September 10, 1999

PHY 711 – Lecture notes on Lagrangian for Electric and Magnetic Fields

For simplicity, consider a Lagrangian for a single particle having the form (in Cartesian coordinates) L(x, y, z, ẋ, ẏ, ż, t) ≡ T − U. The Euler-Lagrange equations have the form:

\frac{d}{dt} (\frac{∂L}{∂ẋ}) − \frac{∂L}{∂x} = 0,  (1)

with similar equations for y and z. We can show that this form is consistent with Newton’s Laws if the potential function U takes the form:

U = U^{0}(x, y, z, t) + U^{EM}(x, y, z, ẋ, ẏ, ż, t),  (2)

where U^{EM} represents the interaction of our particle (having charge q) with an electric field E and magnetic field B where we can represent the fields in terms of the scalar and vector potentials:

E = −∇ϕ − \frac{1}{c} \frac{∂A}{∂t} and B = ∇ × A.  (3)

We must find U^{EM} which is both consistent with the Euler-Lagrange Eq.(1) and with the Lorentz force (written in the x direction):

F_{x} = q(E_{x} + \frac{1}{c}(ṙ × B)⌋_{x}) = −\frac{∂U^{EM}}{∂x} + \frac{d}{dt} (\frac{∂U^{EM}}{∂ẋ} ) .  (4)

We note that the magnetic field terms can be evaluated:

ṙ × (∇ × A)⌋_{x} = ẏ (\frac{∂A_{y}}{∂x} − \frac{∂A_{x}}{∂y} ) − ż (\frac{∂A_{x}}{∂z} − \frac{∂A_{z}}{∂x} ) .  (5)

The right hand side of Eq.(5) (with the addition and subtraction of a convenient term) can be written:

\underbrace{ẋ\frac{∂A_{x}}{∂x} + ẏ \frac{∂A_{y}}{∂x} + ż \frac{∂A_{z}}{∂x}}_{∂(ṙ·A)/∂x} − \underbrace{ẋ\frac{∂A_{x}}{∂x} − ẏ \frac{∂A_{x}}{∂y} − ż \frac{∂A_{x}}{∂z}}_{−dA_{x}/dt+∂A_{x}/∂t},  (6)

where we are assuming that A_{x} = A_{x}(x, y, z, t). Noting that A_{x} = ∂(ṙ · A)/∂ẋ, the electromagnetic force can thus be written:

F_{x} = \underbrace{−q \frac{∂ϕ}{∂x} − \frac{q}{c} \frac{∂A_{x}}{∂t}}_{qE_{x}} +\frac{q}{c} (\underbrace{\frac{∂(ṙ · A)}{∂x} − \frac{d}{dt} \frac{∂(ṙ · A)}{∂ẋ} + \frac{∂A_{x}}{∂t}}_{(ṙ×B)⌋_{x}}) .  (7)

Simplifying this equation, we obtain

F_{x} = −\frac{∂}{∂x} (qϕ − \frac{q}{c}ṙ · A) − \frac{d}{dt} \frac{∂}{∂ẋ} (\frac{q}{c}ṙ · A) .  (8)

Thus, we finally have the result

U^{EM} = qϕ − \frac{q}{c}ṙ · A.  (9)