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[parent] polar coordinate motion example of generalized coordinates (Example)

As an example let us get the equations in polar coordinates for motion in a plane

Here

x = r cosϕ, y = r sin ϕ

˙x2 + ˙y2 = v2 = ˙r2 + r2ϕ˙2

and

m- [ 2 2 ˙2] T = 2 ˙r + r ϕ

∂T- = m r˙ ∂ ˙r

∂T-= mr ˙ϕ2. ∂r

2 δrW = m [¨r − rϕ˙]δr = R δr

if R is the impressed force resolved along the radius vector.

∂T- 2 ˙ ˙= mr ϕ, ∂ϕ

∂T --- = 0. ∂ϕ

d 2 δϕW = m --(r ˙ϕ)δϕ = Φr δϕ dt

if Φ is the impressed force resolved perpendicular to the radius vector.

In a more familiar form

[ ] d2r ( dϕ )2 m --2-− r --- = R, dt dt

( ) m- d- 2d-ϕ r dt r dt = Φ.

"polar coordinate motion example of generalized coordinates" is owned by bloftin.
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Cross-references: radius vector, force, motion

This is version 2 of polar coordinate motion example of generalized coordinates, born on 2008-07-18, modified 2008-07-21.
Object id is 287, canonical name is PolarCoordinateMotionExampleOfGeneralizedCoordinates.
Accessed 1806 times total.

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Physics Classification: 45.20.Jj (Lagrangian and Hamiltonian mechanics)
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