Scattering from rough surfaces: Difference between revisions
en>Bibcode Bot m Adding 1 arxiv eprint(s), 0 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot |
en>Monkbot |
||
| Line 1: | Line 1: | ||
{{About|Bonnet's theorem in classical mechanics|the Bonnet theorem in differential geometry|Bonnet theorem}} | |||
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 | |||
:<math> | |||
v_{\mathrm{combined}} = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}} | |||
</math> | |||
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]]. | |||
==Derivation== | |||
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]]. | |||
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]] | |||
:<math> | |||
F_{k} = \frac{m}{R} v_{k}^{2} | |||
</math> | |||
Adding all these forces together yields the equation | |||
:<math> | |||
\sum_{k=1}^{n} F_{k} = \frac{m}{R} \sum_{k=1}^{n} v_{k}^{2} | |||
</math> | |||
Hence, the combined force-field produces the same orbit if the speed at a point ''P'' is set equal to | |||
:<math> | |||
v_{\mathrm{combined}} = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}} | |||
</math> | |||
==References== | |||
{{Reflist|1}} | |||
[[Category:Classical mechanics]] | |||
{{classicalmechanics-stub}} | |||
Revision as of 22:19, 29 January 2014
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.
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 speeds v1, v2,...,vn at a given point P, then the same orbit will be followed if the speed at point P equals
This theorem was first derived by Adrien-Marie Legendre in 1817,[1] but it is named after Pierre Ossian Bonnet.
Derivation
The shape of an orbit is determined only by the centripetal forces 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.
Let the instantaneous radius of curvature at a point P on the orbit be denoted as R. For the kth force field that produces that orbit, the force normal to the orbit Fk must provide the centripetal force
Adding all these forces together yields the equation
Hence, the combined force-field produces the same orbit if the speed at a point P is set equal to
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
Template:Classicalmechanics-stub
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534