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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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+{{Use Canadian English|date = February 2019}}
-{{context}}
+{{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>
-One of the most studied physical systems in [[Classical Mechanics]] is '''monogenic system'''.  This is because it offers an exceptionally ideal environment for physicists to develop and examine their brilliant ideas and elegent theories.  Monogenic system has excellent mathematical characteristics and is very well suited for mathematical analysis.  It's 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> ;
-where <math>q_i\,\!</math> is generalized coordinate, <math>\dot{q_i}\,\!</math> is generalized velocity, and <math>t\,\!</math> is time.
+<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>
-*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> ；
+where <math>q_i</math> is generalized coordinate, <math>\dot{q_i} </math> is generalized velocity, and <math>t</math> is time.
-*[[Lagrangian mechanics]] often involves monogenic systems.  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>.
+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 display="block">\mathcal{F}_i= - \frac{\partial \mathcal{V}}{\partial q_i}.</math>
 ==See also==
+*[[Scleronomous]]
-*[[Lagrangian mechanics]]
-*[[Hamiltonian mechanics]]
-*[[Holonomic]]
-*[[scleronomous]]
 ==References==
+{{reflist}}
-<references/>
 [[Category:Mechanics]]
@@ Line 27: / Line 29: @@
 [[Category:Lagrangian mechanics]]
 [[Category:Hamiltonian mechanics]]
+[[Category:Dynamical systems]]
-[[zh:單演系統]]