The Principle of Least Action

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The Principle of Least Action

• By Scott Riggs

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Conservative Forces

• If the forces acting on a particle are
  conservative, then the total energy of the
  particle, TV is conserved.
• ?F ? ds 0
• Path Independent
• F -?V
• For example
• V(y) mgh
• F (grad)V -jV(y) -mgj
• (graph)-How does nature know which path to take?

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What is Action?

• Action s ?(KE PE) dt (from t1-t2)
• KE PE is known as the Langragian
• Commonly written as
• L(x,v,t) T(v) V(x)
• The true motion of a particle is the one that
  minimizes the action.
• Nature follows the path where s is smallest

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Example Freefall motion

• Boundary Conditions
• g10, t00 y05
• t11, y10
• 3 Paths that satisfy the BC
• Ya t(y1-y0)/(t1-t0)y0
• Yb -5t25
• Yc 5cos(Pit/2)
• Compute s
• Ya-12.5
• Yb -16.7
• Yc -16.4

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Now you Know

• How to construct a Lagrangian
• How to determine the action
• Find the true path nature takes
• Pretty Amazing!
• Other areas of physics