Variational Principles

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Variational Principles and Lagranges Equations
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• Definitions
• Lagrangian density
• Lagrangian
• Action
• How to find the special value for action
  corresponding to observable ?

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• Variational principle
• Maupertuis Least Action Principle
• Hamilton Hamiltons Variational Principle
• Feynman Quantum-Mechanical Path Integral
  Approach

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• Functionals
• Functional given any function f(x), produces a
  number S
• Action is a functional
• Examples of finding special values of
  functionals using variational approach
• shortest distance between two points on a plane
• the brachistochrone problem
• minimum surface of revolution
• etc.

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• Shortest distance between two points on a plane
• An element of length on a plane is
• Total length of any curve going between points 1
  and 2 is
• The condition that the curve is the shortest
  path is that the functional I takes its minimum
  value

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• The brachistochrone problem
• Find a curve joining two points, along which a
  particle falling from rest under the influence of
  gravity travels from the highest to the lowest
  point in the least time
• Brachistochrone solution the value of the
  functional t y(x) takes its minimum value

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• Calculus of variations
• Consider a functional of the following type
• What function y(x) yields a stationary value
  (minimum, maximum, or saddle) of J ?

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• Calculus of variations
• Assume that function y0(x) yields a stationary
  value and consider all possible functions in the
  form

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• Calculus of variations
• In this case our functional becomes a function
  of a
• Stationary value condition

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Stationary value
1
2
3
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Stationary value
1
2
3
u
dv
u
v
v
du
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Stationary value
1
2
3
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Stationary value
1
2
3
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Stationary value
arbitrary
Trivial ?
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Stationary value
arbitrary
Nontrivial !!! ?
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Shortest distance between two points on a plane
Straight line! ?
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The brachistochrone problem
Scary! ?
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• Recipe
• 1. Bring together structure and fields
• 2. Relate this togetherness to the entire system
• 3. Make them fit best when the fields have
  observable dependencies

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• Back to trajectories and Lagrangians
• How to find the special values for action
  corresponding to observable trajectories ?
• We look for a stationary action using
  variational principle

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• Recipe
• 1. Bring together structure and fields
• 2. Relate this togetherness to the entire system
• 3. Make them fit best when the fields have
  observable dependencies

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• Back to trajectories and Lagrangians
• For open systems, we cannot apply variational
  principle in a consistent way, since integration
  in not well defined for them
• We look for a stationary action using
  variational principle for closed systems

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Stationary value
Nontrivial !!! ?
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• Simplest non-trivial case
• Lets start with the simplest non-trivial result
  of the variational calculus and see if it can
  yield observable trajectories

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Stationary value
Nontrivial !!! ?
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• Euler- Lagrange equations
• These equations are called the Euler- Lagrange
  equations

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• Recipe
• 1. Bring together structure and fields
• 2. Relate this togetherness to the entire system
• 3. Make them fit best when the fields have
  observable dependencies

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• How to construct Lagrangians?
• Let us recall some kindergarten stuff
• On our classical-mechanical level, we know
  several types of fundamental interactions
• Gravitational
• Electromagnetic
• Thats it

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• Gravitation
• For a particle in a gravitational field, the
  trajectory is described via 2nd Newtons Law
• This system can be approximated as closed
• The structure (symmetry) of the system is
  described by the gravitational potential

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• Electromagnetic field
• For a charged particle in an electromagnetic
  field, the trajectory is described via 2nd
  Newtons Law
• This system can be approximated as closed
• The structure (symmetry) of the system is
  described by the scalar and vector potentials

Really???
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Electromagnetic field
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Electromagnetic field
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• Electromagnetic field
• Lorentz force! ?

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• Kindergarten
• Thereby
• In component form

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• How to construct Lagrangians?
• Kindergarten stuff
• The kindergarten equations look very similar
  to the Euler-Lagrange equations! We may be on the
  right track! ?

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Gravitation
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Gravitation
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Electromagnetism
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• Bottom line
• We successfully demonstrated applicability of
  our recipe
• This approach works not just in classical
  mechanics only, but in all other fields of physics

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• Some philosophy
• de Maupertuis on the principle of least action
  (Essai de cosmologie, 1750) In all the
  changes that take place in the universe, the sum
  of the products of each body multiplied by the
  distance it moves and by the speed with which it
  moves is the least that is possible.
• How does an object know in advance
• what trajectory corresponds to a
• stationary action???
• Answer quantum-mechanical path
• integral approach

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• Some philosophy
• Feynman Is it true that the particle doesn't
  just "take the right path" but that it looks at
  all the other possible trajectories? ... The
  miracle of it all is, of course, that it does
  just that. ... It isn't that a particle takes the
  path of least action but that it smells all the
  paths in the neighborhood and chooses the one
  that has the least action ...

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• Some philosophy
• Dyson In 1949, Dick Feynman told me about his
  "sum over histories" version of quantum
  mechanics. "The electron does anything it likes,"
  he said. "It just goes in any direction at any
  speed, forward or backward in time, however it
  likes, and then you add up the amplitudes and it
  gives you the wave-function." I said to him,
  "You're crazy." But he wasn't.

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• Some philosophy
• Philosophical meaning of the Lagrangian
  formalism structure of a system determines its
  observable behavior
• So, that's it?
• Why do we need all this?
• In addition to the deep philosophical meaning,
  Lagrangian formalism offers great many advantages
  compared to the Newtonian approach

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• Lagrangian approach extra goodies
• It is scalar (Newtonian vectorial)
• Allows introduction of configuration space and
  efficient description of systems with constrains
• Becomes relatively simpler as the mechanical
  system becomes more complex
• Applicable outside Newtonian mechanics
• Relates conservation laws with symmetries
• Scale invariance applications
• Gauge invariance applications

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• Simple example
• Projectile motion

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• Another example
• Another Lagrangian
• What is going on?!

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• Gauge invariance
• For the Lagrangians of the type
• And functions of the type
• Lets introduce a transformation (gauge
  transformation)

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Gauge invariance
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Gauge invariance
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Gauge invariance
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• Back to the question How to construct
  Lagrangians?
• Ambiguity different Lagrangians result in the
  same equations of motion
• How to select a Lagrangian appropriately?
• It is a matter of taste and art
• It is a question of symmetries of the physical
  system one wishes to describe
• Conventionally, and for expediency, for most
  applications in classical mechanics

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• Cylindrically symmetric potential
• Motion in a potential that depends only on the
  distance to the z axis
• It is convenient to work in cylindrical
  coordinates
• Then

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• Cylindrically symmetric potential
• How to rewrite the equations of motion in
  cylindrical coordinates?

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• Generalized coordinates
• Instead of re-deriving the Euler-Lagrange
  equations explicitly for each problem (e.g.
  cylindrical coordinates), we introduce a concept
  of generalized coordinates
• Let us consider a set of coordinates
• Assume that the Euler-Lagrange equations hold
  for these variables
• Consider a new set of (generalized) coordinates

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• Generalized coordinates
• We can, in theory, invert these equations
• Let us do some calculations

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• Generalized coordinates
• The Euler-Lagrange equations are the same in
  generalized coordinates!!!

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• Generalized coordinates
• If the Euler-Lagrange equations are true for one
  set of coordinates, then they are also true for
  the other set

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• Cylindrically symmetric potential
• Radial force causes a change in radial momentum
  and a centripetal acceleration

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• Cylindrically symmetric potential
• Angular momentum relative to the z axis is a
  constant

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• Cylindrically symmetric potential
• Axial component of velocity does not change

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• Symmetries and conservation laws
• The most beautiful and useful illustration of
  the structure vs observed behavior philosophy
  is the link between symmetries and conservation
  laws
• Conjugate momentum for coordinate
• If Lagrangian does not depend on a certain
  coordinate, this coordinate is called cyclic
  (ignorable)
• For cyclic coordinates, conjugate momenta are
  conserved

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• Symmetries and conservation laws
• For cyclic coordinates, conjugate momenta are
  conserved

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• Cylindrically symmetric potential
• Cyclic coordinates
• Rotational symmetry Translational
  symmetry
• Conjugate momenta

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• Electromagnetism
• Conjugate momenta

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• Noethers theorem
• Relationship between Lagrangian symmetries and
  conserved quantities was formalized only in 1915
  by Emmy Noether
• For each symmetry of the Lagrangian, there is a
  conserved quantity
• Let the Lagrangian be invariant under
• the change of coordinates
• a is a small parameter. This invariance
• has to hold to the first order in a

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• Noethers theorem
• Invariance of the Lagrangian
• Using the Euler-Lagrange equations

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• Example
• Motion in an x-y plane of a mass on a spring
  (zero equilibrium length)
• The Lagrangian is invariant (to the first order
  in a) under the following change of coordinates
• Then, from Noethers theorem it follows that

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• Example
• In polar coordinates
• The conserved quantity
• Angular momentum in the x-y plane is conserved

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• Example
• For the same problem, we can start with a
  Lagrangian expressed in polar coordinates
• The Lagrangian is invariant (to any order in a)
  under the following change of coordinates
• The conserved quantity from Noethers theorem

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• Back to trajectories and Lagrangians
• How to find the special values for action
  corresponding to observable trajectories ?
• We look for a stationary action using
  variational principle

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Stationary value
1
2
3
u
dv
u
v
v
du
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• More on symmetries
• Full time derivative of a Lagrangian
• From the Euler-Lagrange equations
• If

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• What is H?
• Let us expand the Lagrangian in powers of
  
• From calculus, for a homogeneous function f of
  degree n (Eulers theorem)

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• What is H?
• If the Lagrangian has a form
• Then
• For electromagnetism

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• Conservation of energy
• In the field formalism, the conservation of H is
  a part of Noethers theorem

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• The brachistochrone problem
• Similarly to the H-trick

!!!
Scary! ?
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• The brachistochrone problem
• Change of variables
• Parametric solution
• (cycloid)

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• Scale invariance
• For Lagrangians of the following form
• And homogeneous L0 of degree k
• Introducing scale and time transformations
• Then

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• Scale invariance
• Therefore, after transformations
• If
• Then
• The Euler-Lagrange equations after
  transformations
• The same!

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• Scale invariance
• So, the Euler-Lagrange equations after
  transformations are the same if
• Free fall
• Let us recall

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• Scale invariance
• So, the Euler-Lagrange equations after
  transformations are the same if
• Mass on a spring
• Let us recall

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• Scale invariance
• So, the Euler-Lagrange equations after
  transformations are the same if
• Keplers problem
• Let us recall 3rd Keplers law

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• How about open systems?
• For some systems we can neglect their
  interaction with the outside world and formulate
  their behavior in terms of Lagrangian formalism
• For some systems we can not do it
• Approach to describe the system without leaks
  and feeds and then add them to the description
  of the system

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• How about open systems?
• For open systems, we first describe the system
  without leaks and feeds
• After that we add leaks and feeds to the
  description of the system
• Q Non-conservative generalized forces

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• Generalized forces
• Forces
• 1 Conservative (Potential)
• 2 Non-conservative

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• Generalized forces
• In principle, there is no need to introduce
  generalized forces for a closed system fully
  described by a Lagrangian
• Feynman The principle of least action
• only works for conservative systems
• where all forces can be gotten from a potential
  function. On a microscopic level on the
  deepest level of physics there are no
  non-conservative forces. Non-conservative forces,
  like friction, appear only because we neglect
  microscopic complications there are just too
  many particles to analyze.
• So, introduction of non-conservative forces is a
  result of the open-system approach

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• Degrees of freedom
• The number of degrees of freedom is the number
  of independent coordinates that must be specified
  in order to define uniquely the state of the
  system
• For a system of N free particle there are 3N
  degrees of freedom (3N coordinates)

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• Constraints
• We can impose k constraints on the system
• The number of degrees of freedom is reduced to
  3N k s
• It is convenient to think of the remaining s
  independent coordinates as the coordinates of a
  single point in an s-dimensional space
  configuration space

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• Types of constraints
• Holonomic (integrable) constraints can be
  expressed in the form
• Nonholonomic constraints cannot be expressed in
  this form
• Rheonomous constraints contain time dependence
  explicitly
• Scleronomous constraints do not contain time
  dependence explicitly

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• Analysis of systems with holonomic constraints
• Elimination of variables using constraints
  equations
• Use of independent generalized coordinates
• Lagranges multiplier method

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• Double 2D pendulum
• An example of a holonomic scleronomous
  constraint
• The trajectories of the system are very complex
• Lagrangian approach produces equations of motion
  
• We need 2 independent generalized coordinates
• (N 2, k 2 2, s 3 N k 2)

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• Double 2D pendulum
• Relative to the pivot, the Cartesian coordinates
• Taking the time derivative, and then squaring
• Lagrangian in Cartesian coordinates

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• Double 2D pendulum
• Lagrangian in new coordinates
• The equations of motion

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• Double 2D pendulum
• Special case
• The equations of motion
• More fun at
• http//www.mathstat.dal.ca/selinger/lagrange/doub
  lependulum.html

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• Lagranges multiplier method
• Used when constraint reactions are the object of
  interest
• Instead of considering 3N - k variables and
  equations, this method deals with 3N k
  variables
• As a results, we obtain 3N trajectories and k
  constraint reactions
• Lagranges multiplier method can be applied to
  some nonholonomic constraints

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• Lagranges multiplier method
• Let us explicitly incorporate constraints into
  the structure of our system
• For observable trajectories
• So

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• Lagranges multiplier method
• - constraint reactions
• Now we have 3N k equations for and

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• Application to a nonholonomic case
• A particle on a smooth hemisphere
• One nonholonomic constraint
• While the particle remains on the sphere, the
  constraint is holonomic
• And the reaction from the surface is not zero

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• Application to a nonholonomic case
• Constraint equation in cylindrical coordinates
• New Lagrangian in cylindrical coordinates
• Equations of motion

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• Application to a nonholonomic case
• Constraint equation in cylindrical coordinates
• New Lagrangian in cylindrical coordinates
• Equations of motion

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• Application to a nonholonomic case
• Constraint equation in cylindrical coordinates
• New Lagrangian in cylindrical coordinates
• Equations of motion
• Trivial

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• Application to a nonholonomic case
• Constraint reaction

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• Application to a nonholonomic case
• Constraint reaction
• Reaction disappears when
• The particle becomes airborne