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| In [[physics]], among the most studied physical systems in [[classical mechanics]] are '''monogenic systems'''. A monogenic system has excellent mathematical characteristics and is very well suited for mathematical analysis. It is considered a logical starting point for any serious physics endeavour.
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| In a physical system, if all forces, with the exception of the constraint forces, are derivable from the [[generalized potential|generalized scalar potential]], and this generalized scalar potential is a function of [[generalized coordinate]]s, [[generalized coordinate#Generalized velocities and kinetic energy|generalized velocities]], or time, then, this system is a '''monogenic system'''.
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| Expressed using equations, the exact relationship between [[generalized force]] <math>\mathcal{F}_i\,\!</math> and generalized potential <math>\mathcal{V}(q_1,\ q_2,\ \dots,\ q_N,\ \dot{q}_1,\ \dot{q}_2,\ \dots,\ \dot{q}_N,\ t)\,\!</math> is as follows:
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| :<math>\mathcal{F}_i= - \frac{\partial \mathcal{V}}{\partial q_i}+\frac{d}{dt}\left(\frac{\partial \mathcal{V}}{\partial \dot{q_i}}\right);\, </math>
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| where <math>q_i\,\!</math> is generalized coordinate, <math>\dot{q_i} \, </math> is generalized velocity, and <math>t\,\!</math> is time.
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| If the generalized potential in a monogenic system depends only on generalized coordinates, and not on generalized velocities and time, then, this system is a [[Conservative force|conservative system]].The relationship between generalized force and generalized potential is as follows:
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| ::<math>\mathcal{F}_i= - \frac{\partial \mathcal{V}}{\partial q_i}\, </math> ；
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| [[Lagrangian mechanics]] often involves monogenic systems. If a physical system is both a [[holonomic system]] and a monogenic system, then it is possible to derive [[Lagrangian mechanics|Lagrange's equation]]s from [[d'Alembert's principle]]; it is also possible to derive [[Lagrangian mechanics|Lagrange's equation]]s from [[Hamilton's principle]].<ref name=goldstein2002>{{cite book |last1=Goldstein |first1=Herbert |authorlink1=Herbert Goldstein |last2=Poole | first2=Charles P., Jr. |last3=Safko |first3=John L. |title=Classical Mechanics |edition=3rd |year=2002 |url=http://www.pearsonhighered.com/educator/product/Classical-Mechanics/9780201657029.page |isbn=0-201-65702-3 |publisher=Addison Wesley |location=San Francisco, CA |pages=18–21,45}}</ref>
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| ==See also==
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| *[[Lagrangian mechanics]]
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| *[[Hamiltonian mechanics]]
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| *[[Holonomic system]]
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| *[[Scleronomous]]
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| ==References==
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| {{reflist}}
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| [[Category:Mechanics]]
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| [[Category:Classical mechanics]]
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| [[Category:Lagrangian mechanics]]
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| [[Category:Hamiltonian mechanics]]
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| [[Category:Dynamical systems]]
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Wood Machinist Blomquist from Breton, has many hobbies and interests including motorbikes, ganhando dinheiro na internet and textiles. Found some interesting places following 6 months at Historic Bridgetown and its Garrison.
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