Monogenic forces vs generalized forces

Question

Wikipedia article under generalized forces says

Also we know that the generalized forces are defined as

How can I derive the first equation from the second for a monogenic system?

https://en.wikipedia.org/wiki/Monogenic_system#cite_ref-2

Answer 1

TL;DR: Concerning OP's question (v3), it follows by definition without the use of eq. (2).

1. The first equation is the very definition of a monogenic force, i.e. that the force has a generalized (possibly velocity-dependent) potential.

2. The second equation is the definition of a generalized force.

Eq. (1) is a special case of eq. (2), but not vice-versa.

Example: A friction force is a generalized force that is not monogenic, cf. e.g. this Phys.SE post.