^{1}

^{2}

^{3}

^{2}

^{4}

The effect of resonance on the motion of two cylindrical rigid bodies has been studied in the light of Bhatnagar [1] [2] [3] and under some defined axiomatic restrictions. Here we have calculated variation in Eulerian angles due to resonance in terms of orbital elements and unperturbed Eulerian angles.

Russel [

1) The inertia ellipsoids of two rigid bodies

2) Either the distance between

3) The angular velocities of

4) The mass centers

Bhatnagar [

In our present work, we have proposed to extend the work of Bhatnagar et al. [

1) The inertia ellipsoids

2) The angular velocities of

3) Only the periodic terms are taken and other terms are neglected.

4) The two rigid bodies are symmetrical and cylindrical.

On taking axioms second and fourth under consideration

Let

Let

where,

If

Thus the combination of Equations (1), (2), (3) and (4) yields

Since for cylindrical bodies

The generalized momenta

where,

i.e.

From

From

Introducing

Following Brouwer and Clemenc [

where

From Equation (9), we get

where

The Hamiltonian function is given by

where,

The Canonical equations of motion are given by

The Hamilton-Jacobi Equation for the Hamiltonian

The solution of the above equation is given by

Hence the solution of the problem can be given in term of the Keplerian elements

Here

The set of approximate variational equations may be given by averaging the Hamiltonian

where

Here, we observe that by averaging the Hamiltonian, short-periodic terms are eliminated from the Hamilton-Jacobi equation. An approximate set of variational equations are given by

From the above equations, we get

From Equation (14), we have

Also,

For solving the Equations (17) and (18), we should know

For the solution, we will use the Lagrange’s equation of motion

where

From Equation (6), we get

and

From Equation (12), we have

For

The combination of Equations (19), (20), (21), (22) and (23) gives

This is the required Lagrange’s equation of motion in

Again,

Thus the Lagrange’s equation of motion in

Again,

Similarly for

We have assumed that the angular velocities

In terms of the Eulerian angles, we have

In the case of perturbed motion, let us suppose that

where

Since bodies are cylinders hence

where,

We replace

From Equation (25), we have

From Equation (26), we have

Similarly for the body

From Equation (29), we have

Integrating the Equation (36) and putting the value of

where

Considering Kepler’s equation up to the 1^{st} order approximation

Here we can see that if any one of the denominator vanishes, the motion is indeterminate at the point. It depends on the mean motion and the angular velocity of rotation of the body. There are many points at which resonance will occur but for discussion we have consider only one point

From right hand side of Equation (39), we have

We may write the Equation (39) as

where,

The solution of the equation

is periodic and given by

Let

The Equation (41) may be written as

Then

We want to replace

Using Equations (44) and (45), we get

As

Using Equations (40), (43) and (46), we get

Also from the Equation (46), we get

Obviously the Equations (47) and (48) are linear equations in

So solving these equations for these variables, we get

where,

Also,

As

As

Neglecting higher powers of

Here we observe that

The Equation (53) can be written

Now we are considering here the case in which the critical argument is at the point

Taking

As the first approximation, if we put

This is the equation of motion of a simple pendulum. If co-efficient of

If the oscillation is small, we can take

where

Its solution is given by

where

are given by

of Equation (50) and using the Equations (54) and (57), we get.

where

Now we calculate the libration in the variables

Integrating the Equation (33) and ignoring secular terms, we get

where constants of integration are taken to be zero.

Putting the value of

where

And the perturbed solution for

Obviously in the case of one of the denominator becomes zero, the motion cannot be determined at that point, known as critical point and hence resonance arise at that point. In this case usual method fails to determine the motion, so for the present purpose the present purpose we will use the method as that of

The equation for

On taking the first approximation, we can see that critical argument oscillates about

where

Thus amplitude and period of vibration are given by

The solution for

where

By proceeding exactly same as above case, we can find out the libration in the variables

and the solution up to first order approximation of

Also we see that in the libration in the variable

makes oscillation about the value

The solution of

Solution for

Also when we consider the libration in the variable

argument

The solution of

And the solution for

where

From the Equation (37) it is obvious and

We have from Equation (16),

Integrating the Equation (17) with respect to

Initially at

where,

Again from Equation (18), we have

Initially at

where,

Now we find the time

We have

Let

Again from the Equations (13) and (36), we get

Let

The corresponding change in

In the section of “Equations of motion”, we have derived the perturbed and unperturbed Hamiltonian and the canonical equations of motion with respect to the complete Hamiltonian H where are generalized co-ordinates and are the corresponding generalized momenta. In Section 3, unperturbed solutions can be derived by usual course from the Kepler’s equation of motion. For appropriate variational equation, the required generalized co-ordinates have been calculated in Section 5. In section 6, the effect of resonance has been shown in the solutions of the equations of motion of two cylindrical rigid bodies. In Section 7 and 8, equilibrium points have been calculated in terms of Eulerian angles for both the bodies.. Finally the appropriate variational equation in Section 4 has been completely solved in Section 9.

The tools obtained in different sections of the manuscript can be used to discuss the motion of cable connected two artificial satellites. Thus, we may conclude that this article is highly applicable in Astrophysics and Space Science.

Hassan, M.R., Kumari, B., Hassan, Md.A., Singh, P. and Sharma, B.K. (2016) Effect of Resonance on the Motion of Two Cylindrical Rigid Bodies. International Journal of Astronomy and Astrophysics, 6, 555-574. http://dx.doi.org/10.4236/ijaa.2016.64040