Monogenic system explained

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.

l{F}i

and generalized potential

l{V}(q1,q2, ...,q

N,q
1,q
2, ...,q

N,t)

is as follows:

l{F}i=-

\partiall{V
}+\frac\left(\frac\right);

where

qi

is generalized coordinate,
qi

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:

l{F}i=-

\partiall{V
} .

See also

Notes and References

  1. Web site: J.. Butterfield. 3 September 2004. Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics. https://web.archive.org/web/20181103003631/http://philsci-archive.pitt.edu/1937/1/BetLMLag.pdf. 3 November 2018. 23 January 2015. PhilSci-Archive. 43.
  2. Book: Cornelius. Lanczos. The Variational Principles of Mechanics. University of Toronto Press. 1970. 0-8020-1743-6. Toronto. 30.
  3. Book: Goldstein . Herbert . Herbert Goldstein . Charles P. Poole . Poole . Charles P. Jr. . Safko . John L. . Classical Mechanics . 3rd . 2002 . 0-201-65702-3 . Addison Wesley . San Francisco, CA . 18–21,45.