4. Variational Methods

• Review of Monte-Carlo Techniques

• Variational Methods

  • Fermat's principle
    • Ingredients for Exercise 4.1:

      def refr(x, y):		# return the refractive index at (x, y)
          if x <= 0.5:
      	return 1.0
          else:
      	return 1.5
      
      def transit(y):		# return the light transit time for this y array
          t = 0.0
          for i in range(N):
      	x = xmin + (i+0.5)*dx
      	ni = refr(x, 0.5*(y[i]+y[i+1]))
      	dy = y[i+1] - y[i]
      	ds = math.sqrt(dx*dx + dy*dy)
      	t += ni*ds/C
          return t
      
      .
      .
      
      set up the initial y array
      
      .
      .
      
      deltay = 0.1
      loop:
          i = random.randint(1, N-1)
          y0 = y[i]
          y[i] += random.uniform(-deltay, deltay)
          t = transit(y)
      

    • Exercise 4.1  [Solution]

  • Distribution of charges in a conductor
    • Calculating the potential with numpy
    • Exercise 4.2 [Solution]

    • Calculating the potential and field with numpy
    • Exercise 4.3  [Solution]