

 (5)

where we have used the fact that and are perpendicular to simplify the equation.

Since we are only interested in small perturbations, we expand the Lagrangian in powers of . Using the fact that

 (6)

for small values of , we find that the Lagrangian is

 (7)

Dropping terms of and higher from the Lagrangian gives linearized equations of motion for ,

 (8)

where the in the and equations corresponds to the choice of a solution .

# Normal Mode Frequencies and Stability

Examining the equation first, we see that it has the solution

 (9)

where and are some constants. This shows that a small perturbation in the direction will result in the particle oscillating about with frequency . The solutions and are called normal modes of the perturbation, and is the normal mode frequency.

To see whether perturbations in the and directions result in the same stability as a perturbation in the direction, we will find the normal mode frequencies of the and modes. That is, we will look for solutions to the and equations of motion of the form

 (10)

where , and the exponential has the usual definition

 (11)

and the useful property

 (12)

The solutions and in the case are simply linear combinations of two normal modes with . If is real (has no imaginary part proportional to ), the perturbation has an oscillatory normal mode, like the one in . An imaginary part in results in exponential growth of the perturbation, and will quickly drive us out of the region where our displacement from the stability point can be considered small. Thus, the Lagrange points are stable if all their normal mode frequencies are real.

Inserting the trial solution (10) in the and equations of (8) gives

 (13)

For (13) to have a solution, the matrix must have a zero eigenvalue. This can happen if and only if its determinant is zero. Setting the determinant to zero gives an equation for ,

 (14)

Notice that the has vanished from the equation. The fourth and fifth Lagrange points have the same normal frequencies, as should be expected from symmetry.

Eq. (14) is quadratic in , so we can solve it to obtain

 (15)

There are two possible solutions for , giving four solutions for . This is to be expected, since we started with two second order differential equations. The critical question is whether all the values of are real. If the argument of the square root is negative, then is complex, and so is , leading to instability. If, on the other hand, the argument of the square root is positive, we know it is not more that , since . This gives values of between and , and all values of are real. The condition for stability is

 (16)

This requires that the smaller of the masses of the two large bodies not be more than 3.7% of the larger mass.

This concludes this series of articles. It's been fun, and I hope some of the readers have found them useful.

Ron Steinke 2002-09-27