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Monogenic system

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Type of system in classical mechanics

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).

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.

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 F i {\displaystyle {\mathcal {F}}_{i}} and generalized potential V ( q 1 , q 2 , … , q N , q ˙ 1 , q ˙ 2 , … , q ˙ N , t ) {\displaystyle {\mathcal {V}}(q_{1},q_{2},\dots ,q_{N},{\dot {q}}_{1},{\dot {q}}_{2},\dots ,{\dot {q}}_{N},t)} is as follows:

F i = − ∂ V ∂ q i + d d t ( ∂ V ∂ q i ˙ ) ; {\displaystyle {\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 {\displaystyle q_{i}} is generalized coordinate, q i ˙ {\displaystyle {\dot {q_{i}}}} is generalized velocity, and t {\displaystyle 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:

F i = − ∂ V ∂ q i . {\displaystyle {\mathcal {F}}_{i}=-{\frac {\partial {\mathcal {V}}}{\partial q_{i}}}.}

See also

References

  1. 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.
  2. Cornelius, Lanczos (1970). The Variational Principles of Mechanics. Toronto: University of Toronto Press. p. 30. ISBN 0-8020-1743-6.
  3. 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.
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