2005-2014 Ulm University, Othmar Marti, PIC
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H.2  Beschleunigung

Die Beschleunigung ist in kartesischen Koordinaten

            ( d2x)    (  )
     d2r    | dt22 |    |x|
a =  --2-=  |( ddt2y|) =  (y) = xex +  yey + zez
     dt       d22z      z
              dt
(H.1)

Wir verwenden die Beziehungen

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

und leiten sie zweimal ab. Wir erhalten aus

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

die Gleichungen

= rsin(θ) cos(ϕ) + cos(θ) cos(ϕ)˙
θ - sin(θ) sin(ϕ) ˙
ϕ (H.5)
+ cos(θ) cos(ϕ)θ˙ - r sin(θ) cos(ϕ)˙θ2 - r cos(θ) sin(ϕ)ϕ˙θ˙ + r cos(θ) cos(ϕ)θ
- sin(θ) sin(ϕ)˙
ϕ - r cos(θ) sin(ϕ) ˙
ϕ˙
θ - r sin(θ) cos(ϕ) ˙
ϕ2 - r sin(θ) sin(ϕ) 
ϕ
= rsin(θ) cos(ϕ)
+ ˙
θ[cos(θ)cos(ϕ) + cos(θ)cos(ϕ)]
+ ˙ϕ[- sin(θ)sin(ϕ) - sin(θ) sin(ϕ )]
+ rθ˙2[-  sin(θ) cos(ϕ )]
+ rϕ˙˙θ[- cos(θ)sin(ϕ) - cos(θ)sin(ϕ)]
+ rθ [cos(θ) cos(ϕ)]
+ rϕ˙2[- sin (θ )cos(ϕ)]
+ rϕ [- sin (θ)sin (ϕ)]
= rsin(θ) cos(ϕ) + 2θ˙ cos(θ) cos(ϕ) - 2ϕ˙ sin(θ) sin(ϕ) - r˙θ2 sin(θ) cos(ϕ)
- 2r˙ϕ˙θ cos(θ) sin(ϕ) + rθcos(θ) cos(ϕ) - r˙ϕ2 sin(θ) cos(ϕ) - rϕsin(θ) sin(ϕ)

und

= rsin(θ) sin(ϕ) + cos(θ) sin(ϕ) ˙
θ + sin(θ) cos(ϕ) ˙
ϕ (H.6)
+ cos(θ) sin(ϕ)θ˙ - r sin(θ) sin(ϕ)θ˙2 + r cos(θ) cos(ϕ)˙θϕ˙ + r cos(θ) sin(ϕ)θ
+ 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)
+ ˙
θ[cos(θ)sin(ϕ) + cos(θ)sin (ϕ)]
+ ˙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) + 2θ˙ cos(θ) sin(ϕ) + 2˙ϕ sin(θ) cos(ϕ) - rθ˙2 sin(θ) sin(ϕ)
+ 2r˙θ ˙ϕ cos(θ) cos(ϕ) + rθcos(θ) sin(ϕ) - rϕ˙2 sin(θ) sin(ϕ) + rϕsin(θ) cos(ϕ)

sowie

z = rcos(θ) - sin(θ)˙θ (H.7)
- sin(θ)θ˙ - r cos(θ)˙θ2 - r sin(θ)θ
= rcos(θ) - 2 sin(θ)˙θ - r cos(θ)˙θ - r sin(θ)θ

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

a = ex + ey + z ez (H.8)
= [sin(θ) cos(ϕ )er + cos(θ) cos(ϕ)eθ - sin (ϕ )eϕ]
+ [sin(θ)sin(ϕ)er + cos(θ)sin(ϕ)eθ + cos(ϕ)eϕ]
+ z [cos(θ)er - sin (θ)eθ]
= [x sin (θ)cos(ϕ) + ysin(θ)sin(ϕ) + zcos(θ)] er
+ [x cos(θ)cos(ϕ) + ycos(θ) sin(ϕ) - z sin(θ)] eθ
+ [- x sin(ϕ) + y cos(ϕ)] eϕ

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

ar = sin(θ) cos(ϕ) + sin(θ) sin(ϕ) + zcos(θ) (H.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) + rθcos(θ) cos(ϕ) (H.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) + rϕsin(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)]
+ 2˙
θ[cos(θ)cos(ϕ) sin(θ) cos(ϕ ) + cos(θ) sin(ϕ )sin (θ )sin (ϕ ) - sin(θ)cos(θ)]
+ 2˙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)
+ 2˙θ[                                                        ]
 cos(θ)sin(θ)cos2(ϕ ) + cos(θ)sin(θ)sin2(ϕ) - sin (θ )cos(θ)
+ 2˙
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)
+ 2˙
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 = cos(θ) cos(ϕ) + cos(θ) sin(ϕ) -zsin(θ) (H.11)
= [                   ˙                  ˙                ˙2
 rsin(θ)cos(ϕ ) + 2r˙θ cos(θ)cos(ϕ) - 2˙rϕ sin (θ)sin (ϕ ) - r θ sin (θ)cos(ϕ)
                                                                         ]
- 2rϕ˙˙θcos(θ)sin(ϕ) + rθcos(θ) cos(ϕ) - r ˙ϕ2sin(θ)cos(ϕ) - rϕ 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)
                                                                         ]
+2r θ˙ϕ˙cos(θ)cos(ϕ) + rθ cos(θ)sin (ϕ ) - r ˙ϕ2sin(θ)sin(ϕ) + rϕ sin(θ)cos(ϕ ) cos(θ) sin(ϕ)
-[                                         ]
 rcos(θ) - 2˙r˙θ sin(θ) - r˙θcos(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)]
+ 2˙θ[cos(θ)cos(ϕ) cos(θ )cos(ϕ) + cos(θ)sin (ϕ)cos(θ)sin(ϕ) + sin (θ)sin (θ)]
+ 2˙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)
+ 2˙θ[                                        ]
 cos2(θ)cos2(ϕ) + cos2(θ)sin2(ϕ) + sin2(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[                 2                     2   ]
 - sin(θ)cos(θ)cos (ϕ) - sin(θ)cos(θ)sin (ϕ)
= r [sin (θ )cos(θ) - cos(θ) sin(θ)]
+ 2˙
θ[   2        2   ]
 cos (θ) + sin (θ)
+ r˙θ2[- sin(θ) cos(θ) + cos(θ) sin(θ)]
+ rθ [                ]
 cos2(θ) + sin2(θ)
- r˙
ϕ2[sin (θ )cos(θ)]
= 2θ˙ + rθ- r sin(θ) cos(θ)ϕ˙2

und schliesslich

aϕ = - sin(ϕ) + cos(ϕ) (H.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) + rθ cos(θ )cos(ϕ) - r ˙ϕ2sin(θ)cos(ϕ ) - rϕsin (θ)sin (ϕ) sin(ϕ)
+ [                                                       2
 rsin(θ)sin(ϕ) + 2˙rθ˙cos(θ )sin (ϕ ) + 2 ˙r ˙ϕsin(θ)cos(ϕ) - rθ˙ sin(θ) sin(ϕ )
     ˙˙                                 ˙2                             ]
+2r θ ϕcos(θ)cos(ϕ ) + rθ cos(θ)sin(ϕ) - rϕ sin(θ)sin(ϕ) + rϕ sin(θ) cos(ϕ) cos(ϕ)
= r [- sin(θ)cos(ϕ) sin(ϕ) + sin(θ)sin(ϕ)cos(ϕ)]
+ 2˙
θ[- cos(θ) cos(ϕ )sin (ϕ) + cos(θ )sin (ϕ )cos(ϕ)]
+ 2ϕ˙[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)]
+ rϕ [sin(θ)sin(ϕ)sin(ϕ) + sin (θ )cos(ϕ)cos(ϕ)]
= + 2ϕ˙[                            ]
 sin(θ) sin2(ϕ) + sin(θ)cos2(ϕ)
+ 2r˙ϕθ˙[        2               2   ]
 cos(θ) sin (ϕ ) + cos(θ)cos (ϕ )
+ r
ϕ [         2               2   ]
 sin(θ)sin (ϕ) + sin (θ) cos (ϕ)
= + 2ϕ˙ sin(θ) + 2rϕ˙˙θ cos(θ) + rϕsin(θ)
= [         ]
 rϕ + 2r˙ϕ˙ sin(θ) + 2r˙ϕ˙θ cos(θ)

Zusammenfassend haben wir

a = arer + aθeθ + aϕeϕ (H.13)
= [                   ]
r- rθ˙2 - r sin2(θ)ϕ˙2er
+ [                           ]
 2r˙θ˙+ rθ - r sin(θ) cos(θ )ϕ ˙2eθ
+ [(         )                    ]
  rϕ+ 2 ˙r ˙ϕ sin(θ) + 2rϕ˙θ˙cos(θ)eϕ

H.2.1  Interpretation

Wir teilen die Beschleunigung in drei Komponenten auf

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

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

Im Einzelnen haben wir

ap = r er + rθ eθ + r sin(θ)ϕ eϕ (H.15)
az = -r[ 2     2    2]
 ˙θ + sin (θ) ˙ϕer - r sin(θ) cos(θ)˙ϕ2e θ (H.16)
ac = 2˙
θeθ + 2[            ˙     ]
 r˙sin(θ) + rθ cos(θ )˙
ϕeϕ (H.17)



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2005-2014 Ulm University, Othmar Marti, PIC  Lizenzinformationen