PHY4221 Homework Assignment 1 Solution
(Written by: Leung Shing Chi)
1. (In this problem, we use Einstein’s summation convention, i.e.
, also we use
the following notation:
3
1
ii
i
AA
,
i
ij
j
A
A
x
)
Given the Lagrangian to be
1
(,)
2
q
Lm
v
v
q
r
t
v
A
r
t
c
, we apply the Euler-Lagrange
Equation (ELE) to
L
:
0
dL L
dt v
x
where
1, 2, 3
i
(3 equations in total).
,
i
i
jj
i
Lq
Pm
v
A
vc
qv
xx
c
A
Therefore, we find
,
i
i
i
A
dL d
q
mv
v A
dt v
dt
c
t
j
i
j
(It should be noted that this result
arises as the usage of ordinary derivative) and we take the ELE to be:
,,
,
0
1
i
i
j
i
j
i
j
i
j
A
dq
q
mv
v A
q
v A
dt
c
t
c
A
mv
q
v A
v A
dt
c
t
c
,
j
i
j
j
i
This, in fact, is the solution by observing the following relation among the scalar and vector
potential and the
E
and
B
fields.
,
1
i
jkj
k
i
i
m
nm
n
k i
j
k
ji
j
j j
i
i
A
E
ct
vB vB
vA
vA vA
,
Certainly, to achieve this result, we need some relations between
and
, namely:
ijk
mnk
im
jn
in
jm
Finally, it is now obvious to see the equation of motion (EoM) to be: