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
i
i
i
i
i
A A
A A
,
i
i j
j
A
A
x
)
Given the Lagrangian to be
1
( , )
( , )
2
q
L
mv v
q
r t
v
A r t
c
, we apply the Euler-Lagrange
Equation (ELE) to
L
:
0
i
i
d
L
L
dt
v
x
where
1,2,3
i
(3 equations in total).
,
i
i
i
i
j
j i
i
i
L
q
P
mv
A
v
c
L
q
q
v
x
x
c
A
Therefore, we find
,
i
i
i
A
d
L
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
i
i
j
i j
A
d
q
q
mv
v A
q
v A
dt
c
t
c
A
d
q
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
i
i
j
k
jki
i
m n
mnk
ijk
j
i j
j
j i
i
A
E
c
t
v
B
v B
v
A
v A
v A
,
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:

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