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In a physical system, if all forces, with the exception of the constraint forces, are derivable from the generalized scalar potential, and this generalized scalar potential is a function of generalized coordinates, generalized velocities, or time, then, this system is a monogenic system.

Expressed using equations, the exact relationship between force ${\mathcal {F}}_{i}\,\!$ and generalized potential ${\mathcal {V}}(q_{1},\ q_{2},\ \dots ,\ q_{N},\ {\dot {q}}_{1},\ {\dot {q}}_{2},\ \dots ,\ {\dot {q}}_{N},\ t)\,\!$ is as follows:

{\mathcal {F}}_{i}=-{\frac {\partial {\mathcal {V}}}{\partial q_{i}}}+{\frac {d}{dt}}\left({\frac {\partial {\mathcal {V}}}{\partial {\dot {q_{i}}}}}\right)\,\!

;

where $q_{i}\,\!$ is generalized coordinate, ${\dot {q_{i}}}\,\!$ is generalized velocity, and $t\,\!$ is time.

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 system.

Lagrangian mechanics often involves monogenic systems, which is a more restricted system. Monogenic systems are much easier to analyze theoretically. If a physical system is both a holonomic system and a monogenic system, then it’s possible to derive Lagrange's equations from d'Alembert's principle; it's also possible to derive Lagrange's equations from Hamilton's principle^[1].

Reference

^ Goldstein, Herbert (1980). Classical Mechanics (3rd ed.). United States of America: Addison Wesley. pp. pp. 18-21, 45. ISBN 0201657023. {{cite book}}: |pages= has extra text (help)

[Herb1980-1] Goldstein, Herbert (1980). Classical Mechanics (3rd ed.). United States of America: Addison Wesley. pp. pp. 18-21, 45. ISBN 0201657023. {{cite book}}: |pages= has extra text (help)

[1]

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-If in a physical system, except for the constraint forces, all forces are derivable from the [[generalized potential|generalized scalar potential]], and this generalized scalar potential may be a function of [[generalized coordinate]]s, [[generalized coordinate#Generalized velocities and kinetic energy|generalized velocities]], or time, then, this system is a '''monogenic system'''.
+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'''.
+Expressed using equations, the exact relationship between 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:
-[[Lagrangian mechanics]] often involves monogenic systems.  If a physical system is both a [[holonomic]] system and a monogenic system, then [[Hamilton's principle]] is necessary and sufficient for the validity of [[Lagrangian mechanics|Lagrange's equations]]s<ref name="Herb1980">{{cite book |last=Goldstein|first=Herbert|title=Classical Mechanics|year=1980| location=United States of America | publisher=Addison Wesley| edition= 3rd| isbn=0201657023 | language=English| pages=pp. 35}}</ref>.
+:<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> ;
+where <math>q_i\,\!</math> is generalized coordinate, <math>\dot{q_i}\,\!</math> is generalized velocity, and <math>t\,\!</math> is time.
+*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]].
+*[[Lagrangian mechanics]] often involves monogenic systems, which is a more restricted system.  Monogenic systems are much easier to analyze theoretically.  If a physical system is both a [[holonomic]] system and a monogenic system, then it’s possible to derive [[Lagrangian mechanics|Lagrange's equation]]s from [[d'Alembert's principle]]; it's also possible to derive [[Lagrangian mechanics|Lagrange's equation]]s from [[Hamilton's principle]]<ref name="Herb1980">{{cite book |last=Goldstein|first=Herbert|title=Classical Mechanics|year=1980| location=United States of America | publisher=Addison Wesley| edition= 3rd| isbn=0201657023 | language=English| pages=pp. 18-21, 45}}</ref>.
 ==See also==
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 ==Reference==
 <references/>
-{{stub}}
 [[Category:Mechanics]]
 [[Category:Classical mechanics]]