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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

References

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