Monogenic system

In physics, among the most studied physical systems in classical mechanics are monogenic systems. A monogenic system has excellent mathematical characteristics and is very well suited for mathematical analysis. It is considered a logical starting point for any serious physics endeavour.

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:

${\mathcal {F}}_{i}=-{\frac {\partial {\mathcal {V}}}{\partial q_{i}}}\,$ ；

Lagrangian mechanics often involves monogenic systems. 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.

See also

• Lagrangian mechanics
• Hamiltonian mechanics
• Holonomic system
• Scleronomous