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
[edit]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 and generalized potential is as follows:
where is generalized coordinate, is generalized velocity, and 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:
- .
See also
[edit]References
[edit]- ^ 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.