Monogenic system

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

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

where ${\displaystyle q_{i}}$ is generalized coordinate, ${\displaystyle {\dot {q_{i}}}}$ is generalized velocity, and ${\displaystyle t}$ 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:

$${\displaystyle {\mathcal {F}}_{i}=-{\frac {\partial {\mathcal {V}}}{\partial q_{i}}}.}$$

• Scleronomous