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| {{About|Bonnet's theorem in classical mechanics|the Bonnet theorem in differential geometry|Bonnet theorem}}
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| In [[classical mechanics]], '''Bonnet's theorem''' states that if ''n'' different [[force]] fields each produce the same geometric orbit (say, an ellipse of given dimensions) albeit with different [[speed]]s ''v''<sub>1</sub>, ''v''<sub>2</sub>,...,''v''<sub>''n''</sub> at a given point ''P'', then the same orbit will be followed if the speed at point ''P'' equals
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| :<math>
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| v_{\mathrm{combined}} = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}}
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| </math>
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| This theorem was first derived by [[Adrien-Marie Legendre]] in 1817,<ref>{{cite book | last = Legendre | first = A-M | authorlink = Adrien-Marie Legendre | year = 1817 | title = Exercises de Calcul Intégral | volume = 2 | pages = 382–3 |url = http://archive.org/stream/exercicescalculi02legerich#page/n403/mode/2up | publisher = Courcier | location = Paris}}</ref> but it is named after [[Pierre Ossian Bonnet]].
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| ==Derivation==
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| The shape of an [[orbit]] is determined only by the [[centripetal force]]s at each point of the orbit, which are the forces acting perpendicular to the orbit. By contrast, forces ''along'' the orbit change only the speed, but not the direction, of the [[velocity]].
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| Let the instantaneous radius of curvature at a point ''P'' on the orbit be denoted as ''R''. For the ''k''<sup>th</sup> force field that produces that orbit, the force normal to the orbit ''F''<sub>''k''</sub> must provide the [[centripetal force]]
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| :<math> | |
| F_{k} = \frac{m}{R} v_{k}^{2}
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| </math>
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| Adding all these forces together yields the equation
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| :<math>
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| \sum_{k=1}^{n} F_{k} = \frac{m}{R} \sum_{k=1}^{n} v_{k}^{2}
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| </math>
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| Hence, the combined force-field produces the same orbit if the speed at a point ''P'' is set equal to
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| :<math>
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| v_{\mathrm{combined}} = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}}
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| </math>
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| ==References==
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| {{Reflist|1}}
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| [[Category:Classical mechanics]]
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| {{classicalmechanics-stub}}
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