Question

A material particle with mass M moves under the gravitational effect of the earth. Solve the Hamilton-jacobi equation of the particle.

Answer 1

Solution:

Consider a material particle with mass M moves under gravity. At some instant of time t, let v be the velocity of the body and z be its height above the ground. Then kinetic energy of the body is

Where p being the linear momentum of the particle.

The potential energy (gravitational potential energy) is:

so the Hamiltonian which represents the total energy E of the body is :

......(1)

We may write H as:

........(2)

where S is the Hamilton’s principle function.

so the Hamilton–Jacobi equation is given by:

Using Equation (2), the above equation becomes:

.........(3)

now the general solution of Equation (3) can be written as:

............(4)

From the above, we get

........(5)

Using Equation (5), Equation (3) becomes

Integrating over the variable z, we obtain:

........(6)

Where A is the constant of integration.

Using the result given by Equation (6) in Equation (4), we get:

........(7)

We obtain from the above:

......(8)

Let z = z₀ and p = 0, initially at t = 0.

Then we have

The above gives:

.........(9)

Using Equation (9) in Equation (8), we obtain:

Since at t = 0, z = z₀ we obtain:

The above gives

Hence, we obtain

This is the equation of motion for the particle moves under the gravitational effect of the earth.