Lagrangian and Hamiltonian Dynamics

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Lagrangian and Hamiltonian Dynamics

• Chapter 7
• Claude Pruneau
• Physics and Astronomy

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Minimal Principles in Physics

• Hero of Alexandria 2nd century BC.
• Law governing light reflection minimizes the path
  length.
• Fermats Principle
• Refraction can be understood as the path that
  minimizes the time - and Snells law.
• Maupertuis (1747)
• Principle of least action.
• Hamilton (1834, 1835)

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Hamiltons Principle

• Of all possible paths along which a dynamical
  system may move from one point to another within
  a specified time interval (consistent with any
  constraints), the actual path followed is that
  which minimizes the time integral of the
  difference between the kinetic and potential
  energy.

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Hamiltons Principle

• In terms of calculus of variations

• The d is a shorthand notation which represents a
  variation as discussed in Chap 6.
• The kinetic energy of a particle in fixed,
  rectangular coordinates is of function of 1st
  order time derivatives of the position

• The potential energy may in general be a
  function of both positions and velocities.
  However if the particle moves in a conservative
  force field, it is a function of the xi only.

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Hamiltons Principle (contd)

• Define the difference of T and U as the Lagrange
  function or Lagrangian of the particle.

• The minimization principle (Hamiltons) may thus
  be written

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Derivation of Euler-Lagrange Equations

• Establish by transformation

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Lagrange Equations of Motion

• L is called Lagrange function or Lagrangian for
  the particle.
• L is a function of xi and dxi/dt but not t
  explicitly (at this point)

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Example 1 Harmonic Oscillator

• Problem Obtain the Lagrange Equation of motion
  for the one-dimensional harmonic oscillator.

• Solution
• Write the usual expression for T and U to
  determine L.

• Calculate derivatives.

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Example 1 Harmonic Oscillator (contd)

• Combine in Lagrange Eq.

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Example 2 Plane Pendulum
Problem Obtain the Lagrange Equation of motion
for the plane pendulum of mass m.
l

• Solution
• Write the expressions for T and U to determine L.

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Example 2 Plane Pendulum (contd)

• Calculate derivatives of L by treating ? as if it
  were a rectangular coordinate.

• Combine...

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Remarks

• Example 2 was solved by assuming that ? could be
  treated as a rectangular coordinate and we obtain
  the same result as one obtains through Newtons
  equations.
• The problem was solved by involving kinetic
  energy, and potential energy. We did not use the
  concept of force explicitly.

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Generalized Coordinates

• Seek generalization of coordinates.
• Consider mechanical systems consisting of a
  collection of n discrete point particles.
• Rigid bodies will be discussed later
• We need n position vectors, I.e. 3n quantities.
• If there are m constraint equations that limit
  the motion of particle by for instance relating
  some of coordinates, then the number of
  independent coordinates is limited to 3n-m.
• One then describes the system as having 3n-m
  degrees of freedom.

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Generalized Coordinates (contd)

• Important note if s3n-m coordinates are
  required to describe a system, it is NOT
  necessary these s coordinates be rectangular or
  curvilinear coordinates.
• One can choose any combination of independent
  parameters as long as they completely specify the
  system.
• Note further that these coordinates (parameters)
  need not even have the dimension of length (e.g.
  q in our previous example).
• We use the term generalized coordinates to
  describe any set of coordinates that completely
  specify the state of a system.
• Generalized coordinates will be noted q1, q2, ,
  qn.

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Generalized Coordinates (contd)

• A set of generalized coordinates whose number
  equals the number s of degrees of freedom of the
  system, and not restricted by the constraints is
  called a proper set of generalized coordinates.
• In some cases, it may be useful/convenient to use
  generalized coordinates whose number exceeds the
  number of degrees of freedom, and to explicitly
  use constraints through Lagrange multipliers.
• Useful e.g. if one wishes to calculate forces due
  to constraints.
• The choice of a set of generalized coordinates is
  obviously not unique.
• They are in general (infinitely) many
  possibilities.
• In addition to generalized coordinates, we shall
  also consider time derivatives of the generalized
  coordinates called generalized velocities.

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Generalized Coordinates (contd)
Notation
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Transformation

• Transformation The normal coordinates can be
  expressed as functions of the generalized
  coordinates - and vice-versa.

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Transformation (contd)

• Rectangular components of the velocties depend on
  the generalized coordinates, the generalized
  velocities, and the time.

• Inverse transformations are noted

• There are m3n-s equations of constraint

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Example Generalized coordinates

• Question Find a suitable set of generalized
  coordinates for a point particle moving on the
  surface of a hemisphere of radius R whose center
  is at the origin.

• Solution Motion on a spherical surface implies

• Choose cosines as generalized coordinates.

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Example Generalized coordinates (contd)

• q1, q2, q3 do not constitute a proper set of
  generalized of coordinates because they are not
  independent.
• One may however choose e.g. q1, q2, and the
  constraint equation

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Lagrange Eqs in Gend Coordinates

• Of all possible paths along which a dynamical
  system may move from one point to another in
  configuration space within a specified time
  interval, the actual path followed is that which
  minimizes the time integral of the Lagrangian for
  the system.

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Remarks

• Lagrangian defined as the difference between
  kinetic and potential energies.
• Energy is a scalar quantity (at least in Galilean
  relativity).
• Lagrangian is a scalar function.
• Implies the lagrangian must be invariant with
  respect to coordinate transformations.
• Certain transformations that change the
  Lagrangian but leave the Eqs of motion unchanged
  are allowed.
• E.G. if L is replaced by Ld/dt f(qi,t), for a
  function with continuous 2nd partial derivatives.
  (Fixed end points)
• The choice of reference for U is also irrelevant,
  one can add a constant to L.

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Lagranges Eqs

• The choice of specific coordinates is therefore
  immaterial

• Hamiltons principle becomes

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Lagranges Eqs
s equations m constraint equations

• Applicability
• Force derivable from one/many potential
• Constraint Eqs connect coordinates, may be fct(t)

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Lagrange Eqs (contd)

• Holonomic constraints

• Scleronomic constraints
• Independent of time
• Rheonomic
• Dependent on time

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Example Projectile in 2D

• Question Consider the motion of a projectile
  under gravity in two dimensions. Find equations
  of motion in Cartesian and polar coordinates.

• Solution in Cartesian coordinates

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Example Projectile in 2D (contd)

• In polar coordinates

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Example Motion in a cone

• Question A particle of mass m is constrained
  to move on the inside surface of a smooth cone of
  hal-angle a. The particle is subject to a
  gravitational force. Determine a set of
  generalized coordinates and determine the
  constraints. Find Lagranges Eqs of motion.

z
Solution Constraint
2 degrees of freedom only! 2 generalized
coordinates.
?
y
?
x
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Example Motion in a cone (contd)

• Choose to eliminate z.

L is independent of q.
is the angular momentum relative to the axis of
the cone.
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Example Motion in a cone (contd)

• For r

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Lagranges Eqs with underdetermined multipliers

• Constraints that can be expressed as algebraic
  equations among the coordinates are holonomic
  constraints.
• If a system is subject to such equations, one can
  always find a set of generalized coordinates in
  terms of which Eqs of motion are independent of
  these constraints.
• Constraints which depend on the velocities have
  the form

Non holonomic constraints unless eqs can be
integrated to yield constrains among the
coordinates.
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• Consider

• Generally non-integrable, unless

• One thus has

• Or

• Which yields

• So the constraints are actually holonomic

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Constraints

• We therefore conclude that if constraints can be
  expressed

• Constraints Eqs given in differential form can be
  integrated in Lagrange Eqs using undetermined
  multipliers.
• For

• One gets

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Forces of Constraint

• The underdetermined multipliers are the forces of
  constraint

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Example Disk rolling incline plane
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Example Motion on a sphere
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7.6 Equivalence of Lagranges and Newtons
Equations

• Lagrange and Newton formulations of mechanic are
  equivalent
• Different view point, same eqs of motion.
• Explicit demonstration

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Generalized momentum
Generalized force defined through virtual work dW
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For a conservative system
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7.10 Canonical Equations of Motion Hamilton
Dynamics
Whenever the potential energy is velocity
independent
Result extended to define the Generalized Momenta
Given Euler-Lagrange Eqs
One also finds
The Hamiltonian may then be considered a function
of the generalized coordinates, qj, and momenta
pj
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whereas the Lagrangian is considered a function
of the generalized coordinates, qj, and their
time derivative.
To convert from the Lagrange formulation to the
Hamiltonian formulation, we consider
But given
One can also write
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That must also equal
We then conclude
Hamilton Equations
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Lets now rewrite
And calculate
Finally conclude
If
H is a constant of motion
If, additionally, HUTE, then E is a conserved
quantity.
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Some remarks

• The Hamiltonian formulation requires, in general,
  more work than the Lagrange formulation to derive
  the equations of motion.
• The Hamiltonian formulation simplifies the
  solution of problems whenever cyclic variables
  are encountered.
• Cyclic variables are generalized coordinates that
  do not appear explicitly in the Hamiltonian.
• The Hamiltonian formulation forms the basis to
  powerful extensions of classical mechanics to
  other fields e.g. Beam physics, statistical
  mechanics, etc.
• The generalized coordinates and momenta are said
  to be canonically conjugates because of the
  symmetric nature of Hamiltons equations.

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More remarks

• If qk is cyclic, I.e. does not appear in the
  Hamiltonian, then
• And pk is then a constant of motion.
• A coordinate cyclic in H is also cyclic in L.
• Note if qk is cyclic, its time derivative
  q-dot appears explicitly in L.
• No reduction of the number of degrees of freedom
  in the Lagrange formulation still s 2nd order
  equations of motion.
• Reduction by 2 of the number of equations to be
  solved in the Hamiltonian formulation since 2
  become trivial

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where ?k is possibly a function of t.
One thus get the simple (trivial) solution
The solution for a cyclic variable is thus
reduced to a simple integral as above.
The simplest solution to a system would occur if
one could choose the generalized coordinates in a
way they are ALL cyclic. One would then have s
equations of the form Such a choice is
possible by applying appropriate transformations
this is known as Hamilton-Jacobi Theory.
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Some remarks on the calculus of variation
Hamiltons Principle
Evaluated
The above integral becomes after integration by
parts
Which gives rise to Euler-Lagrange equations
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Alternatively, Hamiltons Principle can be
written
Which evaluates to
Consider
Integrate by parts
The variation may then be written
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Hamilton Equations