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In classical mechanics, a physical system is termed a monogenic system if the force acting on the system can be modelled in a particular, especially convenient mathematical form. The systems that are typically studied in physics are monogenic. The term was introduced by Cornelius Lanczos in his book The Variational Principles of Mechanics (1970).^[1]^[2]

In Lagrangian mechanics, the property of being monogenic is a necessary condition for certain different formulations to be mathematically equivalent. If a physical system is both a holonomic system and a monogenic system, then it is possible to derive Lagrange's equations from d'Alembert's principle; it is also possible to derive Lagrange's equations from Hamilton's principle.^[3]

Mathematical definition

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 generalized 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. The relationship between generalized force and generalized potential is as follows:

${\mathcal {F}}_{i}=-{\frac {\partial {\mathcal {V}}}{\partial q_{i}}}.$

References

^ J., Butterfield (3 September 2004). "Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics" (PDF). PhilSci-Archive. p. 43. Archived from the original (PDF) on 3 November 2018. Retrieved 23 January 2015.
^ Cornelius, Lanczos (1970). The Variational Principles of Mechanics. Toronto: University of Toronto Press. p. 30. ISBN 0-8020-1743-6.
^ Goldstein, Herbert; Poole, Charles P. Jr.; Safko, John L. (2002). Classical Mechanics (3rd ed.). San Francisco, CA: Addison Wesley. pp. 18–21, 45. ISBN 0-201-65702-3.

[1] J., Butterfield (3 September 2004). "Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics" (PDF). PhilSci-Archive. p. 43. Archived from the original (PDF) on 3 November 2018. Retrieved 23 January 2015.

[2] Cornelius, Lanczos (1970). The Variational Principles of Mechanics. Toronto: University of Toronto Press. p. 30. ISBN 0-8020-1743-6.

[goldstein2002-3] Goldstein, Herbert; Poole, Charles P. Jr.; Safko, John L. (2002). Classical Mechanics (3rd ed.). San Francisco, CA: Addison Wesley. pp. 18–21, 45. ISBN 0-201-65702-3.

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-{{Context|date=October 2009}}
+{{Use Canadian English|date = February 2019}}
+{{Short description|Type of system in classical mechanics}}
+In [[classical mechanics]], a physical system is termed a '''monogenic system''' if the force acting on the system can be modelled in a particular, especially convenient mathematical form. The systems that are typically studied in physics are monogenic. The term was introduced by [[Cornelius Lanczos]] in his book ''The Variational Principles of Mechanics'' (1970).<ref>{{cite web|last1=J.|first1=Butterfield|date=3 September 2004|title=Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics|url=http://philsci-archive.pitt.edu/1937/1/BetLMLag.pdf|archive-url=https://web.archive.org/web/20181103003631/http://philsci-archive.pitt.edu/1937/1/BetLMLag.pdf|archive-date=3 November 2018|access-date=23 January 2015|website=PhilSci-Archive|page=43}}</ref><ref>{{cite book|last1=Cornelius|first1=Lanczos|title=The Variational Principles of Mechanics|publisher=[[University of Toronto Press]]|year=1970|isbn=0-8020-1743-6|location=Toronto|page=30}}</ref>
+In [[Lagrangian mechanics]], the property of being monogenic is a necessary condition for certain different formulations to be mathematically equivalent. 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 Lagrange's equations from [[Hamilton's principle]].<ref name=goldstein2002>{{cite book |last1=Goldstein |first1=Herbert |author-link1=Herbert Goldstein |author2-link=Charles P. Poole |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>
-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.
+==Mathematical definition==
 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 [[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:
+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:
-:<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>
+<math display="block">\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.
+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]].The relationship between generalized force and generalized potential is as follows:
+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:
-::<math>\mathcal{F}_i= - \frac{\partial \mathcal{V}}{\partial q_i}\, </math> ；
+<math display="block">\mathcal{F}_i= - \frac{\partial \mathcal{V}}{\partial q_i}.</math>
-*[[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="Herb1980">{{cite book |last=Goldstein|first=Herbert|title=Classical Mechanics|year=1980| location=United States of America | publisher=Addison Wesley| edition= 3rd| isbn=0-201-65702-3 | pages=18–21, 45}}</ref>.
 ==See also==
-*[[Lagrangian mechanics]]
-*[[Hamiltonian mechanics]]
-*[[Holonomic]]
 *[[Scleronomous]]
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 [[Category:Hamiltonian mechanics]]
 [[Category:Dynamical systems]]
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