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] 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
{\displaystyle {\mathcal {F}}_{i}=-{\frac {\partial {\mathcal {V}}}{\partial q_{i}}}+{\frac {d}{dt}}\left({\frac {\partial {\mathcal {V}}}{\partial {\dot {q_{i}}}}}\right);}
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: