Least Action, Lagrangian

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Least Action, Lagrangian Hamiltonian Mechanics

• A very brief introduction to some very
  powerful ideas
• foolproof way to find the equations of motion
  for complicated dynamical systems
• equivalent to Newtons equations
• provides a framework for relating conservation
  laws to symmetries
• the ideas may be extended to most areas of
  fundamental physics (special and general
  relativity, electromagnetism, quantum mechanics,
  quantum field theory)
• NOT FOR EXAMINATION

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Lagrange and Hamilton

• Joseph-Louis Lagrange (1736-1810) Giuseppe
  Lodovico Lagrangia
• Sir William Rowan Hamilton (1805-1865)

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Why?

• Newtons laws are vector relations
• For complicated situations maybe hard to identify
  all the forces, especially if there are
  constraints

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Hints

• We have already exploited the energy conservation
  equation, especially for conservative forces

or

• Note that the energy equation relates scalar
  quantities

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More
For many simple systems When averaged over a
path This leads to the idea that (the
action) evaluated along a path may take a
minimum or stationary value
falling constant force
PE
KE
t
For another, more formal, approach see the
appendix on DAlembert Hamilton
SHM
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Least Action by brute force
y
Numerical integration Fixed distance R and time
tR Horizontal speed u0R/tR Trajectory is of the
form Note Vary v0 - calculate For paths B,
C, D A has v0 lt 0 (!) C is path with S
stationary With
D
C
B
x
R
O
A
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Path integrals

• L T - V
• is the Lagrangian, a function
• of
• Minimise the action integral w.r.t. variations in
  the path?
• Calculus of variations?
• No, use Eulers geometrical approach.
  Approximate S as sum
• xi at start of each segment
• for segment

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Euler I
x

• Key observation every section of action
  integral or sum must be stationary
• Consider two linearised sections of sum, with
  point at time t2 moved to N

N
O
M
t
1 2 3
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Euler II
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From action to Newton
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Generalised coordinates
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Example Newtonian gravity
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Sliding blocks
B
Mass A M, mass B m Find the initial
acceleration of A No friction
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Pendulum with oscillating support
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Symmetry and conserved quantities

• The Lagrangian approach provides a useful
  alternative to a direct formulation using
  Newtons equations
• However it also provides the framework in which
  fundamental questions about the nature of forces
  and interactions can studied.
• In particular the very close relationship between
  a symmetry of the Lagrangian and a conserved
  quantity
• By symmetry we mean an operation eg coordinate
  rotation that leaves the Lagrangian unchanged

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Noethers theorem
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Examples
Free particle and translational invariance
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Rotational Invariance
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Conservation of energy
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The Hamiltonian Hamiltons Eqs of Motion
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Summary

• Lagrangian LT-V plus Euler-Lagrange equations
  give a convenient way of generating equations of
  motion for complicated dynamical problems
• Noethers theorem provides a mechanism for
  finding conserved quantities that follow from
  symmetry of the Lagrangian
• The Hamiltonian is an alternative formulation,
  useful in formal treatments, and with an analogue
  in Schrödingers equation of quantum mechanics.
• The method of least action can be extended to
  many other areas of physics
• to learn more try
• Classical Mechanics Short Option S7 next year

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Sources and further reading

• Any advanced text on classical mechanics
• Kibble Berkshire 5th Ed, IC Press 2005
• Goldstein, Poole Safko 3rd Ed, AW 2002
• Fowles Cassidy, Harcourt Brace 1993
• Cowan, RKP 1984
• Feynman Lectures on Physics, Vol II 19-1
• Feynman, Leighton Sands, AW 1964
• For many interesting, almost evangelical,
  articles on the principles of least action
• http//www.eftaylor.com/leastaction.html
• (Prof Edwin Taylor (MIT) and collaborators)

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Appendix DAlembert, Hamilton Least Action
B
Hamilton considers variation in paths between
fixed points at A and B such that
varied coordinates are evaluated at the same
time, so dt 0. Note that all paths must have
the same start and end points in space and time.
variation of path at fixed t
A
tB
tA
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Problems
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