Transcript Physics 321 Hour 19 More Lagrangian Examples

Physics 321
Hour 19
More Lagrangian Examples
Lagrangian Strategy
• Determine the number of degrees of freedom
• Write down T and U in terms of convenient
coordinates
• Write equations of constraint – constraints
between variables
• Be sure the variables left are independent of
each other
• Use
𝜕ℒ
𝜕𝑞
=
𝑑 𝜕ℒ
𝑑𝑡 𝜕𝑞
• Add friction and “lossy” forces by hand
Lagrange Multipliers
• Use only when you need to know forces of
constraint
• Write the equations of constraint in the form
𝑓 𝑞𝑗 , 𝑞𝑘 = 0
• Then the forces of constraint are
𝜕𝑓
𝜕𝑓
𝐹𝑗 = λ
, 𝐹𝑘 = λ
𝜕𝑞𝑗
𝜕𝑞𝑘
• And the equations of motion are
𝜕ℒ
𝑝𝑗 + 𝐹𝑗 =
𝜕𝑞𝑗
• Use the constraints to reduce the # of equations
of motion
Example
pendulum on incline.nb