Question

Q1: A particle of mass m and electric charge q moves in a plane under the influence of a central potential V(r) and a constant uniform magnetic field B, perpendicular to the plane, generated by a vector potential A=1/2B×r A) Write the Lagrangian of the system. B) Find the Hamiltonian. C) Find the equations of motion aid solve them. L=T−(ϵq​)A⋅∇L=T−qv+cq​A⋅∇​

Answer 1

The Lagrangian of the system is L = T - qV(r) + qA·∇. The Hamiltonian is found using the Legendre transform, H = p·v - L. The equations of motion can be derived using the Euler-Lagrange equation and can be solved to find the particle's motion.

Step-by-step explanation:

The Lagrangian of the system can be written as L = T - qV(r) + qA·∇ where T is the kinetic energy of the particle, V(r) is the central potential, q is the electric charge of the particle, and A is the vector potential. The Hamiltonian, which is the total energy of the system, can be found using the Legendre transform: H = p·v - L where p is the momentum of the particle and v is its velocity. The equations of motion for the particle can be derived using the Euler-Lagrange equation: ∂L/∂q - d/dt(∂L/∂v) = 0. Solving these equations will give the motion of the particle under the influence of the central potential and magnetic field.