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Classical Mechanics
Lagrangian mechanics
Use this for “In what order do the steps of Lagrange’s method go, for a bead sliding on a spinning hoop?”
Put in order
Put the steps of the Lagrangian method in order.
1
?
2
?
3
?
4
?
5
?
Assumptions
•
The constraints are holonomic: each one fixes a relation among the coordinates, so the coordinates left over can each change freely.
•
Every force that does work comes from a potential V; the constraint forces do no work and never appear.
•
A coordinate missing from L is cyclic: its momentum ∂L/∂q̇ stays the same for the whole motion.
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More problem types
An Atwood machine with a heavy pulley, by Lagrange
Use this for “Blocks of 4 kg and 6 kg hang over a 3 kg disk pulley. Use the Lagrangian to find their acceleration, then find both tensions.”
A bead on a spinning hoop
Use this for “A bead slides on a 0.25 m hoop spun at 9 rad/s about its vertical diameter. At what angle does it rest, and how fast does it rock about that point?”