Monogenic system

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).{{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}}{{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}}

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.{{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}}

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

Category:Mechanics

Category:Classical mechanics

Category:Lagrangian mechanics

Category:Hamiltonian mechanics

Category:Dynamical systems

Monogenic system

Mathematical definition

See also

References