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[parent] example of dynamics of a particle: free_motion

(Example)

Examples for Free Motion

(a) Polar Coordinates in a Plane

Let us get the equations in polar coordinates for motion in a plane. Here

x = rcosϕ, y = r sin ϕ,

˙x2 + y˙2 = ˙r2 + r2 ˙ϕ2,

and

( ) T = m- ˙r2 + r2ϕ˙2 . 2

Then

∂T- = m r˙, ∂T- = mr ϕ˙2. ∂ ˙r ∂r

If R is the impressed force resolved along the radius vector,

( 2) δrW = m ¨r − r ˙ϕ δr = R δr.

Also

∂T- 2 ˙ ∂T- ˙ = mr ϕ, ∂ ϕ = 0, ∂ ϕ

and, if Φ is the impressed force resolved perpendicular to the radius vector,

d ( 2 ) δϕW = m -- r ϕ˙ δϕ = Φr δϕ. dt

In more familiar form,

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

( ) m d 2d ϕ -r dt r -dt = Φ.

(b) Cylindrical Coordinates

In cylindrical coordinates, where

x = r cosϕ, y = rsin ϕ, z = z,

m ( 2 2 2 2) T = -- ˙r + r ˙ϕ + ˙z . 2

We have

∂T- = m r˙, ∂T- = mr ϕ˙2, ∂ ˙r ∂r

∂T- = mr2 ϕ˙, ∂T- = 0, ∂ ˙ϕ ∂ ϕ

∂T- = m z˙, ∂T- = 0. ∂ ˙z ∂z

Hence

( ) m ¨r − rϕ˙2 δr = R δr,

( ) m -d r2ϕ˙ δ ϕ = Φr δϕ, dt

m ¨zδz = Zδz,

or

[ 2 ( )2 ] m d-r-− r dϕ- = R, dt2 dt

( ) m- d- 2d-ϕ r dt r dt = Φ,

d2z m ---2 = Z. dt

(c) Spherical Coordinates

In spherical coordinates where

x = r cos𝜃, y = rsin𝜃 cosϕ, z = r sin 𝜃sinϕ,

[ ] m- 2 2 ˙2 2 2 ˙2 T = 2 ˙r + r 𝜃 + r sin 𝜃 ϕ .

Then

∂T- = m ˙r, ∂r˙

∂T ( 2 2 2) ---= mr 𝜃˙ + sin 𝜃ϕ˙ , ∂r

∂T- 2 ˙ ˙ = mr 𝜃, ∂𝜃

∂T- 2 ˙2 ∂ 𝜃 = mr sin 𝜃cos 𝜃ϕ ,

∂T- = mr2 sin2 𝜃ϕ˙, ∂T- = 0. ∂ ˙ϕ ∂ ϕ

Thus

[ ( ) ] δ W = m ¨r − r ˙𝜃2 + sin2 𝜃ϕ˙2 δr = R δr, r

[ ] δ𝜃W = m d-(r2 ˙𝜃) − r2sin𝜃 cos𝜃ϕ˙2 δ𝜃 = Θr δ𝜃, dt

and

( ) δϕW = m d- r2sin2𝜃 ˙ϕ δϕ = Φr sin𝜃 δϕ. dt

Or

[ { ( ) ( ) } ] d2r d𝜃 2 2 dϕ 2 m --2-− r --- + sin 𝜃 --- = R, dt dt dt

[ ( ) ( )2 ] m- -d 2d𝜃- 2 d-ϕ r dt r dt − r sin 𝜃 cos𝜃 dt = Θ,

m d ( dϕ ) -------- r2 sin2 𝜃--- = Φ. rsin𝜃 dt dt

Source

William Elwood Byerly, An Introduction to the Use of Generalized Coördinates in mechanics and Physics, Ginn and Company, 1916. Chapter I, “Introduction.”

The 1916 source work is in the public domain in the United States.


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Cross-references: domain, work, mechanics, radius vector, force, motion

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Physics Classification: 45. (Classical mechanics of discrete systems)

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