Physics: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).[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 [math]\displaystyle{ \mathcal{F}_i }[/math] and generalized potential [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ q_i }[/math] is generalized coordinate, [math]\displaystyle{ \dot{q_i} }[/math] is generalized velocity, and [math]\displaystyle{ 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 system. The relationship between generalized force and generalized potential is as follows:
- [math]\displaystyle{ \mathcal{F}_i= - \frac{\partial \mathcal{V}}{\partial q_i} }[/math] .
See also
References
- ↑ J., Butterfield (3 September 2004). "Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics". p. 43. http://philsci-archive.pitt.edu/1937/1/BetLMLag.pdf.
- ↑ 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. http://www.pearsonhighered.com/educator/product/Classical-Mechanics/9780201657029.page.
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