Hamiltonian Formalism

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Hamiltonian Formalism
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• Legendre transformations
• Legendre transformation

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• What is H?
• Conjugate momentum
• Then
• So

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What is H?
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What is H?
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• What is H?
• If
• Then
• Kinetic energy
• In generalized coordinates

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• What is H?
• For scleronomous generalized coordinates
• Then
• If

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• What is H?
• For scleronomous generalized coordinates, H is a
  total mechanical energy of the system (even if H
  depends explicitly on time)
• If H does not depend explicitly on time, it is a
  constant of motion (even if is not a total
  mechanical energy)
• In all other cases, H is neither a total
  mechanical energy, nor a constant of motion

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• Hamiltons equations
• Hamiltonian
• Hamiltons equations of motion

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• Hamiltonian formalism
• For a system with M degrees of freedom, we have
  2M independent variables q and p 2M-dimensional
  phase space (vs. configuration space in
  Lagrangian formalism)
• Instead of M second-order differential equations
  in the Lagrangian formalism we work with 2M
  first-order differential equations in the
  Hamiltonian formalism
• Hamiltonian approach works best for closed
  holonomic systems
• Hamiltonian approach is particularly useful in
  quantum mechanics, statistical physics, nonlinear
  physics, perturbation theory

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Hamiltonian formalism for open systems
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• Hamiltons equations in symplectic notation
• Construct a column matrix (vector) with 2M
  elements
• Then
• Construct a 2Mx2M square matrix as follows

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• Hamiltons equations in symplectic notation
• Then the equations of motion will look compact
  in the symplectic (matrix) notation
• Example (M 2)

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• Lagrangian to Hamiltonian
• Obtain conjugate momenta from a Lagrangian
• Write a Hamiltonian
• Obtain
  from
• Plug
  into the Hamiltonian to make it a
  function of coordinates, momenta, and time

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• Lagrangian to Hamiltonian
• For a Lagrangian quadratic in generalized
  velocities
• Write a symplectic notation
• Then a Hamiltonian
• Conjugate momenta

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• Lagrangian to Hamiltonian
• Inverting this equation
• Then a Hamiltonian

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Example electromagnetism
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Example electromagnetism
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• Hamiltons equations from the variational
  principle
• Action functional
• Variations in the phase space

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• Hamiltons equations from the variational
  principle
• Integrating by parts

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• Hamiltons equations from the variational
  principle
• For arbitrary independent variations

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• Conservation laws
• If a Hamiltonian does not depend on a certain
  coordinate explicitly (cyclic), the corresponding
  conjugate momentum is a constant of motion
• If a Hamiltonian does not depend on a certain
  conjugate momentum explicitly (cyclic), the
  corresponding coordinate is a constant of motion
• If a Hamiltonian does not depend on time
  explicitly, this Hamiltonian is a constant of
  motion

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• Higher-derivative Lagrangians
• Let us recall
• Lagrangians with i gt 1 occur in many systems and
  theories
• Non-relativistic classical radiating charged
  particle (see Jackson)
• Diracs relativistic generalization of that
• Nonlinear dynamics
• Cosmology
• String theory
• Etc.

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• Higher-derivative Lagrangians
• For simplicity, consider a 1D case
• Variation

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Higher-derivative Lagrangians
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Higher-derivative Lagrangians
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• Higher-derivative Lagrangians
• Generalized coordinates/momenta

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• Higher-derivative Lagrangians
• Euler-Lagrange equations
• We have formulated a higher-order Lagrangian
  formalism
• What kind of behavior does it produce?

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Example
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Example
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• Example
• H is conserved and it generates evolution it
  is a Hamiltonian!
• Hamiltonian linear in momentum?!?!?!
• No low boundary on the total energy lack of
  ground state!!!
• Produces runaway solutions the system becomes
  highly unstable - collapse and explosion at the
  same time

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• Runaway solutions
• Unrestricted low boundary of the total energy
  produces instabilities
• Additionally, we generate new degrees of
  freedom, which require introduction of additional
  (originally unknown) initial conditions for them
• These problems are solved by means of
  introduction of constraints
• Constraints restrict unstable behavior and
  eliminate unnecessary new degrees of freedom

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9.1

• Canonical transformations
• Recall gauge invariance (leaves the evolution of
  the system unchanged)
• Lets combine gauge invariance with Legendre
  transformation
• K is the new Hamiltonian (Kamiltonian ?)
• K may be functionally different from H

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9.1

• Canonical transformations
• Multiplying by the time differential
• So

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9.1

• Generating functions
• Such functions are called generating functions
  of canonical transformations
• They are functions of both the old and the new
  canonical variables, so establish a link between
  the two sets
• Legendre transformations may yield a variety of
  other generating functions

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9.1

• Generating functions
• We have three additional choices
• Canonical transformations may also be produced
  by a mixture of the four generating functions

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9.2

• An example of a canonical transformation
• Generalized coordinates are indistinguishable
  from their conjugate momenta, and the
  nomenclature for them is arbitrary
• Bottom-line generalized coordinates and their
  conjugate momenta should be treated equally in
  the phase space

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9.4

• Criterion for canonical transformations
• How to make sure this transformation is
  canonical?
• On the other hand
• If
• Then

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9.4

• Criterion for canonical transformations
• Similarly,
• If
• Then

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9.4

• Criterion for canonical transformations
• So,
• If

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9.4

• Canonical transformations in a symplectic form
• After transformation
• On the other hand

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9.4

• Canonical transformations in a symplectic form
• For the transformations to be canonical
• Hence, the canonicity criterion is
• For the case M 1, it is reduced to (check
  yourself)

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9.3

• 1D harmonic oscillator
• Let us find a conserved canonical momentum
• Generating function

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9.3

• 1D harmonic oscillator
• Nonlinear partial differential equation for F ?
• Lets try to separate variables
• Lets try

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9.3

• 1D harmonic oscillator
• We found a generating function!

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9.3
1D harmonic oscillator
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9.3
1D harmonic oscillator
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9.5

• Canonical invariants
• What remains invariant after a canonical
  transformation?
• Matrix A is a Jacobian of a space transformation
• From calculus, for elementary volumes
• Transformation is canonical if

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9.5

• Canonical invariants
• For a volume in the phase space
• Magnitude of volume in the phase space is
  invariant with respect to canonical
  transformations

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9.5

• Canonical invariants
• What else remains invariant after canonical
  transformations?

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9.5

• Canonical invariants
• For M 1
• For many variables

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9.5

• Poisson brackets
• Poisson brackets
• Poisson brackets are invariant with respect to
  any canonical transformation

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9.5

• Poisson brackets
• Properties of Poisson brackets

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9.5

• Poisson brackets
• In matrix element notation
• In quantum mechanics, for the commutators of
  coordinate and momentum operators

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9.6
Poisson brackets and equations of motion
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9.6

• Poisson brackets and conservation laws
• If u is a constant of motion
• If u has no explicit time dependence
• In quantum mechanics, conserved quantities
  commute with the Hamiltonian

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9.6

• Poisson brackets and conservation laws
• If u and v are constants of motion with no
  explicit time dependence
• For Poisson brackets
• If we know at least two constants of motion, we
  can obtain further constants of motion

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9.4

• Infinitesimal canonical transformations
• Let us consider a canonical transformation with
  the following generating function (e small
  parameter)
• Then

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9.4

• Infinitesimal canonical transformations
• Multiplying by dt
• Then

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9.4

• Infinitesimal canonical transformations
• Infinitesimal canonical transformations
• In symplectic notation

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9.6

• Evolution generation
• Motion of the system in time interval dt can be
  described as an infinitesimal transformation
  generated by the Hamiltonian
• The system motion in a finite time interval is a
  succession of infinitesimal transformations,
  equivalent to a single finite canonical
  transformation
• Evolution of the system is a canonical
  transformation!!!

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9.9

• Application to statistical mechanics
• In statistical mechanics we deal with huge
  numbers of particles
• Instead of describing each particle separately,
  we describe a given state of the system
• Each state of the system represents a point in
  the phase space
• We cannot determine the initial conditions
  exactly
• Instead, we study a certain phase volume
  ensemble as it evolves in time

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9.9

• Application to statistical mechanics
• Ensemble can be described by its density a
  number of representative points in a given phase
  volume
• The number of representative points does not
  change
• Ensemble evolution can be thought as a canonical
  transformation generated by the Hamiltonian
• Volume of a phase space is a constant for a
  canonical transformation

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9.9

• Application to statistical mechanics
• Ensemble is evolving so its density is evolving
  too
• On the other hand
• Liouvilles theorem
• In statistical equilibrium

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10.1

• HamiltonJacobi theory
• We can look for the following canonical
  transformation, relating the constant (e.g.
  initial) values of the variables with the current
  ones
• The reverse transformations will give us a
  complete solution

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10.1

• HamiltonJacobi theory
• Let us assume that the Kamiltonian is
  identically zero
• Then
• Choosing the following generating function
• Then, for such canonical transformation

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10.1

• HamiltonJacobi theory
• HamiltonJacobi equation
• Conventionally Hamiltons principal
  function
• Partial differential equation
• First order differential equation
• Number of variables M 1

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10.1

• HamiltonJacobi theory
• Suppose the solution exists, so it will produce
  M 1 constants of integration
• One constant is evident
• We chose those M constants to be the new momenta
• While the old momenta

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10.1

• HamiltonJacobi theory
• We relate the constants with the initial values
  of our old variables
• The new coordinates are defined as
• Inverting those formulas we solve our problem

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10.1

• Have we met before?
• Remember action?

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10.1

• Hamiltons characteristic function
• When the Hamiltonian does not depend on time
  explicitly
• Generating function (Hamiltons characteristic
  function)

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10.3

• Hamiltons characteristic function
• Now we require
• So
• Detailed comparison of Hamiltons characteristic
  vs. Hamiltons principal is given in a textbook
  (10.3)

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10.3

• Hamiltons characteristic function
• What is the relationship between S and W ?
• One of possible relationships (the most
  conventional)

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10.6

• Periodic motion
• For energies small enough we have periodic
  oscillations (librations) green curves
• For energies great enough we msy have periodic
  rotations red curves
• Blue curve separatrix trajectory bifurcation
  transition from librations to rotations

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10.6

• Action-angle variables
• For either type of periodic motion let us
  introduce a new variable action variable (dont
  confuse with action!)
• A generalized coordinate conjugate to action
  variable is the angle variable
• The equation of motion for the angle variable

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10.6

• Action-angle variables
• In a compete cycle
• This is a frequency of the periodic motion

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10.2
Example 1D Harmonic oscillator
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10.2
Example 1D Harmonic oscillator
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10.6

• Action-angle variables for 1D harmonic oscillator
• Therefore, for the frequency

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10.4

• Separation of variables in the Hamilton-Jacobi
  equation
• Sometimes, the principal function can be
  successfully separated in the following way
• For the Hamiltonian without an explicit time
  dependence
• Functions Hi may or may not be Hamiltonians