On the Shoulders of Giants

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On the Shoulders of Giants

• An Introduction to Classical Mechanics

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• If I have seen further it is by standing on the
  shoulders of giants.
• Isaac Newton, Letter to Robert Hooke, February 5,
  1675English mathematician physicist (1642 -
  1727)

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• Quantum Mechanics (QM) is based on classical
  mechanics. It combines classical mechanics with
  statistics and statistical mechanics.
• For native English-speakers, it is somewhat
  unfortunate that it uses the word quantum. A
  better English word which describes the thrust of
  this approach would be pixel.

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Lights! Camera! Action!

• 2nd Century BC
• Hero of Alexandria found that light, traveling
  from one point to another by a reflection from a
  plane mirror, always takes the shortest possible
  path.
• 1657
• Pierre de Fermat reformulates the principle by
  postulating that the light travels in a path that
  takes the least time!
• In hindsight, if c is constant then Hero and
  Fermat are in complete agreement.
• Based on his reasoning, he is able to deduce both
  the law of reflection and Snells law (nsinQ n
  sinQ)

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An Aside

• Fermat is most famous for his last theorem
• Xn Yn Zn where n2 and
• On his deathbed, he wrote
• And n arrgh! Im having a heartattack!
• His last theorem was only solved by computer in
  the last 10 years

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Now we wait for the Math

• 1686
• The calculus of variations is begun by Isaac
  Newton
• 1696
• Johann and Jakob Bernoulli extend Newtons ideas

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Now we can get back

• 1747
• Pierre-Louise-Moreau de Maupertuis asserts a
  Principle of Least Action
• More Theological than Scientific
• Action is minimized through the Wisdom of God
• His idea of action is also kind of vague
• Action (todays definition)
• Has dimensions of length x momentum or energy x
  time
• Hmm p x or Et seems familiar

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To the Physics

• 1760
• Joseph Lagrange reformulates the principle of
  least action
• The Lagrangian, L, is defined as LT-V where T
  kinetic energy of a system and Vpotential energy
  of a system

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Hamiltons Principle

• 1834-1835
• William Rowan Hamiltons publishes two papers on
  which it is possible to base all of mechanics and
  most of classical physics.
• Hamiltons Principle is that a particle follows a
  path that minimizes L over a specific time
  interval (and consistent with any constraints).
• A constraint, for example, may be that the
  particle is moving along a surface.

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Lagranges Equations
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Lagranges Equations
And I can add zero to anything and not change the
result
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Expanding to 3 Dimensions

• Since x, y, and z are orthogonal and linearly
  independent, I can write a Lagranges EOM for
  each. In order to conserve space, I call x, y,
  and z to be dimensions 1, 2, and 3.
• So
• Amusingly enough, 1, 2, 3, could represent r, q,
  f (spherical coordinates) or r, q, z
  (cylindrical) or any other 3-dimensional
  coordinate system.

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Example Simple Harmonic Oscillator

• Recall for SHO V(x) ½ kx2 and let T1/2 mv2
• Hookes Law F-kx

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Tip

• The trick in the Lagrangian Formalism of
  mechanics is not the math but the proper choice
  of coordinate system.
• The strength of this approach is that
• Energy is a scalar and so is the Lagrangian
• The Lagrangian is invariant with respect to
  coordinate transformations

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Two Conditions Required for Lagranges Equations

• The forces acting on the system (apart from the
  forces of constraint) must be derivable from a
  potential i.e. F-dU/dx or some similar type of
  function
• The equations of constraint must be relations
  that connect the coordinates of the particles and
  may be functions of time.

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Your Turn

• Projectile
• Go to the board and work a simple projectile
  problem in cartesian coordinates. Dont worry
  about initial conditions yet.

• Now do the same in polar coordinates.
• Hint

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Introducing the Hamiltonian

• First, any Lagrangian which describes a uniform
  force field is independent of time i.e. dL/dt0.

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Introducing the Hamiltonian

• Hmmm H for Hornblower or Hamilton?

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Introducing the Hamiltonian
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H is only E when

• It is important to note that H is equal to E only
  if the following conditions are met
• The kinetic energy must be a homogeneous
  quadratic function of velocity
• The potential energy must be velocity independent
• While it is important to note that there is an
  association of H with E, it is equally important
  to note that these two are not necessarily the
  same value or even the same type of quantity!

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Making Simple Problems Difficult with the
Hamiltonian

• Most students find that the Lagrangian formalism
  is much easier than the Hamiltonian formalism
• So why bother?

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Making Simple Problems Difficult with the
Hamiltonian

• First, we need to define one more quantity
  generalized momenta, pj

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SHO with the Hamiltonian

• Big deal, right?
• But look what we did
• Lf(q,dq/dt,t)
• Hf(q,p,t)
• So our mechanics all depend on momentum but not
  velocity
• Recall light has constant velocity, c, but a
  momentum which is phc/l !

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The Big Deal

• So if we are going to define mechanics for light,
  it does not make any sense to use the Lagrangian
  formulation, only the Hamiltonian!

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That Feynman Guy!

• Richard Feynman thought that Lagrangian mechanics
  was too powerful a tool to ignore.
• Feynman developed the path integral formalism of
  quantum mechanics which is equivalent to the
  picture of Schroedinger and Dirac.
• So which is better? Both and Neither
• There seems to be no undergraduate treatment of
  path integral formalism.

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Hamiltons Equations of Motion

• Just like Lagrangian formalism, the Hamiltonian
  formalism has equations of motion. There are
  two equations for every degree of freedom
• They are

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Finishing the SHO
Hookes Law again!
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Symmetry

• Note that Hamiltons EOM are symmetric in
  appearance i.e. that q and p can almost be
  interchanged!
• Because of this symmetry, q and p are said to be
  conjugate

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Definition of Cyclic

• Consider a Hamiltonian of a free particle i.e.
  Hf(p) then dp/dt0 i.e. momentum is a
  constant of the motion
• Now in the projectile problem, U-mgy and for
  x-component, Hf(px) only!
• Thus, px constant and the horizontal variable, x
  is said to cyclic!
• A more practical definition of cyclic is
  ignorable and modern texts sometimes use this
  term.

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Definition of canonical

• Canonical is used to describe a simple, general
  set of something such as equations or
  variables.
• It was first introduced by Jacobi and rapidly
  gained common usuage but the reason for its
  introduction remained obscured even to
  contemporaries
• Lord Kelvin was quoted as saying Why it has been
  so called would be hard to say

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Poisson Brackets
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Kronecker Delta

• di,k1 if ik
• di,k0 if i?k

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Back to Fish

• Consider two continuous functions g(q,p) and
  h(q,p)
• If g,h0 then h and g are said to commute In
  other words, the order of operations does not
  matter
• If g,h1 then quantities are canonically
  conjugate
• A look ahead we will find that canonically
  conjugate quantities obey the Uncertainty
  principle

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Properties of Fish
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Levi-Civita Notation
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Levi-Civita Notation