Question

The Lagrange of a charge particle moving in an electromagnetic field is given by L=1/2mv^2+q (V.A-phi). How do I derive an expression for the Hamiltonian and the corresponding Hamiltonian equation?

Answer 1

To derive the expression for the Hamiltonian and the corresponding Hamiltonian equations, we start with the Lagrangian:

L = (1/2)mv^2 + q(V - A·Φ),

where m is the mass of the particle, v is its velocity, q is its charge, V is the electric potential, A is the vector potential, and Φ is the magnetic flux.

The Hamiltonian (H) is defined as the Legendre transformation of the Lagrangian. To obtain it, we follow these steps:

1. Define the generalized momenta: p = ∂L/∂v = mv + qA.

2. Write the expression for the Hamiltonian: H = Σ(pv) - L, = p·v - L, = (mv + qA)·v - [(1/2)mv^2 + q(V - A·Φ)], = (mv)^2 + qv·A - (1/2)mv^2 - qV + qA·Φ, = (1/2)mv^2 + qv·A - qV + qA·Φ.

Therefore, the Hamiltonian is given by:

H = (1/2)mv^2 + qv·A - qV + qA·Φ.

To derive the Hamiltonian equations, we use the canonical equations of motion:

dq/dt = ∂H/∂p, dp/dt = -∂H/∂q.

Taking the derivatives, we have:

dq/dt = ∂H/∂p = v,

dp/dt = -∂H/∂q = -∂[(1/2)mv^2 + qv·A - qV + qA·Φ]/∂q, = -(-qv·∂A/∂q - q∂V/∂q + q∂(A·Φ)/∂q), = -(-qv·0 - q∂V/∂q + q(∂A/∂q)·Φ + qA·∂Φ/∂q), = q∂V/∂q - q(∂A/∂q)·Φ - qA·∂Φ/∂q.

These equations, dq/dt = v and dp/dt = q∂V/∂q - q(∂A/∂q)·Φ - qA·∂Φ/∂q, are the Hamiltonian equations of motion for the charge particle moving in an electromagnetic field.