Question

Consider a double pendulum, with equal masses and equal rod lengths. In addition, the lower pendulum bob carries an electric charge and there is a uniform magnetic field directed perpendicular to the plane in which the two masses oscillate. This can be accommodated in the Lagrangian formalism by a velocity dependent potential proportional to x₂dy₂/dt where "2" denotes the lower particle . Find the canonical momenta for this system. Then find the corresponding Hamiltonian and write Hamilton’s equations for this system.

Answer 1

Let us consider a double pendulum as shown in the figure. Let at any moment the position of the bobs are denoted by and .

Taking the point of suspension of the first bob as origin clearly from the figure,

Therefore,

The kinetic energy of the pendulum is given by,

The potential energy of the pendulum in presence of the magnetic field is given by,

where the second term on the right hand side is due to the magnetic field and 'a' is the proportionality constant.

Therefore the Lagrangian can be written after some simple calculations as,

Therefore canonical momenta are given by,

Now Hamiltonian is given by,

Therefore Hamilton's equation is given by,