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G.2  Beschleunigung

Die Beschleunigung ist in kartesischen Koordinaten

( d2x) ( ) d2r | dt2| ¨x a = ----= |( d2y2|) = |( ¨y|) = ¨xex + ¨yey + ¨zez dt2 ddt2z ¨z dt2
(G.1)

Wir verwenden die Beziehungen

x = r sin(θ) cos(ϕ) (G.2)
y = r sin(θ) sin(ϕ) (G.3)
z = r cos(θ) (G.4)

und leiten sie zweimal ab. Wir erhalten aus

ẋ = ṙ sin(θ) cos(ϕ) + r cos(θ) cos(ϕ) ˙ θ - r sin(θ) sin(ϕ) ˙ ϕ
ẏ = ṙ sin(θ) sin(ϕ) + r cos(θ) sin(ϕ)˙θ + r sin(θ) cos(ϕ)ϕ˙
ż = ṙ cos(θ) - r sin(θ) ˙ θ

die Gleichungen

ẍ =¨rsin(θ) cos(ϕ) + ṙ cos(θ) cos(ϕ) ˙ θ -ṙ sin(θ) sin(ϕ) ˙ ϕ (G.5)
+ ṙ cos(θ) cos(ϕ)˙θ - r sin(θ) cos(ϕ)˙θ2 - r cos(θ) sin(ϕ)ϕ˙˙θ + r cos(θ) cos(ϕ)¨θ
-ṙ sin(θ) sin(ϕ)ϕ˙ - r cos(θ) sin(ϕ)˙ϕθ˙ - r sin(θ) cos(ϕ)ϕ˙2 - r sin(θ) sin(ϕ)¨ϕ
= ¨rsin(θ) cos(ϕ)
+ ṙ˙θ[cos(θ)cos(ϕ) + cos(θ)cos(ϕ )]
+ ṙ˙ϕ[- sin(θ)sin(ϕ) - sin(θ)sin(ϕ)]
+ r˙ θ2[- sin (θ)cos(ϕ)]
+ r˙ϕ˙θ[- cos(θ) sin(ϕ ) - cos(θ)sin(ϕ)]
+ r¨ θ [cos(θ)cos(ϕ)]
+ r˙ϕ2[- sin(θ)cos(ϕ)]
+ r¨ ϕ [- sin(θ)sin(ϕ)]
= ¨rsin(θ) cos(ϕ) + 2ṙ˙θ cos(θ) cos(ϕ) - 2ṙ˙ϕ sin(θ) sin(ϕ) - r˙θ2 sin(θ) cos(ϕ)
- 2r˙ϕθ˙ cos(θ) sin(ϕ) + r¨θcos(θ) cos(ϕ) - rϕ˙2 sin(θ) cos(ϕ) - r¨ϕsin(θ) sin(ϕ)

und

ý =¨rsin(θ) sin(ϕ) + ṙ cos(θ) sin(ϕ)˙ θ + ṙ sin(θ) cos(ϕ) ˙ ϕ (G.6)
+ ṙ cos(θ) sin(ϕ)θ˙ - r sin(θ) sin(ϕ)˙θ2 + r cos(θ) cos(ϕ)˙θϕ˙ + r cos(θ) sin(ϕ)¨θ
+ ṙ sin(θ) cos(ϕ)ϕ˙ + r cos(θ) cos(ϕ)θ˙ϕ˙ - r sin(θ) sin(ϕ)˙ϕ2 + r sin(θ) cos(ϕ)¨ϕ
= ¨rsin(θ) sin(ϕ)
+ ṙ˙θ[cos(θ)sin(ϕ) + cos(θ)sin(ϕ)]
+ ṙ˙ϕ[sin(θ)cos(ϕ) + sin(θ)cos(ϕ)]
- r˙ θ2 sin(θ) sin(ϕ)
+ r˙θ ˙ϕ[cos(θ)cos(ϕ ) + cos(θ)cos(ϕ )]
+ r cos(θ) sin(ϕ)¨ θ
- r˙ϕ2 sin(θ) sin(ϕ)
+ r sin(θ) cos(ϕ)¨ ϕ
= ¨rsin(θ) sin(ϕ) + 2ṙθ˙ cos(θ) sin(ϕ) + 2ṙϕ˙ sin(θ) cos(ϕ) - r˙θ2 sin(θ) sin(ϕ)
+ 2r˙θ ˙ϕ cos(θ) cos(ϕ) + r¨θcos(θ) sin(ϕ) - rϕ˙2 sin(θ) sin(ϕ) + rϕ¨sin(θ) cos(ϕ)

sowie

¨z = ¨rcos(θ) -ṙ sin(θ) ˙ θ (G.7)
-ṙ sin(θ)˙θ - r cos(θ)θ˙2 - r sin(θ)¨θ
= ¨rcos(θ) - 2ṙ sin(θ) ˙ θ - r cos(θ)˙ θ - r sin(θ) ¨ θ

Wir setzen in die Gleichung G.1 die Gleichungen G.8, G.9, G.10, G.5, G.6 und G.7 ein und ordnen nach er, eθ und eϕ.

a = ẍex + ýey + ¨z ez (G.8)
= ẍ[sin(θ)cos(ϕ)er + cos(θ)cos(ϕ)e θ - sin(ϕ)eϕ]
+ ý[sin (θ )sin (ϕ )er + cos(θ) sin (ϕ )eθ + cos(ϕ )eϕ]
+ ¨z [cos(θ)er - sin(θ)eθ]
= [¨xsin(θ)cos(ϕ ) + ¨y sin (θ )sin (ϕ ) + z¨cos(θ)] er
+ [¨xcos(θ)cos(ϕ ) + ¨y cos(θ)sin(ϕ) - ¨zsin(θ)] eθ
+ [- ¨x sin (ϕ) + ¨ycos(ϕ)] eϕ

Der Übersichtlichkeit halber berechnen wir nun die drei Komponenten er, eθ und eϕ getrennt. Wir beginnen mit er.

ar =ẍ sin(θ) cos(ϕ) + ý sin(θ) sin(ϕ) + ¨zcos(θ) (G.9)
= [ ¨rsin(θ)cos(ϕ ) + 2r˙θ˙cos(θ)cos(ϕ) - 2˙rϕ˙sin (θ )sin (ϕ ) - r ˙θ2sin (θ)cos(ϕ)
- 2r˙ϕ˙θ cos(θ) sin(ϕ) + rθ¨cos(θ) cos(ϕ) (G.10)
˙2 ¨ ] - rϕ sin(θ) cos(ϕ) - rϕsin(θ)sin(ϕ) sin(θ) cos(ϕ)
+ [ ˙ ˙ ¨rsin(θ)sin(ϕ) + 2˙rθcos(θ) sin(ϕ ) + 2r˙ϕ sin(θ)cos(ϕ)
- r˙θ2 sin(θ) sin(ϕ) + 2r˙θ ˙ϕ cos(θ) cos(ϕ)
2 ] +r cos(θ) sin(ϕ )¨θ - rϕ˙ sin(θ)sin(ϕ) + r¨ϕsin(θ)cos(ϕ ) sin(θ) sin(ϕ)
+ [ ˙ ˙2 ¨ ] ¨rcos(θ) - 2˙rθsin(θ) - rθ cos(θ) - rθ sin(θ) cos(θ)
= ¨r [sin (θ) cos(ϕ)sin(θ)cos(ϕ) + sin (θ)sin (ϕ)sin(θ)sin(ϕ) + cos(θ)cos(θ)]
+ 2ṙ˙ θ[cos(θ)cos(ϕ) sin(θ) cos(ϕ ) + cos(θ) sin(ϕ )sin (θ )sin (ϕ ) - sin(θ)cos(θ)]
+ 2ṙ˙ϕ[- sin(θ)sin(ϕ) sin(θ)cos(ϕ ) + sin(θ)cos(ϕ) sin (θ) sin(ϕ )]
+ r ˙ θ2[- sin(θ) cos(ϕ)sin(θ)cos(ϕ) - sin (θ)sin (ϕ)sin(θ)sin(ϕ) - cos(θ)cos(θ)]
+ 2r˙θ ˙ϕ[- cos(θ)sin(ϕ) sin(θ) cos(ϕ) + cos(θ) cos(ϕ)sin(θ)sin(ϕ)]
+ r ¨ θ [cos(θ )cos(ϕ)sin(θ)cos(ϕ) + cos(θ)sin(ϕ) sin(θ) sin(ϕ) - sin(θ)cos(θ)]
+ rϕ˙2[- sin (θ)cos(ϕ)sin(θ)cos(ϕ ) - sin(θ)sin(ϕ)sin(θ)sin(ϕ)]
+ r ¨ ϕ [- sin(θ)sin (ϕ)sin(θ)cos(ϕ) + sin (θ)cos(ϕ)sin(θ)sin(ϕ)]
= ¨r [ ] sin2(θ)cos2(ϕ) + sin2(θ) sin2(ϕ) + cos2(θ)
+ 2ṙ˙θ[ ] cos(θ)sin(θ)cos2(ϕ ) + cos(θ)sin(θ)sin2(ϕ) - sin (θ )cos(θ)
+ 2ṙ˙ ϕ[ 2 2 ] - sin (θ)sin(ϕ) cos(ϕ) + sin (θ)cos(ϕ )sin (ϕ )
+ r ˙ θ2[ 2 2 2 2 2 ] - sin (θ )cos (ϕ) - sin (θ)sin (ϕ) - cos (θ)
+ rθ¨ [ ] cos(θ)sin(θ)cos2(ϕ) + cos(θ)sin(θ)sin2(ϕ) - sin(θ) cos(θ )
+ rϕ˙2[ ] - sin2(θ)cos2(ϕ) - sin2(θ) sin2(ϕ )
+ rϕ¨ [ ] - sin2(θ)sin(ϕ)cos(ϕ ) + sin2(θ)cos(ϕ) sin(ϕ)
= ¨r [ ] sin2(θ) + cos2(θ)
+ 2ṙ˙ θ[cos(θ)sin(θ) - sin(θ) cos(θ )]
+ rθ˙2[ ] - sin2 (θ ) - cos2(θ)
+ rθ¨ [cos(θ )sin (θ ) - sin(θ)cos(θ)]
+ r ˙ ϕ2[ 2 ] - sin (θ)
= ¨r- r˙θ2 - r sin 2(θ)˙ϕ2

und

aθ =ẍ cos(θ) cos(ϕ) + ý cos(θ) sin(ϕ) -¨zsin(θ) (G.11)
= [ ¨rsin(θ)cos(ϕ ) + 2r˙θ˙cos(θ)cos(ϕ) - 2˙rϕ˙sin (θ)sin (ϕ ) - r ˙θ2sin (θ)cos(ϕ)
] - 2rϕ˙˙θcos(θ)sin(ϕ) + r¨θcos(θ) cos(ϕ) - r ˙ϕ2sin(θ)cos(ϕ) - r¨ϕ sin (θ )sin (ϕ ) cos(θ) cos(ϕ)
+ [ ¨rsin(θ)sin(ϕ) + 2˙r˙θcos(θ) sin(ϕ ) + 2r˙ϕ˙sin(θ)cos(ϕ) - r˙θ2sin(θ)sin(ϕ)
2 ] +2r θ˙ϕ˙cos(θ)cos(ϕ) + r¨θ cos(θ)sin (ϕ ) - r ˙ϕ sin(θ)sin(ϕ) + r¨ϕ sin(θ)cos(ϕ ) cos(θ) sin(ϕ)
-[ ˙ ˙ ¨ ] ¨rcos(θ) - 2˙rθ sin(θ) - rθcos(θ) - rθ sin(θ) sin(θ)
= ¨r [sin (θ )cos(ϕ)cos(θ)cos(ϕ ) + sin(θ)sin(ϕ)cos(θ) sin(ϕ ) - cos(θ) sin(θ)]
+ 2ṙ˙ θ[cos(θ)cos(ϕ) cos(θ )cos(ϕ) + cos(θ)sin (ϕ)cos(θ)sin(ϕ) + sin (θ)sin (θ)]
+ 2ṙ˙ϕ[- sin(θ)sin(ϕ) cos(θ )cos(ϕ) + sin(θ) cos(ϕ )cos(θ)sin(ϕ)]
+ r˙ θ2[- sin(θ) cos(ϕ)cos(θ)cos(ϕ ) - sin(θ)sin(ϕ)cos(θ) sin(ϕ ) + cos(θ)sin(θ)]
+ 2r˙ϕ˙θ[- cos(θ)sin(ϕ) cos(θ)cos(ϕ) + cos(θ)cos(ϕ)cos(θ) sin(ϕ )]
+ r¨ θ [cos(θ)cos(ϕ)cos(θ) cos(ϕ ) + cos(θ) sin(ϕ )cos(θ)sin(ϕ) + sin(θ) sin(θ)]
+ r˙ϕ2[- sin (θ)cos(ϕ)cos(θ) cos(ϕ ) - sin(θ)sin(ϕ) cos(θ )sin (ϕ )]
+ r¨ϕ [- sin(θ)sin(ϕ)cos(θ)cos(ϕ ) + sin(θ)cos(ϕ) cos(θ)sin (ϕ)]
= ¨r [ ] sin(θ)cos(θ)cos2(ϕ) + sin (θ)cos(θ)sin2(ϕ) - cos(θ)sin (θ)
+ 2ṙ˙θ[ 2 2 2 2 2 ] cos (θ)cos (ϕ) + cos (θ)sin (ϕ) + sin (θ)
+ r˙ θ2[ 2 2 ] - sin (θ)cos(θ)cos (ϕ) - sin (θ)cos(θ)sin (ϕ) + cos(θ)sin (θ )
+ r¨ θ [ 2 2 2 2 2 ] cos (θ)cos (ϕ) + cos (θ)sin (ϕ ) + sin (θ)
+ r˙ϕ2[ ] - sin(θ)cos(θ)cos2(ϕ) - sin(θ)cos(θ)sin2(ϕ)
= ¨r [sin (θ )cos(θ) - cos(θ) sin(θ)]
+ 2ṙ˙ θ[ 2 2 ] cos (θ) + sin (θ)
+ r˙θ2[- sin(θ) cos(θ) + cos(θ) sin(θ)]
+ r¨θ [ 2 2 ] cos (θ) + sin (θ)
- r˙ ϕ2[sin (θ )cos(θ)]
= 2ṙθ˙ + r¨θ- r sin(θ) cos(θ)ϕ˙2

und schliesslich

aϕ = -ẍ sin(ϕ) + ý cos(ϕ) (G.12)
= -[ ¨rsin(θ)cos(ϕ ) + 2r˙θ˙cos(θ)cos(ϕ) - 2˙rϕ˙sin (θ )sin (ϕ ) - r ˙θ2sin (θ)cos(ϕ)
] - 2rϕ˙˙θcos(θ)sin(ϕ) + r¨θ cos(θ )cos(ϕ) - r ˙ϕ2sin(θ)cos(ϕ ) - rϕ¨sin (θ)sin (ϕ) sin(ϕ)
+ [ ¨rsin(θ)sin(ϕ) + 2˙rθ˙cos(θ )sin (ϕ ) + 2 ˙r ˙ϕsin(θ)cos(ϕ) - rθ˙2 sin(θ) sin(ϕ )
2 ] +2r θ˙˙ϕcos(θ)cos(ϕ ) + rθ¨cos(θ)sin(ϕ) - r ˙ϕ sin(θ)sin(ϕ) + r¨ϕ sin(θ) cos(ϕ) cos(ϕ)
= ¨r [- sin(θ)cos(ϕ) sin(ϕ) + sin(θ)sin(ϕ)cos(ϕ)]
+ 2ṙ˙θ[- cos(θ) cos(ϕ )sin (ϕ) + cos(θ )sin (ϕ )cos(ϕ)]
+ 2ṙϕ˙[sin(θ)sin(ϕ) sin(ϕ) + sin(θ)cos(ϕ) cos(ϕ )]
+ r˙θ2[sin(θ) cos(ϕ)sin(ϕ) - sin (θ) sin (ϕ )cos(ϕ)]
+ 2r˙ϕθ˙[cos(θ)sin(ϕ) sin(ϕ ) + cos(θ)cos(ϕ )cos(ϕ)]
+ r¨ θ [- cos(θ)cos(ϕ) sin(ϕ ) + cos(θ)sin(ϕ) cos(ϕ)]
+ r˙ϕ2[sin (θ)cos(ϕ)sin(ϕ) - sin (θ)sin (ϕ)cos(ϕ)]
+ r¨ ϕ [sin(θ)sin(ϕ)sin(ϕ) + sin (θ )cos(ϕ)cos(ϕ)]
= + 2ṙϕ˙[ ] sin(θ) sin2(ϕ) + sin(θ)cos2(ϕ)
+ 2r˙ϕθ˙[ ] cos(θ) sin2(ϕ ) + cos(θ)cos2(ϕ )
+ r¨ϕ [ 2 2 ] sin(θ)sin (ϕ) + sin (θ) cos (ϕ)
= + 2ṙ ˙ ϕ sin(θ) + 2r ˙ ϕ˙ θ cos(θ) + r¨ ϕsin(θ)
= [ ] r¨ϕ + 2r˙ϕ˙ sin(θ) + 2r˙ϕ˙θ cos(θ)

Zusammenfassend haben wir

a = arer + aθeθ + aϕeϕ (G.13)
= [ ] ¨r - r˙θ2 - rsin2(θ) ˙ϕ2er
+ [ ] 2˙r˙θ + r¨θ - rsin(θ)cos(θ) ˙ϕ2eθ
+ [( ) ] r¨ϕ + 2˙rϕ˙ sin (θ) + 2r ˙ϕ˙θcos(θ)eϕ

G.2.1  Interpretation

Wir teilen die Beschleunigung in drei Komponenten auf

a = ap + az + ac
(G.14)

Dies ist in der angegebenen Reihenfolge die Parallelbeschleunigung, die den Betrag der Geschwindigkeit erhöht, die Zentripetalbeschleunigung und die Coriolis-Beschleunigung.

Im Einzelnen haben wir

ap = ¨r er + r¨ θ eθ + r sin(θ)¨ ϕ eϕ (G.15)
az = -r[ ] ˙θ2 + sin2(θ) ˙ϕ2er - r sin(θ) cos(θ)˙ϕ2e θ (G.16)
ac = 2ṙ˙θeθ + 2[ ] r˙sin (θ ) + rθ˙cos(θ)˙ϕeϕ (G.17)


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