Monogenic system

In classical mechanics, a physical system is termed a monogenic system if the force acting on the system can be modelled in an especially convenient mathematical form (see mathematical definition below). In physics, among the most studied physical systems are monogenic systems.

In Lagrangian mechanics, the property of being monogenic is a necessary condition for the equivalence of different formulations of principle. 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.^[1]

The term was introduced by Cornelius Lanczos in his book The Variational Principles of Mechanics (1970).^[2]^[3]

Monogenic systems have excellent mathematical characteristics and are well suited for mathematical analysis. Pedagogically, within the discipline of mechanics, it is 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 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}}}

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References

^ 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ J., Butterfield (3 September 2004). "Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics" (PDF). PhilSci-Archive. p. 43. Retrieved 23 January 2015.
^ Cornelius, Lanczos (1970). The Variational Principles of Mechanics. Toronto: University of Toronto Press. p. 30. ISBN 0-8020-1743-6.

[goldstein2002-1] 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.{{cite book}}: CS1 maint: multiple names: authors list (link)

[2] J., Butterfield (3 September 2004). "Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics" (PDF). PhilSci-Archive. p. 43. Retrieved 23 January 2015.

[3] Cornelius, Lanczos (1970). The Variational Principles of Mechanics. Toronto: University of Toronto Press. p. 30. ISBN 0-8020-1743-6.

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Mathematical definition

See also

References