Classical Mechanics: Relation between general velocity and general potential function for velocity-dependent potential How is the general force derived from the general potential for a velocity-dependent potential $U = U(q_j,\dot{q_j})$?
$$Q_j=-\frac{\partial U}{\partial q_j}+ \frac{\mathrm{d}}{\mathrm{dt}}(\frac{\partial U}{\partial \dot{q_j}}). \tag{1}$$
What I understand

*

*In a conservative vector field, the force could be found through the gradient of the scalar potential, i.e. $F = -\nabla V$.

*General force in non-velocity-dependent potentials is given by $Q_j = -\frac{\partial V}{\partial q_j}$, in a similar manner as of that in bullet point 1, where $V$ is a conservative vector field, $Q_j$ is the general force and $q_j$ a general coordinate.

My attempt to derive Eqn. (1)

*

*Recall that the general potential is a function of general coordinate and the general velocity, $U = U(q_j,\dot{q_j})$


*Apply $Q_j  = -\nabla U$, and get
$$
\begin{aligned}
Q_j &= -\nabla U(q_j,\dot{q_j})\\
&=-(\frac{\partial}{\partial q_j}+\frac{\partial}{\partial \dot{q_j}})U(q_j,\dot q_j)\\
&=-\frac{\partial U}{\partial q_j}-\frac{\partial U}{\partial \dot q_j}
\end{aligned} \tag{2}
$$
where the second term on R.H.S. does not match Eqn. (1)
My question

*

*What are the things/concepts that I am missing?

*How is Eqn.(1) derived?

*How to intuitively/physically understand the second term on R.H.S. in Eqn.(1)?

 A: *

*1 & 2: Eq. (1) is not derived. It is a defining property of a generalized velocity-dependent potential.


*


*The form of eq. (1) mimics the Euler-Lagrange operator, so that we can bring Lagrange equations
$$ \frac{d}{dt}\frac{\partial T}{\partial \dot{q}^j}-\frac{\partial T}{\partial q^j}~=~Q_j, \qquad j~\in \{1,\ldots, n\}, \tag{L}$$
in the form of Euler-Lagrange equations
$$ \frac{d}{dt}\frac{\partial (T-U)}{\partial \dot{q}^j}-\frac{\partial (T-U)}{\partial q^j}~=~0, \qquad j~\in \{1,\ldots, n\}. \tag{EL}$$
A: Let us start from the general form of Euler-Lagrange equations for a curve (I will discuss all this issue in local coordinates thoug global approaches are possible)
$$\mathbb{R} \ni t \mapsto (t, q(t), \dot{q}(t)) \in \mathbb{R}^{2n+1}$$
$$\left.\frac{d}{dt}\left( \frac{\partial T|_R(t, q,\dot{q})}{\partial \dot{q}^k}\right)\right|_{(t,q(t),\dot{q}(t)} - \left.\frac{\partial T|_R(t,q,\dot{q})}{\partial q^k}\right|_{(t,q(t),\dot{q}(t)} = Q_k|_R(t,q(t), \dot{q}(t))\quad \mbox{where}\quad \frac{dq^k}{dt} = \dot{q}^k\:, \quad k=1,\ldots, n\:.\tag{1}$$
Above, $T|_R$ is the kinetic energy of the system with respect to some fixed reference frame $R$ written as a function of the $n$ Lagrangian coordinates $q^k$ and their formal time-derivatives $\dot{q}^k$ once we have ``solved'' the possible (ideal) $c$ constraints on the configurations of the physical system made of, say $N$ material points, so that $n= 3N-c$.
The known  functions $Q_k(t,q, \dot{q})$ are obtained out of the forces acting on the system of $N$ points in the reference frame $R$.
$$Q_k|_R(t,q,\dot{q}) = \sum_{i=1}^N \frac{\partial \vec{x}_i}{\partial q^k} \cdot \vec{F}_i|_R(t, \vec{x}_1,\ldots,\vec{x}_N, \vec{v}_1|_R, \ldots \vec{v}_N|_R)\:,$$
Here,  $\vec{x}_i(t,q^1,\ldots, q^n)$ is the position vector of the ith material point in the reference frame $R$ written as a function of time and the Lagrangian coordinates and $\vec{v}_i|_R(t,q^1,\ldots, q^n, \dot{q}^1,\ldots, \dot{q}^n)$ is the velocity of that material point in $R$.
It is important to specify the reference frame $R$ since, if $R$ is not inertial, the $Q_k|_R$ also include the contribution of the inertial pseuodoforces.
We are also assuming that we know the functional form of these forces $\vec{F}_i|_R$ and thus $Q_k|_R:= Q_k|_R(t,q, \dot{q})$ are $n$ known functions (differently form the functional form of the reactive forces due to the ideal constraints which are embodied in the Lagrangian formalism).
If the forces $\vec{F}_i|_R$ are conservative in $R$, i.e., $$\vec{F}_i|_R(\vec{x}_1,\ldots,\vec{x}_N) = -\nabla_{\vec{x}_i} U|_R(\vec{x}_1,\ldots,\vec{x}_N)\:,$$
we immediately have that
$$Q_k|_R(t,q) = -\frac{\partial }{\partial q^k} U|_R(t,q^1,\ldots,q^n)\:,$$
where we have adopted the non completely rigorous notation, but very effective
$$U|_R(t,q^1,\ldots,q^n) := U|_R(\vec{x}_1(t,q^1,\ldots, q^n), \ldots, \vec{x}_N(t,q^1,\ldots, q^n))\:.$$
Euler-Lagrange's equations specialize to
$$\frac{d}{dt}\left( \frac{\partial T|_R(t, q,\dot{q})}{\partial \dot{q}^k}\right) - \frac{\partial T|_R(t,q,\dot{q})}{\partial q^k} = - \frac{\partial U|_R}{\partial q^k} \:.$$
Since,  trivially,
$$\frac{\partial}{\partial \dot{q}^k} U|_R(t,q^1,\ldots,q^n)  =0\:,$$
we can  re-write those equations as
$$\frac{d}{dt}\left( \frac{\partial T|_R(t, q,\dot{q})- U|_{R}(t,q)}{\partial \dot{q}^k}\right) - \frac{\partial T|_R(t,q,\dot{q})- U|_{R}(t,q)}{\partial q^k} = 0\:,$$
namely
$$\frac{d}{dt}\left( \frac{\partial L|_R(t, q,\dot{q})}{\partial \dot{q}^k}\right) - \frac{\partial L|_R(t,q,\dot{q})}{\partial q^k} = 0\:, \tag{2}$$
where we have introduced the Lagrangian referred to $R$,
$$L|_R(t, q,\dot{q}) := T|_R(t, q,\dot{q}) - U|_R(t, q)\:.\tag{3}$$
It is clear that the procedure can be generalized further encompassing forces, if any, whose functional form may depend also on the velocities like this
$$Q_k|_R(t,q, \dot{q}) = \frac{d}{dt}\left( \frac{\partial U|_R(t, q,\dot{q})}{\partial \dot{q}^k}\right) - \frac{\partial U|_R(t,q,\dot{q})}{\partial q^k}\tag{4}$$
for some generalized potential
$$U|_R:= U|_R(t,q, \dot{q})\:.$$
Also in that case, starting from (1) one sees immediately that E.L's equations take the form (2) if we have defined a Lagrangian as in (3).
Due to several issues arising from the implementation of this recipe, one eventually sees that a generalized potential can be only linear in the variables $\dot{q}^k$. (As a matter of fact, that is due to the structure of the EL equations, without this condition the existence and uniqueness theorem may fail, but I do not want to enter here into the details of this technical issue).
The most general permitted form of a generalize potential is therefore
$$U(t,q,\dot{q}) = B(t,q)+ \sum_{k=1}^n A_k(t,q)\dot{q}^k \tag{5}\:.$$
A natural question pops out now.
Is this nothing but an elegant mathematical extension of the formalism without physical content?
Nope! There are at least two very important cases.

*

*The Lorentz force acting on a charge due to generic external electromagnetic fields. The E.L. equations can be written using a well-known generalized potential.


*All types of inertial forces when $R$ is not inertial and its motion is known with respect to an inertial reference frame $R_0$. Also in this case there is a generalized potential giving rise to all inertial pseudoforces.
