Transcript 投影片 1

3. Classical Physics in Galilean and
Minkowski Spacetimes
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
The Action Principle in Galilean Spacetime
Symmetries and Conservation Laws
The Hamiltonian
Poisson Brackets and Translation Operators
The Action Principle in Minkowski Spacetime
Classical Electrodynamics
Geometry in Classical Physics
Purpose: Set out classical mechanics in a form suitable for generalization to
general relativity & quantum mechanics.
Two basic ideas:
1.
Action (contains all dynamical information of the system ).
2.
Symmetry ~ conservation law.
Comment:
Not all systems can be described by an action, e.g. systems with dissipation.
However, all known fundamental (microscopic) interaction can.
3.1.
The Action Principle in Galilean Spacetime
Euler-Lagrange Equations:
Assumptions:
1. State of a microscopic system is uniquely specified by
(Equations of motion are 2nd order in time)
2. Least action principle.
t2
The action S of a system is defined as S   dt L .
 q , q    q1,
, q1,
L  L  q, q, t  = Lagrangian.
t1
Least action principle: Actual evolution of the system minimizes S, i.e., δS = 0.
2
 L

 L

L
L
  dt    q 
  q    dt 
 qi 
 qi 

q

q

q

q


i
 i

t1
t1
t2
t2
 S   dt  L
t1
t
2
2
t2
L
L d
 d L 

L
dt

q

dt

q


q

dt
t qi i t qi dt i q i   dt q   qi
i
i 

1
1
t1
t1
t2
t
t
2
 d L 
 d L 
   dt 

q


dt
 q
 i



 dt q 
 dt qi 
t1
t1
t2
t

t2
 L d L 

 qi  0

 qi dt qi 
 S   dt 
t1
L d L

0
qi dt qi
Euler-Lagrange equations
L d L

0
q dt q
Generalized momentum pi conjugate to qi is defined as
L
pi 
or
qi
L
p
p
For L = T  V, the Euler-Lagrange equations become Newton’s equations:
dpi L
V


 fi
dt qi
qi
or
dp L
V


f
dt q
q
Spacetime Symmetries
Invariance under Time Translations:
Galilean time has its own metric
dτ2 = g(t) dt2
where t = time coordinate,
g(t) = metric tensor,
and
 = linear measure of the time manifold,
i.e.,
d = "length" of time between 2 points at t and t + dt.
t = aτ+b →
g(τ) = 1/a2
Any t with g(t) = const is called a proper time coordinate.
Newton's 1st law can be used to check if t is a proper time.
By definition, the Lagrangian of a free particle is the same for all proper time t,
L0  x, x, t  t  L0  x, x, t   L0  x, x 
i.e., L0 is invariant under proper time translations.
↔
t = aτ+b
Invariance under Spatial Translations:
Space in Galilean spacetime is Euclidean.
→ in Cartesian coordinates, g = I.
g is invariant under any spatial translation → Euclidean space is homogenous.
In an inertial frame, L0 is invariant under spatial translations:
L0  x, x   L0  x  x, x   L0  x 
↔
x is an inertial frame
Invariance under Rotations:
The Euclidean metric tensor is invariant under any spatial rotations.
→ Euclidean space is isotropic.
In an inertial frame, L0 must be invariant under any rotations:
L0  Rx   L0  x   L 0  x 
where R is a rotation operator and
x
xx
Invariance under Galilean Transformations:
Coordinates of S and S', moving with relative velocity v, are related by
x = x  v t
t=t
If S is an inertial system in which Newton's 1st law holds, so is S'.
→
Newtonian physics is invariant under Galilean transformations.
It obeys Galilean relativity.
For a free particle with Lagrangian
L0  x   L0  X 
where
X 
1 2
x
2
the Euler-Lagrange equations are
d   X d L0 
d   L0 
d  d L0 
0 
  d t  x d X    x

d t  x 

 dt  d X 
where
d 2 L0
d  d L0 
  d L0 

  x 
  xx d X 2
dt  d X 
x  d X 
d  d L0 
d L0
d 2 L0
0x
x 
 x x  x 
 x
dX
dt  d X 
dX
dX2
d L0
X
 1


x

x

x
x x  2

x
d L0
dX
 x x  x 
d 2 L0
dX
2
0
Under the Galilean transformation
x  x  x  v
x  x  x
1
1 2
X  X   x  v   x  v   X  x  v  v
2
2
The Euler-Lagrange equation can remain unchanged iff
d 2 L0
dX
→
d L0
dX
L0 
2
0
 const  m
1
m x 2  const
2
Lagrangian of free particle is determined by spacetime symmetries.
Lagrangian
L0 = T is determined solely by the symmetries of the Galilean spacetime.
L of a system of non-interacting particles must be a sum of Ti of all particles.
If the particles interact via a potential V x i  ; x i 
x  x  ; x 
V  V x  x  ; x  x 
1. Invariance under spatial translation requires V  V
2. Invariance under Galilean transformation requires
, then
i
j
i
i
j
i
j
3. Invariance under rotation requires V to depend only on rotational scalars such as
x
→
i
 x j    xk  xl  ,
L
i
L
 x i  x j    x k  x l    x m  x n 




1
mi xi2  V ri j ;rk l 
2
where
etc
ri j  x j  xi
1
m x 2  V  x  is not of this form since V is produced by external sources.
2
3.2.
Symmetries and Conservation Laws
Conservation of Energy:
Invariance under time translation →
→
d L d qi  L d qi  L


dt
d t  qi d t  qi
 qi
L  L q, q 
or
L
0
t
L
L
L
L
 q
 q
 qi
q
q
 qi
 qi
d  L 
d   L  d qi  L

 qi 
 qi


d t   qi 
d t   qi  d t  qi
→
 dE
d  L
q

L
0
 i

d t   qi
 dt
where
E  qi
L
L
L
 L  q
q
 qi
 energy of the system.
Invariance of time translation
→
conservation of energy
( for any isolated system )
Noether's Theorem
Consider transformation
or
with
qi  qi   fi
qi  qi   fi
q  q  f
q  q  f
f  f  q, q,t 
 0
L
L 
L q   f , q   f , t  L  q, q, t      f 
 f   O  2 
q 
 q


L is invariant under the transformation if
L  q   f , q   f , t   L  q, q, t 
dL
 lim
0


0
d

0
→
i.e.,
d  L 
L
d  L 
d

f


f


f

pf 





d t  q 
q
d t  q 
dt
L
 f  p  f  F  const
q
(Noether's theorem)
L
L
f 
f  0
q
q
Examples
Conservation of Total Linear Momentum for a system of particles :
Let
xj → xj + εa.
If L is invariant, the Noether's theorem gives
a  p
j
 a  P  const
j
i.e., the total momentum P = Σj pj is conserved if L is invariant spatial translations.
Conservation of Total Angular Momentum:
See Exercise 3.1.
Miscellaneous
Interpretations
A symmetry transformation can be interpreted in 2 ways.
Active point of view
Portion of the system being studied is switched from x to x + a.
Passive point of view
Origin of the coordinate system is switched by –a.
Finite Transformations
Finite transformations = (path-dependent) integral of infinitesimal transformation.
Conservation under infinitesimal transformations
→ conservation over finite version of same transformations.
3.3.
The Hamiltonian
Lagrangian formulism: state variables =
q, q
p
Hamiltonian formulism : state variables = (q , p )
L
q
L
 q  L  q, q   p  q  L q, q 
( Legendre transformation )
q
H is a function defined for every value of (q , p ).
But E is defined only on the actual trajectories.
H  q, p  
Note:
L
L
L
dL 
dq 
dq 
dq  pdq
q
q
q
d H  qdp  pdq  d L
 qdp 
→
H
q
p
 qdp  pdq 
L
H
H
dq 
dp 
dq
q
p
q
L
dq  pdq
q
H
L
d L
 p


q
q
d t q
Hamilton's equations
3.4.
Poisson Brackets and Translation Operators
Lagrangian and Hamiltonian formulisms allow theories beyond Newtonian mechanics.
E.g.,
statistical mechanics,
quantum mechanics.
d A  q, p   A
A
 A H  A H

q 
p 
 A, H P



dt
q
p
q  p  p q
(on trajectory)
A, BP
H  i H ,
→
 A B  A B




q  p  p q
P
Poisson bracket
 H  H  
i


 
 q  p  p q 
dA
 iH A
dt
or
Liouville operator
dA
i
 H A
dt
(q,p) is restricted to the actual trajectories (q(t) ,p(t)) by means of the density function:
  q, p,t    q  q  t    p  p  t    qi  qi  t    pi  pi t 
i
→
A  t   A q  t  , p  t    dq  dp   q, p, t  A  q, p 
   q, p, t 
dA
  d q d p
A  q, p   i H A   dq  dp   q, p, t  i H A  q, p 
dt
t
  dq  dp i H   A   dq  dp  i H   A
 H  A H  A 
d
q
d
p
i
H

A


d
q
d
p


     

 

q

p

p

q


    A     A  
   dq  dp    H

 H
 0


p  p 
q  
 q 
→
dA

  dq  dp
A    dq  dp i H   A
dt
t
→
i

H 
t
Liouville
equation
dA
 H A
dt
i
→
Translation Operators
n

t
n
1
n

A
 0
At   exp it H  A0    i tH  A  0  
n 0 n !
n 0 n !

 exp(i t H) is a temporal translation operator,
and H is a generator of time translation.
n

1
n
1  
f  x  a     a   f  x     i a  P  f  x  =exp ia P  f  x 
x 
n 0 n !
n 0 n ! 

 exp(i a · P) is a spatial translation operator,
and P is a generator of spatial translation.
N
 P 
P  
P 
P  i  





i





p j x j 
j 1  x j p j
j 1 p j x j
n
 
 i  

j 1  p j
N
 
pk  

 x j
k 1

N

 i 
j 1 x j 
N

 i P, P

x
j 1
j
N
P  i 
3.5.
The Action Principle in Minkowski Spacetime
A Minkowski spacetime is defined as a spacetime manifold in which a class of
inertial Cartesian coordinate systems exists such that
g  x   
 1

  1
0

for
   0
    1, 2,3
 
x
Distance between 2 points in the manifold is called a proper time interval, .
For infinitesimal separations,
If
g  x   
d 2  g  x  dx dx
x0
then time is defined as t 
c
while spatial position is given by the 3-D vector x = xi, i = 1,2,3.
A coordinate system is called inertial if it can be obtained from an inertial Cartesian
coordinate system by a transformation that leaves the form of g unchanged.
Isometries
Isometry = Symmetry transformation that leaves the form of g unchanged.
 g ' ' dx  'dx '  g ' '  '   ' dx  dx
ds2  g dx  dx
g  g ' ' 
→
Λ 
 g 



or

'
 g
 Λ1 
g ' '    '  ' g
g  Λ
g  Λ g Λ
 g  
T

'

T
Matrix form:
where
'
 g  ' '

   '
T
 Λ 

1
g Λ 1
   '
 x

 x
 x
 '
x
 Λ 
T
1
Λ is an isometry ↔ g and g' have the same functional form.
Isometries convert one inertial system into another.


  ' 
Examples of Isometries
Translations:
x '  x  a
dx  '  dx 
ds2  g dxdx  g dx  'dx '  g ' 'dx  'dx '

g  g ' '
Lorentz Transformation:
x  '   '  x  with
det    '  1
Rotation about the x1 axis by an angle  :
Proper
Lorentz group
Improper
1
0
Λ
0

0
0
0
1
0
0 cos 
0  sin 
0 
0 
sin  

cos  
Boost along x1 by v :
 cosh 
  sinh 
Λ
 0

 0
 sinh 
cosh 
0
0
0
0
1
0
0
0 
0

1
sinh    
v

c
cosh   

1
1  2
Poincare Transformation:
'
x 
'


x a
'
with
det    '  1
Proper
Poincare group
Improper
Tensors
Let f(x) be a scalar function of the Cartesian coordinates xμ.
Under a Poincare transformation
→
'
dx  
'

dx

x  '   '  x   a  '
 ' f 
f
f



'
 x '
 x
   '   f
An object V that transforms like dx is called a contravariant 4-vector.
An object V that transforms like  f is called a covariant 4-vector.
Objects with m upper and n lower indices and transform like
T11''
m '
n '
 1 '1
m ' m 1 1 '
n n ' T11
m
n
are called mixed 4-tensors of rank m+n.
Caution: 4-tensors are not necessarily true tensors since only  with
constant elements are considered.
A contraction of all tensor indices is a Lorentz scalar invariant under all
Lorentz transformations. E.g., ημν U μV ν
Lagrangian of a Free Particle
Motion of particle = path xμ(τ) in Minkowski spacetime (  = proper time ).
 is a scalar →

d
x
x 
d
is a 4-vector called the 4-velocity.
Principle of relativity → EOMs have same form in every inertial frame.
i.e., they are invariant under all isometric (Poincare) transformations.
→ Action is a Lorentz scalar and translationally invariant.
For a free particle:
S0   d L0  x x

→

 L0
0

x

  d L0  X 
 L0  X d L0
 

x
x d X
1
X    x  x
2
1
1
X 1 
 
 


 



x


x



x


x
 x


x
x






 







2
2
x
2 x
d
d
 d L0  d X d 2 L0
 d X   d d X 2


2
  L0  d  d L0 
d L0
d d L0
d
L
d
X
d
L0
0

x

x

x
 x
 x
0


 
 

2
d
X
d

d
X

x
d

d
X
dX
d d X




d L0
2
x  0
or
On the actual path of the particle, X = ½ c →
x
0
dX
d
d
A simple choice is
1
L0  m X   m  x  x
2
1
 m  c2  v 2 
2
for v << c.
Energy-Momentum 4-Vectors
The canonical momentum p  
L
 x
conjugate to xμ
are called an energy- momentum 4-vector or 4-momentum.
In Cartesian coordinates, η     diag 1, 1, 1, 1
so that for a free particle,
L
p     m  x  m  c, x
x
The 1-form pμ is conserved due to translational invariance.
Vector version:
where
p    p  mx   m  c, x
   
   
The velocity of a particle with coordinates
d

dt
1
u2

1 2

c
→

d
x
x 
d
p   p , p   m x   m  c, u

0

 ct, x , x , x    ct, x 
1
2
3
d t d x

d d t
p0 
is u 
   c, u
E
c
p p  m2  x  x  m 2 2  c 2  u 2   m 2c 2
dx
dt
Lagrangian of Many Free Particles
N
For a system of N non-interacting particles
n  t, x    3  x  xi  t  
1
S    d i mi xi xi
2
i 1
j  t, x   
i
i
d xi  t  3
  x  xi  t  
dt
n
   j  0 (number of total particles is conserved )
Equation of continuity:
t
d x0
c
dt
→
j

 x    cn  x  , j  x  
d xi  t  3

  x  xi  t  
dt
i
dxi  i  4
 c  d i
  x  xi  i  
d i
i
Equation of continuity:
 j   0
Prove it!
The associated flow of quantity A carried by the particles is defined as
d x   i  4
jA  x   c  d i Ai
  x  xi  i  
d i
i

Setting A to be the electric charge, we obtain the electromagnetic current.
Setting A to be the 4-momentum, we get the (energy-momentum) stress tensor


d
x

d
x
i  4




i
T  x   c  d i mi
  x  xi  i  
d i
d i
i
T is symmetric ( Tμν = Tνμ ) and conserved ( ν T μν= 0 ).
A perfect fluid is a fluid that has a rest frame in which its density is spatially
uniform and the average velocity of its particles is zero.
Its stress tensor is (see exercise 3.4),
T  T    diag   , p, p, p 
where  is the energy density and p the pressure.
3.6.
Classical Electrodynamics
Maxwell's Equations in Heaviside-Lorentz units:
Gauss' law:
Faraday's law:
 E  e
E
1 B
0
c t
B  0
No monopoles:
Ampere's law:
B 
1 E 1
 je
c t c
Note: some GUTs (see chapter 12) suggest the existence of magnetic monopoles.
The homogeneous eqs are satisfied automatically by setting
E   
1 A
c t
Introducing the 4-vector
B  A
A  , A
with
A  , A
we define the anti-symmetric field strength 4-tensor as
F   A   A
F     A   A
In Cartesian coordinates, we have

    1
 1 
    0 , i    t , i   
, 
 x x   c
  c t

F     A   A
F
0j
→
1  A j 

 j  E j
c t  x
F   F  
 0
 1
E
 2
E
 3
E
F    F  
1 

  
,  
 c t

F    F    0
 Am
 A j  Ai
k
F   i  j   i j k  k l m
l   i j k B
x
x x
ij
 E1
E 2
0
 B3
B3
B2
0
B1
E3 
2 
B 
 B1 

0 
Caution: index conventions
not strictly observed here.
The inhomogeneous Maxwell eqs can be combined as   F  
where
je   ce , je 
is the 4-current density.
1 
je
c
Field Lagrangian
The Maxwell's equations can be derived from the least action principle.
S   dt L   dt  d 3x L 
1
4
d
xL

c
1
1
L  x    F  x  F   x   je  x  A  x 
4
c
Since d 4 x is a Lorentz scalar, so are S and .
F
L
1
  je
A
c
  F  F   
    A 

 

  F  F  
    A 
A
0
Aμ(x) is a 4-vector dynamic
variable with label x.
F
    A 
      
             F   F      
         F   F             F   F   F   F   4F 

L
L
 
0
A
    A 
becomes
1 
 je    F   0
c
Interaction with a Charged Particle
For a single particle of charge q
je  x   cq  d x     4  x  x   
1
q 4
4

 2  d x je  x  A  x     d x  d x     4  x  x    A  x 
c
c
q
   d x    A  x   
c
S   d L   d 4 x Lint   d 4 x L
where
1
L   m  x  x
2
L
1
F  x  F   x 
4c
  d L   d Lint   d 4 x L
q 
x A  x 
c
1
Lint   je  x  A  x 
c
Lint  
Equations of Motion
The dynamic variables are xμ(τ) and Aμ(x) .
The x in Aμ(x) has 2 interpretations.
1. x is just a label when Aμ(x) is treated as a dynamic variable, as in  and int .
2. x is a dynamic variable when Aμ(x) is a potential acting on a particle, as in Lint.
Analogous interpretation also applies to jeμ(x) .
In the variation of each dynamic variable, all others should be taken as fixed.
Thus, for the particle degrees of freedom xμ(τ), we set δAμ= 0 so that δ = 0
q 
 1

 
 S   d   L  Lint    d    m  x x  x A  x  
c
 2

q

  q  A


   d  m  x  A  x   x  x
x 

c
c x



 d
   d  
 d
q

 q  A  

m  x  c A  x   c x x    x

q
q  A
m  x  A  x
0

c
c x

dx A
A 

d
d x
dA
Using
q 
q 
m  x  x    A   A   x F
c
c
q 
q
q
q
mx  x F
mx   x F   x F    x F 
c
c
c
c
we have
→
 x  A
 0
 ct 
 1
 1
x
qE


m 2 
 x  c  E2
 3
 3
x
 
E

 E1
E 2
0
 B3
B3
B2
0
B1
Ex


 E 3   ct 

 1
3 2
2 3

B 2    x1 
q  cE t  B x  B x 

2
1
c  cE 2t  B 3 x1  B1 x 3 
B    x 


 3
2 1
1 2
cE
t

B
x

B
x 
0    x3 

E   E 1 , E 2 , E 3  is a 3-D Euclidean vector.
t
dt
 
d
x
dx
u
d
d

mc


dt

1
dp
 

dt

2
dp
 

dt

3
dp
 
dt

p  
1
1
2
t
u
c2
d
d

d
dt
u
d
 dp
x
 u  
d
m dt



Eu



 q  cE 1  B 3u 2  B 2u 3 

  2
3 1
1 3
 c  cE  B u  B u 
 3
2 1
1 2

cE

B
u

B
u 




q



m

x

A
L

L

int 
 
c
x
dx
dt
p  m u
Rate of work and Lorentz force
 2 d
 mc dt

 dp
 dt

 Eu 


  q
1
E  u  B

c



p  m x 
q 
A
c
Gauge Transformation
Under a gauge transformation
F   A   A
A  x   A  x   A  x     x 
is invariant:
F   A   A    A      A       A   A  F
Gauge invariance of F can be traced to its antisymmetry.
1 
1
1
je  x     L  x      je  x        je
c
c
c
1
"~" relates s that give the same EOMs,
L  x      je
c
e.g., s that differ by a divergence.
L  x  L x  L  x 
  be gauge invariant →  conserved current
  je  0
Symmetry of gauge invariance → Conservation of electric charges.
3.7.
Geometry in Classical Physics
3.7.1.
More on Tensors
3.7.2.
Differential Forms, Dual Tensors and Maxwell's Equations
3.7.3.
Configuration Space and Its Relatives
3.7.4.
The Symplectic Geometry of Phase Space
3.7.1. More on Tensors
A rank (mn) tensor T is a multi-linear function that
maps m 1-forms and n vectors to a number in K.
T U  V ,
  T U ,
  T U, ,   ,    T V , ,   , 
   T U, , ,    T V , ,,    T V , , , 
,   ,
,,
where U ,V
are vectors and
, 
1-forms.
A vector is a linear function that maps a 1-form to a number.
A 1-form is a linear function that maps a vector to a number.
In a coordinate system { xa },
Tab  T  ea ,
T U ,
,   Tab U a
, eb 
where ea are basis vectors and e b
basis 1-forms.
b
Vectors and 1-Forms
d
V 
= tangent to curve C(λ).
d
d
df
V  f  
df 
d
d
Let  f  d f = gradient to scalar field f.
→
 d  df
 f V   d f  
d
 d 
 
Given a coordinate system {xa} , the natural bases for vectors and 1-forms are



e



and
 a
a
a
x


e
a
 dx a
These so-called coordinate bases satisfy
    xa

d x  b   b   ba  b d x a
x
 x  x
a



respectively.

V  V ea  V
 xa
a
a
  a ea  a d xa
 b  
b a
a
b
a  
 V   a d x V


V



V


V
d
x
a
b
a
a
b 
 b

x

x




a


b
a
b
V    V

d
x

V

d
x
b
b
 xa
 xa
a




 V   V  
 V a b  ab  V a a
Under a coordinate transformation, x a  x a '
a'

x
a'
a
d xa  d xa '  a d x a   a d x
x
 a 'a ba ' 
→
x x
 xa  xa '
a'
b


 xa   a 
 a'  a' a
a'
a
a

x
x
x
x x
 xb
 a   ab
x
  a ' d xa '  a a a 'a 'b d xb  a ba d xb  a d xa
Direct Product
The direct product of a ( mn ) tensor S with a (m n  ) tensor
T is a ( m + m n + n  ) tensor S  T such that
S  T u1,
 S u1,
→
, un , 1,
S  T cd
ab
TT
ab
cd
ab
cd
T
, m ; v1,
, vn ,1,
, m 
, m  T  v1,
, vn ,1,
, m 
, un , 1,
ef
gh
ab
 Scd
Tgehf
dx dx 
c
d
 

 T c , d ,
 x x


 a b
x
x
a
b
; d x ,d x ,



3.7.2. Differential Forms, Dual Tensors and Maxwell's Equations
A p-form is a totally antisymmetric tensor of rank ( 0p ):
  ,U , ,V ,

a a
0
→
   
C pn 
,V ,
,U ,
n!
p ! n  p !


a b
 
b a
independent p-forms in an n-D manifold.
 only one n-form, which must be proportional to the Levi- Civita tensor density
 abc
1

  1
0

even permutation of 1,2,3,
if
abc… 
odd permutation of 1,2,3,
otherwise
Wedge Product Between 1-Forms
A totally antisymmetrized version of the direct product is called the wedge product .
The wedge product between two 1-forms is defined as
   U ,V    U  V    U   V 
      
→
       
   a b  a b   ab
For a 2-form


1
  ab d x  d x  a b d x a  d x b  ba d x b  d x a
2
1
1
a
b
b
a
 a b d x  d x  a b d x  d x  a b d x a  d x b
2
2!
a

b

ab  ba
The set of C n2 independent 2-forms is a basis for the 2-forms in an n-D manifold.
Wedge Product Between Arbitrary Forms
Wedge (exterior) products are associative:
            
p-form

1
a1 a2
p!
ap
   
d x a1  d x a2 
dx
ap
The wedge product of p-form ω and q-form σ is therefore
1
  
a1
p ! q!

ap
b
1

   V1 , ,V p q  
dx 
a1
bq
dx dx 
ap

1
P

 VP1 ,



p ! q! P
b1
dx
 
bq
    
,VP p   VP p 1 ,
where P is a permutation of the p+q labels of the vectors.
pq
,VP p  q 

Cross Product
3-D vectors with Cartesian coordinates:
u  v   u 2v 3  u 3v 2 , u 3v1  u1v 3 , u1v 2  u 2v1



dx 2  dx 3  u , v   dx 2  dx 3  dx 3  dx 2  u , v 
2 3
3 2
 dx 2  u  dx 3  v   dx 3  u  dx 2  v   u v  u v

1
u v  u v , u v  u v , u v  u v    abcdx b  dx c  u , v   a
2
2 3
3 2
3 1
1 3
1 2
2 1
The rank 3 Levi-Civita symbol εabc is defined only for Cartesian coordinates.
Raising index using the Euclidean metric tensor gab = δab gives
1 ab
  g  bcd d x c  d x d  u , v 
2
a
→
u v  w
This works only because 2-forms in 3-D transform like 1-forms.
Dual Tensors

B  dx 1 
Consider the set of basis p-forms in an d-D manifold
d
There are only C p 
a
dx
ap
d!
independent p-forms in B.
 d  p ! p !
Similarly, p-vectors can be constructed with the help of the wedge product.
Cpd  Cdd p
→ dim pV of p-vectors = dim d-p*V of (d–p)-forms
  bijection between pV and d-p*V ( they are dual to each other ).
V
1 a1
V
p!

Thus
ap
ea 1 
 ea p
is dual to
W  *V    
is the volume d-form with components  a 1
Wa1
ad  p
 *Va1

ad  p
ad
1
p d  p
  
w  x   b1
p!
1
w  x   a1
p!
ad  p b1
bp V
b1 b p
p d  p 
 w  x   a1
bp a 1
ad  p V
1
 V 
p!
ad
b1 bp

Exterior Derivatives
The exterior derivative is a derivation that increases the degree of a form by 1.
Furthermore, we require, for forms β and γ of arbitrary degrees, that
1. Linearity:


d     d   d
2. (Generalized) Leibniz rule:

  
d     d         d 
p
 
d d  0
3. Nilpotence:
Reminder: the Leibniz rule for a derivation d is d  f g    d f  g  f  dg 
A derivation that satisfies property (2) is called an antiderivation.
Combining (1) and (2) gives


d a   b  ad   bd 
for any constants a and b.
p-forms
Taking a scalar function f as a 0-form, we have a 1-form
Consistency check:
Note that d xa
 x1 a
1
a
1
d x  a d x  a d x  d x
x
df 
f
a
d
x
 xa
1
is a 1-form while the ordinary derivative dxa is a 1-vector.
Also, most authors (Lawrie is one) use the symbol d to denote d
For a 1-form   a dx

d   d a dx a
   d   dx
a
a
 
    a d dx a 
0
a b
a
dx

dx
x b
a a
b
a
b
b
a
dx

dx



dx

dx



dx

dx
a, b
b, a
x b
 b, a dx a  dx b   1 a , b  b, a  dx a  dx b
2

 d 
ab
  a , b  b, a 
 
d dx a  0
a
For a p-form

1
d 
da1
p!
 d 
ba1 a p

ap

1
a1
p!
 dx 
a1

  p  1 [ b a1
x
a1
dx

ap
 dx
ap ]
If ω is itself the exterior derivative
of another (p1)-form
ap
 dx

ap
1

a1
p!
  
p
a p ,b

 p  1 a
1
1

 a1
 p  1!
dx b  dx a1 
a p 1
dx a1 
1
a
 a1 a p 1 , cb dx b  dx c  dx a1   dx p 1
 p  1!
1
a

 a1 a p 1 ,bc dx c  dx b  dx a1   dx p 1  0
 p  1!
Thus, the nilpotent assumption is consistent.
ap
a p , b
1
a
 a1 a p 1 ,b dx b  dx a1   dx p 1
  d 
 p  1!
1
a
dd 
 a1 a p 1 ,bc dx c  dx b  dx a1   dx p1
 p  1!

 dx
 dx
a p 1
E3
a  a j dx j  a1dx1  a2dx2  a3dx3


da  d a j dx j  da j  dx j  a j ddx j  a j , k dx k  dx j
 a1, k dx k  dx1  a2, k dx k  dx 2  a3, k dx k  dx3
 a1, 2 dx2  dx1  a1, 3dx3  dx1  a2,1dx1  dx2  a2, 3dx1  dx3  a3,1dx1  dx3  a3, 2 dx2  dx3
  a1, 2  a2, 1  dx 2  dx1   a3, 1  a1, 3  dx1  dx 3   a2, 3  a3, 2  dx 3  dx 2





*da   a1, 2  a2,1  * dx 2  dx1   a3,1  a1, 3  * dx1  dx3   a2, 3  a3, 2  * dx 3  dx 2
Dual of the volume n-form:


* dx j  dx k 


1 i jk
 ei  e j  ek
3!

1
 dx j  dx k
2!





1 il m
 ei  el  em dx j  dx k
2!3!
1 il m
 ei  el  em dx j  dx k
3!



1 il m j k
jkm
em
  i  l em P23  
3!
*da   a1, 2  a2, 1    e3    a3, 1  a1, 3    e2    a2, 3  a3, 2    e1 
*da   a1, 2  a2, 1    e3    a3, 1  a1, 3    e2    a2, 3  a3, 2    e1 
 a a 
 a a 
 a a 
  23  23  e1   13  13  e2   12  21  e3
x 
 x
 x x 
 x x 
*a  *  a ei   a *  ei 
i
i
1 3 j
 a P1  i  j k l e k  e l
3!
i
 a
1
 j k l e j  e k  e l  ei 
3!
1
  j k l a j e k  el
2
 ai

 
 
1
1
j
i
k
l
j
k
l
j
k
l
d * a   j k l d  a j e k  e l    j k l a, i dx  e  e  a de  e  a e  de
2
2

1
1
d * a   jkl a, ji dx i  dx k  dx l  123 a,11 dx1  dx 2  dx 3  132 a,11 dx1  dx 3  dx 2
2
2
1
  312 a,33 dx 3  dx1  dx 2   321 a,33 dx 3  dx 2  dx1
2
1
  231 a,22 dx 2  dx 3  dx1   213 a,22 dx 2  dx1  dx 3
2
  a,11  a,22  a,33  dx1  dx 2  dx 3





d * a   a,11  a,22  a,33  dx1  dx 2  dx 3  a, jj      a  
  dx1  dx 2  dx3 

*da    a
dd  0

Similarly
1
 j k l dx j  dx k  dxl
3!
is the volume n-form.
 
dda  dda       a  
d * *da    
2 3 2 
    a   0
 
*d d         0
Poincare Lemma
A p-form ω is said to be closed if
d  0
A p-form ω is said to be exact if there exists a (p1)-form σ such that
dd  0
  d
 an exact form is also closed, but not vice versa.
Poincare lemma: Any closed form in a n-D manifold is exact in any region
that is homeomorphic to the open unit ball Sn1 [see §4.19 of Schutz for proof].
Rn is homeomorphic to Sn1
 a closed form is exact in any region describable by a single coordinate patch.
 Every closed form is exact locally but not necessarily so globally.
The study of this is called the cohomology theory.
Maxwell Equations
 0
 E
F   F    1
  E2

  E3
Faraday 2-form
E1
E2
0
 B3
B3
0
 B2
B1
E3 
B2 
 B1 

0 
Reminder: E and B are not the spatial parts of some vectors in Minkowski space.
 Either E j or Ej can be used to denote the jth component of E.
F   F
 dF 

F0 j  E j
Fi j  i j k Bk
 3F  ,  

3
F ,   F ,   F ,   F ,   F ,   F ,  

3!
 F ,   F ,   F ,
 dF 
 
dF
 dF 
0i j
i jk

 F ,   F ,   F ,
 F0i , j  Fi j , 0  Fj 0, i  Ei, j   i j k Bk , 0  E j , i   i j k   E  1 B 
c t k

 Fi j , k  Fj k , i  Fk i , j  i j m Bm, k   j k m Bm, i   kim Bm, j
 i j m Bm, k   m j k Bm, i  im k Bm, j
 Homogeneous Maxwell equations:
F       F 
  i j k Bm, m   i j k B
dF  0
 0
E
F   F     1
 E2

 E3
 E1
 E2
0
 B3
B3
0
 B2
B1
 E3 
B2 
 B1 

0 
*F 
1
1
2 42
     F     F 
2!
2!
1
*
F

  2     F  
d * F 
d * F 
 
0i j

0

E1

F
 E2

 E3
 E1
 E2
0
 B3
B3
0
 B2
B1
1
    dx   dx  dx  dx 
4!
0
B
*F   1
 B2

 B3
 E3 

B2 
 B1 

0 
 B2
0
 E3
E3
0
 E2
E1
 B3 
E2 
 E1 

0 
 F  E  B, B  E
 *F ,   *F  ,   *F  ,
d * F 
1 E 

  i j k   B 

c t k

j   j     c  , j
 B1
i jk
1 4 1
* j  
 * j 0 i j   0 i j  j    i j k j k
Non-homogenous Maxwell equations:
  j     j 
  i j k   E
* j       j
* j i j k   i j k  j   i j k j 0   i j k c
1
d *F  * j
c
3.7.3. Configuration Space and Its Relatives
Newtonian dynamics of N particles with Galilean relativity:
Positions of all particles at time t = point in the configuration space Q.
Q = manifold with 3N generalized coordinates { qi } as coordinates.
State of a system = point in a 6N-D manifold TQ.
TQ = tangent bundle with Q as base & tangent space T at a point in Q as fibre.
[See §2.11, Schutz, for a definition of a fibre bundle].
A point in TQ has coordinates ( { qi } , { vi } ),
where { vi } are coordinates in the fibre at point { qi } in Q.
The actual velocities of the particles are denoted as
 q t  
i
Evolution of system is represented a curve in TQ.
A vector field over Q is called a cross section of the bundle.
Lagrangian L({qi}, {vi} ) is a scalar field on TQ.
Let
L
1
gi j  q  vi v j  V  q 
2
Generalized momentum conjugate to
gi j (q) = metric tensor field on Q.
qi
is
L
pi  i
v
 gi j  q v j
pi is a 1-form obtained by lowering the indices of vi.
The set of all pi at a point P in Q forms a vector space called the cotangent space TP* .
The fibre bundle formed using T* as typical fibre is called a cotangent bundle T*Q.
3.7.4. The Symplectic Geometry of Phase Space
Symplectic Manifold:
The fibre bundle T*Q is called a phase space in Hamiltonian dynamics.
The state of the system is given by a point in T*Q with coordinates
    ,

1
, 6N  
q , p 
i
Symplectic form:
i
  q1 ,
, q 3 N , p1 ,
If Ω is used as volume form for dual operations,
the bundle T*Q is called a symplectic manifold.
  dqi  dpi
A vector field V is written as
V V

 
Vi
The underbars serve as a reminder that V i and V
i
By definition:
, p3 N 
V i  dqi V 



V
i
qi
pi
are components of a vector.
V i  dpi V 
Canonical 1-form
Ω is exact:
  d
where
  pi dqi is called the canonical 1-form.
History of system is represented by a curve with tangent vector d/dt.
d
Components of the velocity vectors of the particles = qi  t   dqi  
 dt 

d
i
i d 

p
q

p
dq
i
i

 
 dt 
 dt 

=H+L
~ Legendre transform between H & L.
~ Transform between TQ and T* Q .
Symplectic 2-form
Ω is a 2-form → * associates vectors and 1-forms
For a vector V:
 
 V   V    *V
V

   V 


If Ω is non-degenerate, i.e. the 2-vector     exists with    
V 
we can associate a 1-form uniquely with a vector: V   

exists iff
  
det   0
For the phase space described by coordinates
 0 I
  

 I 0 

q ,
1
, q 3 N , p1 ,
where I is the 3N3N unit matrix and
, p3 N 
det   1
A degenerate symplectic 2-form implies mismatch between the numbers of
independent coordinates and momenta.
Example: Electromagnetism where the gauge degrees of freedom are “unphysical”.
Hamiltonian description of such systems require special techniques.
Hamiltonian Dynamics
The Hamiltonian vector field VA
is defined by
In components :
associated with a scalar field


A  A qi , pi 
 VA   dA   *VA

 j 
  A
A
j
dq  dpi VA

V

dq

dp j

Aj
j
 q j

p j  q
p j


i
A j A
V dp j  V A j dq  j dq 
dp j
q
p j
j
A
→
j
A
V 
p j
A 
A 
VA 
 j
 A,
j
p j q q p j
Hamilton’s equations:
V A j 
j
A
 A, BP
P
d
  H ,
dt
P
 VH

A
q j
A B A B
  VA , VB 
 j

j
q p j p j q
H 
H 

p j q j q j p j
Symmetries
The study of (continuous) symmetries is best conducted using mathematical
techniques devised by S. Lee, namely, Lie derivatives, Lie group, and Lie algebra.
An introduction of these can be found in Schutz.
First, the Hamiltonian vector fields form a Lie algebra:
VA , VB   VA, B
P
Hence,
A, BP  0
[see Ex 5.6, Schutz]
VA , VB   0

By definition, the Lie derivative of a vector field W 
along another vector field V 
If
d
d
is
d
d
L VW  V , W 
V ,W   0 , W is said to be Lie-dragged by V and
  
 ,

   
is a coordinate basis for vectors.
Consider VH , whose integral curves are possible trajectories of the system.
VA , VH   0
→
↓
L V H VA  0
→
A, HP  0
→
dA
0
dt
→ A is conserved
integral curves of VA , or simply A, is “Lie-dragged”
(the same) along the trajectories of the system.
↓
L V A VH  0
→
H is invariant along the integral curves of VA ,
i.e., H has a symmetry.