Monogenic system

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One of the most studied physical systems in Classical Mechanics is monogenic system. This is because it offers an exceptionally ideal environment for physicists to develop and examine their brilliant ideas and elegent theories. Monogenic system has excellent mathematical characteristics and is very well suited for mathematical analysis. It's 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’s possible to derive Lagrange's equations from d'Alembert's principle; it's also possible to derive Lagrange's equations from Hamilton's principle^[1].