Lecture 1 Least Action Principle

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Lecture 1Least Action Principle
Updated on 29 Aug 2008
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OverviewTowards the Path Integraland QM
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• Newtons laws

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Phase along a path
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All paths P possible
Path integral
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Path integral
Schrödinger equation
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Contents

• Newtonian mechanics
• LAP in Newtonian mechanics
• Derive equations of motion
• General form EL equations
• Advantages of LAP
• LAP for relativistic free particle
• LAP for particle in EM field

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Newtonian Mechanics
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Path x(t)

• Dynamics of all paths satisfying

Which is the correct one?
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Newtonian Mechanics
Which is the correct one?
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Least action principle

• 1. Consider all paths P
• 2. For each P, calculated a number S(P)
• which is additive (to be explained)
• 3. S(P) min ? correct path

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Concept of path

• Given end points (x1, t1), (x2, t2)

There is only one correct path
But let us consider any other path x(t) as well
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Path P x(t) x(t1) x1 x(t2) x2
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LAP in Newtonian mechanics
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Go back to Newtonian physics

• Guess the form of S
• Obtain equation of motion from S

Example
Particle mass m 1D x(t) Potential V(x)
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Additivity

• If a path is made up of many segments

S(P) S(P1) S(P2) ???
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Additivity

• Consider a small segment of path

Centred at (t, x)
Length (Dt, Dx)
Action DS ? LDt
This is the definition of Lagrangian
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Ansatz

• But
• L T ? V is NOT general definition
• S ? ?dt L is general definition

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function
functional
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Example

• A stone of mass m
• Thrown upwards from ground at t 0
• Returns to ground at t T
• What is trajectory x(t)?

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• Guess x(t) a t (T ? t)
• Determine a

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Exercise Calculate S in term of a. What value
of a makes it minimum?
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• But this derivation is correct only if we know
  x(t) is of the given form
• What if, eg, x(t) a sinpt/T?
• Need a method which does not assume a specific
  form of x(t)

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Derive Equations of Motion
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Derive equations of motion

• Let x (t) be correct path

Let x (t) h (t) be a neighboring path
h very small h (t1) h (t2) 0
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As a shorthand, we say (to 1st order in h)
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• DS S for small segment

dS difference between neighboring paths
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More General Form
Euler Lagrange Equation
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• More general form and simplify notation

Write h as dx
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AB x(t) CD x(t) h (t)
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• Euler-Lagrange equation

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Hamiltonian
The strange form will be understood later
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Hamiltonian equation of motion
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H is conserved
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Advantages of LAP
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Advantages of using paths

• Path P independent of coordinates
• Min S(P) will select the same P in any
  coordinate system

Principle of least action will be
automatically invariant
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Generalized coordinates

• Euler-Lagrange equation has same form in all
  coordinates

Circular motion
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• But Euler-Lagrange equations are the same!

centrifugal
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LAP forRelativistic Free Particle
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• S SDS
• DS for Dxµ (Dt, Dx, Dy, Dz) ?
• Invariant, linear

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For small ?
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LAP fora Particle in an EM Field
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• Assume there is a 4-vector field Am(x)
• Assume the effect is to add an interaction term
  to S

Assume Am(x) enters SI linearly
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• For a small segment of path DSI is
• linear in Am
• linear in Dxm (additivity)
• a 4-scalar DSI q Am(x) Dxm
• e will turn out to be charge

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Lorentz force law follows from this natural form
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• L ? KE ? PE !

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Euler-Lagrange equation

• Exercise!

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• We shall encounter a problem later
• H involves A
• But A is arbitrary to some extent
• ? ? ?