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Module I
Review of Classical Mechanics
1.
Coordinates and Trajectories
Two of the primary goals of classical physics is to describe the location of an
object in space and how the object's location will change with time.
A.
Coordinates
A set of numbers that uniquely describes mathematically the spatial
location of an object is called the object's coordinates.
Some important points about coordinates:
a) An object has an actual physical location independent of our describing it
mathematically. Thus, no particular coordinate system is required by the
physical world. We choose the coordinate system to simplify mathematical
calculations.
b) Between any two coordinate systems there are a unique set of equations
(transformation equations) that allows us to take measurements made in one
system to predict measurements made in the other system.
c) Since there is no preferred coordinate system, the true mathematical
expressions of the laws of the physical world should be unchanged in form
under coordinate transformations (i.e. Invariant under coordinate changes).
d) The origin for any coordinate system is arbitrary. Since the value for a
coordinate depends on the origin chosen, you should specify your origin in
every problem.
B.
Trajectories
The set of values that an object's coordinates take over an interval of time is
called a trajectory.
The central problem in classical mechanics is to find the equations of motion and
then solve these equations to find the object's trajectory (i.e. how the bodies
position changes with time).
We will see later that the very concept of a trajectory is impossible in quantum
mechanics (Chapter 2).
This leads us to the following questions (answered in Chapter 3):
C.
1.
What quantities should be used to describe a quantum system?
2.
What do we use instead of trajectories to mathematically describe the time
evolution of a quantum system?
3.
What replaces the equations of motion in solving quantum systems?
4.
What is the connection that links the results of quantum mechanics to
those of classical physics?
Constraints
In many physics problems, the values of an object coordinates are limited in some
manner. An example is a particle moving in the x-y plane. In this case, the
particle's z-coordinate is limited to some constant values. Such a condition is
known as a constraint condition.
Constraints will reduce a system's degrees of freedom and the number of
independent coordinates needed to specify the system.
Holonomic and Non-Holonomic Constraint
If the constraint condition can be written as an equation with the right
hand side equal to zero then the constraint is said to be holonomic
otherwise the constraint is said to be non-holonomic.
EXAMPLE: A particle moves in the x-y plane with z = 3. The constraint condition can
be written as z  3  0 so it is a holonomic constraint.
z
z=3
y
x
EXAMPLE: A particle slides along the outer surface of a cylinder of radius R before
leaving the cylinder as shown below. The constraint condition can be written either as
x
2
2
 y R0
r  R  0.
This is a non-holonomic constraint.
y
R
r
x
D.
Generalized Coordinates
It is always possible to write the location of an object in terms of cartessian,
cylindrical, or polar coordinates. In more complicated problems, it is often more
convenient to use a different set of coordinates due to the form of the constraint
equations and forces.
Consider a lure attached to a fishing line as it unwraps from the spool as shown
below:
l

r

You can describe the location of the lure by using either the cylindrical
coordinates (r,) or the length of the line, l, and the angle the line makes with the
spool, . You could also describe the motion in terms of r and  or in terms of l
and . A wise choice can greatly simplify the math.
Independent coordinates that serve to uniquely determine the orientation and
location of a system in physical space are called generalized or canonical or
good coordinates.
E.
Generalized Velocity
A generalized velocity is the time derivative of a generalized coordinate.
II.
Newtonian Mechanics
The essence of Newton's mechanics is contained in Newton's 2nd Law which
states that the cause of a change in a physical system's linear momentum is a net
external force.


dp
F 
dt
EXT
For some problems, the application of Newton's Mechanics presents several
difficulties including 1) complicated vector mathematics and 2) determination of
the functional form of all forces.
III.
Lagrangian Mechanics
An alternative mechanics to Newton's was developed in Europe by Leibniz,
Bernouli, Lagrange, and Hamilton. This mechanics is based upon scalar functions.
It is easier to apply in many problems involving constraints and more importantly
laid the theoretical foundation for the development of quantum mechanics.
A.
Lagrangian - L
The Lagrangian, L, for a conservative system is defined to be the difference
between the kinetic energy and the potential energy.
LT V
B.
Hamilton's Principle of Least Action
Of all the possible paths for a physical system, the system will take the path that
minimizes the integral of the Lagrangian with respect to time.
δ  L dt  0
t2
t1
y
y
A
A
B
B
t
at
C.
The figures above shows three possible paths that a ball can follow from point A
time t1 to point B at time t2. In fact, the ball can follow an infinite number of
different paths as we vary the generalized coordinates and velocities arbitrarily
between the time t1 and t2. The only path constraints are that the generalized
coordinates and velocities of the ball must be path independent at points A and B.
Hamilton's principle of least action says that of all these paths the ball will follow
the path that minimizes the time integral of the Lagrangian.
Lagrangian Equations of Motion
Hamilton's principle of least action allows us to develop an alternative system to
Newton's mechanics for obtaining the equations of motion for a classical system.
A classical system with n generalized coordinates and only holonomic constraints
must obey n general equations of motion of the form
L d L

 0,
q dt q
k
k 1, ......, n
k
where q k and q k are the kth generalized coordinate and generalized velocity.
t
Proof: We start with Hamilton's principle of least action
δ  L dt  0
t2
t1
In creating path variations, we only adjust the generalized coordinates and velocity and
not time. These spatial displacements are sometime called virtual displacements since
they occur in zero time intervals. We can therefore bring δ inside the integral.
 δL dt  0
t2
t1
For a conservative system with only holonomic constraints, the Lagrangian is at most a
function of the generalized coordinates and the generalized velocities. From Calculus, we
have that
δL  
n
k 1
L
L
δq 
δq
q
q
k
k
n
t2
k 1 t1
k
 L
L
δq 
δq
q
 q

k
k
k
k
k

 dt  0

Since each generalized coordinate is independent, each term in the sum must be zero.
Furthermore, the variation of a generalized velocity is related to the change of its
generalized coordinate according to Calculus by the relationship
 dq
δq  δ
 dt
k
k
 d
  δq
 dt
k

Thus, for the k-th generalized coordinate, we have the requirement that
L
L d
δq dt  0
δq dt  

q
q dt
t2
t2
k
t1
k
k
t1
k
The second integral can now be rewritten using integration of parts as
L d
L


δq
dt

δq

q dt
q
t2

t2
k
t1
k
t2
k
k
t1
d  L 

 δq dt
dt  q 
k
k
t1
However, the paths are indistinguishable at the endpoints (i.e. δq k  0 at t1 and t2).
Thus, we have
L d
d  L 


δq
dt


 δq dt

 
q dt
dt  q 
t2
t2
k
t1
k
t1
k
k
Substituting this back into our previous work, we find that for each term of the sum (i.e.
each generalized coefficient), the following condition must hold
 L d  L 
 
 δq dt  0

 q dt  q 
t2
k
t1
k
k
Since δq k and dt are arbitrary changes to two independent variables, the term in the
bracket must be zero. Thus, we have Lagrangian's equation of motion for the k-th
generalized coordinate.
Q.E.D.
D.
Lagrangian Mechanics Solution Method
1.
Choose a coordinate system
2.
Write down any constraint equations
3.
`Use the constraint equations to reduce the number of coordinates
4.
Write the kinetic energy of the system in terms of the coordinates
5.
For conservative systems, write the potential energy
6.
Find the Lagrangian for the system
7.
Find the equations of motion using the Lagrangian-Euler equations
8.
Solve the equations of motion
E.
Conjugate Momentum - p k
The conjugate momentum associated with a generalized coordinate is found by
p 
k
L
q
k
IV.
Hamiltonian Mechanics
Another version of mechanics was developed by Hamiltan who also was
instrumental in developing the mathematics of vector's for Newtonian Mechanics.
A.
Hamiltonian - H
The Hamiltonian of a system is defined as the function
n
H   p k q k  L
k 1
Special Case:
For most system's studied at the undergraduate level, the Lagrangian and
therefore the Hamiltonian is not an explicit function of time. In this case, the
Hamiltonian is the mechanical energy of the system and can be found by:
HT V
The Hamiltonian plays a central role in our study of Quantum Mechanics. It is
the Hamiltonian that provides the bridge between Classical Mechanics and
the development of Quantum Mechanics. We will also see shortly that the
Hamiltonian is always the energy operator in Quantum Mechanics even when the
proceeding formula is invalid for classical physics.
B.
Hamilton's Equations of Motion
In Hamiltonian mechanics, a system with n generalized coordinates has 2n 1st
order differential equations instead of n 2nd order Lagrange-Euler differential
equations.
H
q
H
q 
p
p  
k
k
k
k
It should be noted that the Hamiltonian is written in terms of the generalized
coordinates and their associated conjugate momentum while the Lagranian is
written in terms of the generalized coordinates and their associated velocities.
C.
Conservation of Conjugate Momentum
If the Hamiltonian doesn't contain a particular generalized coordinate then the
generalized coordinate is said to be ignorable and according to our first equation
the time rate-of-change of the associated conjugate momentum is zero. Thus, the
conjugate momentum must be constant (i.e. It is conserved).
Thus, Hamilton's first equation embodies and generalizes Newton's 2nd Law
including your previous work on conservation of linear and angular momentum.
D.
Your Textbook
The textbook spends considerable effort demonstrating that Hamiltonian
mechanics gives the same equations of motion as Newtoian mechanics for several
simple systems. The only real problem with the book's approach is that he doesn't
provide any systematic method for finding the conjugate momentums. He has
chosen special examples from classical mechanics where he knows the conjugate
momentums from his freshman physics work. I believe that his presentation fails
to convey the real power of Hamilton's work. The Hamiltonian and the
generalization of the conjugate momentum play an essential role for handling
difficult problems including the interaction of radiation with matter.
Liboff does give a short discussion of the Lagrangian mechanics connection in
Chapter 7.
E.
Hamiltonian Solution Steps
1.
Chose a system of generalized coordinates
2.
Construct the Lagrangian
3.
Use the Lagrangian to find the conjugate momentum for each coordinate
4.
Write the Hamiltonian in terms of the generalized coordinates and associated
conjugate momentum.
5.
Use Hamilton's equations to find the differential equation's of motion.
Before Class Exercise #1:

In the x-y coordinate system, the position of the ball shown below is r  4 m î  3 mĵ
v
y
u
x
1.
Write the position vector for the ball in the u-v orthogonal coordinate system.
2.
Write the two transformation equations needed to convert spatial measurements in
the x-y frame into measurements in the u-v frame.
Before Class Exercise #2:

The location of a particle in cylindrical coordinates is given by r  ρ ρ̂  z k̂ .

Show that the velocity of the particle is v  ρ ρ̂  ρ  ˆ  z k̂ .
Hint: Write ρ̂ and ̂ in terms of î and ˆj .
Class Example #1:
Use Lagrangian mechanics to find the equation of motion for a 1-dimensional simple
harmonic oscillator constructed using a mass and spring.
k
m
Class Example #2:
Write the equation of motion for an Attwood machine whose pulleys has a moment of
inertia I and a radius R.
x
L-x
m1
m2
Class Example #3:
For our previous spring mass problem, write the system's Hamiltonian and use
Hamiltonian mechanics to get the system's equation of motion.