Hamiltonian equations of motion question

[Pyroadept]

Homework Statement

Question 1 on the following page: http://www.maths.tcd.ie/~frolovs/Mechanics/PS10.pdf

It's the second part I'm stuck on ('Explain why the equations of motion do not...')

Homework Equations

The Attempt at a Solution

I first found equations for x_1 'dot' and x_2 'dot' (using p = dL/dx'dot')
and rearranged to get L(x_i,p_i)

Then H(p,x) = p_1.x_1'dot' + p_2.x_2'dot' - L(x_i,p_i)

and Hamilton's equations of motion:
p_i 'dot' = -dH/dx_i
q_i 'dot' = dH/dp_i

I don't know how to solve the second part though. Is it dependent on my answers or is it something about equations of motion in general?

Thanks for any help!

 

[mighty2000]

Thank you for your question. The second part of the question asks you to explain why the equations of motion do not uniquely determine the motion of the system. This is a common issue in classical mechanics and is known as the "two-body problem."

The reason for this is that the equations of motion, which you have correctly derived, do not take into account the initial conditions of the system. In other words, the equations of motion alone cannot tell us the exact positions and velocities of the particles at a given time. This is because there are an infinite number of possible initial conditions that could result in the same equations of motion.

To fully determine the motion of the system, we need to also know the initial positions and velocities of the particles. This is usually done by specifying the initial conditions, such as the initial positions and velocities of the particles, and then solving the equations of motion to determine the motion of the system.

I hope this explanation helps. If you have any further questions, please don't hesitate to ask.