Problem 5-1: Spherical pendulum
Write the Lagrangian for a spherical pendulum (a bob of mass \(m\) attached to a mass-less rigid rod of length \(l\)) using spherical coordinates and identify all conserved quantities. A spherical pendulum is the generalized case of the simple pendulum when the mass is not constrained to swing in a plane.
Problem 5-2: Two masses on a spring
Two masses, \(m_1\) and \(m_2\), are connected by a spring of rest length \(l\) and spring constant \(k\). The two masses are constrained to move in one dimension, along the x-axis, on a friction-less surface.
a) Give the Lagrangian for the system and write the equations of motion for the two masses
b) Show that the total linear momentum in the x-direction, \(P_x\), is conserved (\(P_x=m_1v_1+mv_2\))
c) List all conserved quantities for the system.
Problem 5-3: Arbitrary potential
Calculate the conserved quantities for the Lagrangian:
where \(a\) and \(b\) are constants, and V() is some unknown function of the linear combination \(aq_1-bq_2\).
Problem 5-4: Spring pendulum with two masses
The pendulum in figure 5-4 is constructed with two masses, \(m_1\) and \(m_2\), a spring of constant \(k\) and rest length \(d\). Mass \(m_2\) is fixed at the end of a mass-less rigid rod of length \(l\), while \(m_1\) can slide without friction along the rod and is connected to the spring.
a) Give the Lagrangian for the system and write the equations of motion for the two masses
b) List all conserved quantities for the system.