Lagrangian and Hamiltonian Dynamics

[yukawa]

1. (from Marion 7-29)

A simple pendulum consist of a mass m supended by a massless spring with unextended length b and spring constant k. The pendulum's point of support rises vertically with constant acceleration a. Find the Lagrange equation of motion.

Does the motion of the mass confided in a plane? Or i need to consider the motion in 3D?
Away from the gravitational potential energy, do i also need to consider the elastic potential energy in the spring? Is it equal to 1/2 k * b^2?

2. (from Marion 7-33)

Finding the Hamiltonian equation of motion of a double atwood machine
http://thumb12.webshots.net/t/64/564/6/16/84/2610616840102234032jMZtum_th.jpg

Using the generalized coordinates x and y in the figure.
i found:
p_x = m_1 * dx/dt +m_2*(dy/dt - dx/dt) + m_3 *(dx/dt + dydt)
p_y = m_2*(dy/dt - dx/dt) + m_3 *(dx/dt + dydt)
H = T + U =1/2*m_1*(dx/dt)^2 + 1/2 * m_2 *(dy/dt - dx/dt)^2 + 1/2*m_3 *(dx/dt + dydt)^2 - m_1 *g*x -m_2*g*(l_1 - x +y ) - m_3*g*(l_1+l_2-x-y)

How can i write H in terms of the generalized momentum (p_x and p_y) and coordinates only ?

 

[Jhero]

Dear Marion,

Thank you for your interesting questions regarding the Lagrange and Hamiltonian equations of motion for a simple pendulum and a double Atwood machine.

For the simple pendulum, the Lagrange equation of motion can be written as:

m(b+aθ) + kθ - mgcosθ = ma

Where m is the mass of the pendulum, b is the length of the unextended spring, a is the vertical acceleration of the point of support, k is the spring constant, g is the acceleration due to gravity, and θ is the angle of displacement.

This equation assumes that the motion of the pendulum is confined to a plane. However, if the motion is not confined to a plane, then the equation would need to be modified to include the motion in the third dimension.

In addition to the gravitational potential energy, it is important to consider the elastic potential energy in the spring. The elastic potential energy can be calculated as 1/2 * k * b^2, as you mentioned.

For the double Atwood machine, the Hamiltonian equation of motion can be written as:

H = T + U = 1/2 * m1 * (dx/dt)^2 + 1/2 * m2 * (dy/dt - dx/dt)^2 + 1/2 * m3 * (dx/dt + dy/dt)^2 - m1 * g * x - m2 * g * (l1 - x + y) - m3 * g * (l1 + l2 - x - y)

To write this equation in terms of the generalized momentum and coordinates, we can substitute the expressions for the generalized momentum (p_x and p_y) into the equation. This would result in a Hamiltonian equation that is solely in terms of the generalized coordinates and their derivatives.

I hope this helps answer your questions. Please let me know if you need any further clarification or assistance.