Download Lagrange's Methods III: Symmetries and Hamiltonians in Physics 505 Lecture 7 (Autumn 2005) and more Study notes Mechanics in PDF only on Docsity!
Lecture 7: The Methods of Lagrange III – Symmetries and Hamiltonians
We want to discuss again the question of the connections between the symmetry
properties of a mechanical system, i.e ., the invariance of the Lagrangian (and the
equations of motion) under a change of variables, and the existence of conserved
quantities ( i.e. , constants of the motion). As a starting point we return to the
Lagrange equation, written in generalized coordinates for a conservative system,
k k
d L L
dt q q
(7.1)
For a system of generalized coordinates such that the Lagrangian is independent of
(at least) one of the coordinates, say
n
q , it follows that the corresponding canonical
momentum is independent of time, i.e ., is a constant of the motion,
constant.
k
k k
k
L d L d
p
q dt q dt
p
(7.2)
This is just the familiar statement in Cartesian coordinates that translational
invariance in
k
r
implies conservation of
k
p
.
Next we want to consider this result in the somewhat more general language of the
general coordinate transformations of the last lecture. In particular, consider
coordinate transformations (of our f unconstrained degrees of freedom) that are
parameterized in terms of continuous, differentiable parameters ( e.g ., rotations in
terms of the rotation angles, i.e ., just the Lie groups we discussed earlier),
1
, , ,.
j j f
q F q q
(7.3)
We represent the inverse transformation by
1
, , ,
j j f
q F q q
and the identity
transformation by
1
0 , , ,.
j j f j
q F q q q
(7.4)
In general such a transformation will yield a new form for the Lagrangian in the sense
that the new Lagrangian (where may be a “vector” of parameters)
L q q t , , L q q t , , L q , q , t
(7.5)
is a different function of the new coordinates than the old Lagrangian was of the old
coordinates. Now connect this transformation to our previous studies of infinitesimal
variations by considering an infinitesimal transformation near the origin in ,
0
k
k k k
q
dq q q d
(7.6)
where it is important to recognize that this is a “real”, not virtual transformation
(hence the notation of d s
instead of s
) and there is no constraint that it vanish at
the endpoints in time. [Recall from our brief introduction to group theory that the
quantity in the square brackets is an element of the algebra, a linear combination of
generators, times the untransformed coordinates.] The corresponding change in the
action (to first order in the parameter change) is
2
1
2
1
0
0
.
t
t
t
k k
k k t
dL
dA d dt
d
q q L L
d dt
q q
(7.7)
If we now use the fact that
k
q
is taken to be a solution of Lagrange’s equation for
any value, we have
Thus G is a constant of the motion, i.e ., it is conserved.
So far we have only assumed that the action is invariant under these coordinate
transformations. If the Lagrangian itself is invariant, i.e ., the functional dependence
on the coordinates and velocities is the same before and after the transformations,
L q q t , , L q q t , , , we can make a connection between this discussion and the
earlier discussion associated with Eq. (7.2). We can now perform a further change of
variables such that one of the new variables is equal to the transformation parameter,
j
q . The new Lagrangian is necessarily independent of
j
q , 0
j
L q
, and the
corresponding canonical momentum is conserved. It is easy to verify that this
canonical momentum is just the quantity G. This is how we can understand the
connection between translational invariance and momentum conservation; rotational
invariance and angular momentum conservation.
In fact, the invariance of the action under transformations, which is all we really need
here, does not require the full invariance of the Lagrangian. If a coordinate
transformation described by a continuous parameter leaves the Lagrangian invariant
except for a total time derivative,
L L d dt , the action still is invariant (under
variations that vanish at the temporal endpoints of Eq. (7.7)) and the corresponding G
is conserved. This connection between an invariance of the Lagrangian (up to a total
derivative) under coordinate transformations described by continuous, differentiable
parameters (Lie groups) and the existence of conserved quantities (Eq. (7.10)) is
usually called Noether’s Theorem (after Emmy Noether). Further the invariance of
the Lagrangian is typically related to some geometrical symmetry, e.g ., rotational
symmetry, translational symmetry, etc. of the physical system. The concepts of
symmetry, invariance and conservation laws are unavoidably connected and a major
component of the physics advances of the last 100 years. It is also possible that the
underlying symmetry is associated with a space other than the usual 3-dimensional
configuration space, i.e ., some “internal” space.
A well known example of both an internal space symmetry and a Lagrangian that
changes by a total time derivative is provided by electromagnetism. Here we focus
on the motion of a charged particle in an “external” field ( i.e ., ignore the back
reaction of the charged particle on the sources of the fields). Recall that Maxwell’s
equations look like
4 ,
1
,
0,
1 4
,
E
B
E
c t
B
E
B j
c t c
(7.11)
with the external change density, j
the external changed current and no magnetic
monopoles. The content of these equations (in free space) is most efficiently
expressed by writing the electric and magnetic fields in terms of a vector and a scalar
potential, A
and . We have
1
,
.
A
E
c t
B A
(7.12)
The corresponding Lagrangian for a particle of electric charge Q and mass m has the
following form
2
EM
m Q
L r Q r t r A r t
c
(7.13)
where it is important to note that the potential (the second and third terms) is
dependent on the velocity of the particle. Applying the Lagrange equations to this
Lagrangian we find
We recognize the right-hand-side of the last equation as the desired Lorentz force of
electromagnetism, confirming that we have the correct Lagrangian. We know that
the electric and magnetic fields (the physical quantities) are invariant under gauge
transformations defined by a single scalar function of the form
A A
c t
(7.19)
Thus the equations of motion, Eq. (7.18), are also invariant under such a
transformation. Is the Lagrangian? Let us check. The only component that changes
is the potential
Q
U Q r A
c t c
Q Q d
U r U
c t c dt
(7.20)
Thus the Lagrangian changes by a total time derivative
EM EM
Q d
L L
c dt
(7.21)
under a gauge transformation, which we have already noted does not change the
physics. (Since the equations of motion derive from the study of virtual
displacements that vanish at the end points in time, a change in the action of the form
2
1
,
t
t
r t
does not contribute to the virtual variation of the action and hence to the
equations of motion.) As you may know this gauge transformation (corresponding to
a change of phase for the electrically charged fields) is described by the group U(1)
and invariance leads, via Noether, to conversed electric charge and currents.
The last topics to be discussed in this lecture are Hamilton’s canonical equations.
The goal is to switch from second order differential equations a la Newton and
Lagrange to first order differential equations. The subtext is that we will be
refocusing our attention from configuration space alone (the
k
q ) to phase space
involving both the generalized coordinates and the canonical momenta, the canonical
variables. As at the start of this lecture we wish to consider a conservative system
described by a Lagrangian of f unconstrained generalized coordinates and velocities,
1 1
, , , , , ,.
f f
L q q q q t T U
(7.22)
With the canonical momenta defined as in Eq. (7.2) (
k k
p L q ), we can use the
Legendre transform to construct the Hamiltonian as a function of the canonical
variables
q p ,
1 1
1
f
f f k k
k
H q q p p t p q L
(7.23)
i.e ., we are to think of the Hamiltonian as a function of the 2 f + 1 variables
1 1
, , , , , ,
f f
q q p p t. As usual (now) we can analyze this function by looking at
small variations on both sides of Eq. (7.23)
1
1
1
1
1
,
,
f
k k
k
k k
f
k k
k
f
k k k k k k
k k k
f
k k k k k k k
k k
f
k k k k
k
H H H
dH dq dp dt
q p t
d p q dL
L L L
dp q p dq dq dq dt
q q t
d L L
dp q p dq dq p dq dt
dt q t
L
dp q p dq dt
t
(7.24)
where the “metric” is time independent and symmetric, kl lk
m m
. It follows that
1
1
,
1
.
2
f
j jk k
k j
f
k k
k
T
p m q
q
T p q
(7.28)
Thus the Hamiltonian is just the total mechanical energy
1
f
k k
k
H p q L T T U T U E
(7.29)
Since, by assumption, T has no explicit time dependence, if U is also free of explicit
time dependence, then the total mechanical energy E is conserved.
As a simple example consider the usual Cartesian description of a single point
particle,
2 2 2
2 2 2
x y z
x y z
x y z
m
L x y z U x y z
L
p mx p my p mz
x
H p x p y p z L
p p p U
m
(7.30)
Hamilton’s equations are then
y
x z
x y z
p
p p
x y z
m m m
U U U
p p p
x y z
(7.31)
i.e ., just the usual definitions and Newton equations. The results are a bit more
interesting in spherical coordinates where
2
2
2
2 2 2
2 2 2
2
2
3 2 3
2 sin
sin
cos
sin
r
r
r
r
H p r p p L
p
p
p U r
m r r
p
p p
r
m mr mr
p p U U U
p p p
mr r mr
(7.32)
An important feature of the Hamiltonian formalism is the similarity to the equations
of fluid flow. This connection helps us to visualize the solutions of Hamilton’s
equations as a flow through phase space. To see the connection, consider the flow of
an incompressible fluid in 2-dimensions. The continuity equation is
V 0
t
(7.33)
where is the fluid density and
,
x y
V V V
is the velocity field describing the
motion of the fluid. From the fact that the density is assumed to be constant in space
and time (incompressible) it follows that the velocity field is divergence free. Like
the case of the magnetic field, it follows that we can define the velocity field as the
curl of vector field, which points in the direction orthogonal to the 2-D motion. We
can write
