The Principle of Least Action
By Scott Riggs
Conservative Forces
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If the forces acting on a particle are conservative, then the total
energy of the particle, T+V is conserved.
 ∫F ۰ ds = 0
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Path Independent
F = -∆V
For example
– V(y) = mgh
– F – (grad)V = -jV’(y) = -mgj
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(graph)-How does nature know which path to take?
What is Action?
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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
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Boundary Conditions
g=10, t0=0 y0=5
t1=1, y1=0
3 Paths that satisfy the BC
Ya= t(y1-y0)/(t1-t0)+y0
Yb= -5t^2+5
Yc= 5cos(Pi*t/2)
Compute s
Ya=-12.5
Yb = -16.7
Yc = -16.4
Now you Know
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How to construct a Lagrangian
How to determine the action
Find the true path nature takes
Pretty Amazing!
Other areas of physics