Physics 321 Hour 18 Lagrange Multipliers

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Transcript Physics 321 Hour 18 Lagrange Multipliers 

Physics 321
Hour 18
Lagrange Multipliers
Lagrange Multipliers
β€’ Use only when you need to know forces of
constraint
β€’ Write the equations of constraint in the form
𝑓 π‘žπ‘— , π‘žπ‘˜ = 0
β€’ Then the forces of constraint are
πœ•π‘“
πœ•π‘“
𝐹𝑗 = Ξ»
, πΉπ‘˜ = Ξ»
πœ•π‘žπ‘—
πœ•π‘žπ‘˜
β€’ And the equations of motion are
πœ•β„’
𝑝𝑗 =
+ 𝐹𝑗
πœ•π‘žπ‘—
β€’ Use the constraints to reduce the # of equations
of motion
Outline
β€’ Calculus of Variations
β€’ Lagrange Multipliers
β€’ Using Lagrange Multipliers
β€’ Only to find forces of constraint!!!
β€’ Atwood Machine II
β€’ Start with β€œnormal” solution
β€’ Ball on ball (.nb)
β€’ One constraint
β€’ Two constraints
Examples
ball on ball.nb