Q:

a question on Lagrange's equation when the time derivative of the generalized co-ordinates is constant 

Consider a system whose generalized co-ordinates are $q_i$ and is under the constraints $\dot{q_i} = K_i \forall i = 1,2,3,...$ where $K_i$ are constants. I have a problem in writing the Lagrange's equation for this system as $\dot{q_i}$ is a constant. What would be $\frac{\partial L}{\partial \dot{q_i}}$ ?

A:

I will assume that we are talking about a classical (as opposed to a quantum) system.
Well, Karsus Ren is of course right that $\dot{q}^i=K^i$ implies that the solution $q^i(t)$ is an affine function of time $t$. 
I guess the heart of the question (v2) is not so much the solution itself, but more that if the Lagrangian $L=L(q,\dot{q},t)$ is given, and we have constraints $\dot{q}^i=K^i$, should we substitute $\dot{q}^i\to K^i$ in (1) none, (2) all, or (3) some of the $\dot{q}$ appearances in $L=L(q,\dot{q},t)$ before differentiating wrt. $\dot{q}^i$? 
The answer is that it doesn't matter. Normally we treat constraints by introducing Lagrange multipliers $\lambda_i$ and a new Lagrangian
$$ \tilde{L}(q,\dot{q},\lambda,t) = L(q,\dot{q},t) + \lambda_i (\dot{q}^i-K^i). $$
The Lagrange eqs. wrt. $\lambda_i$ yield the constraints $\dot{q}^i=K^i$.
This is the evolution equation for $q^i$.
On the other hand, the Lagrange eqs. wrt. $q^i$ yield
$$\frac{\partial L}{\partial q^i} = \frac{d}{dt} \left[ \frac{\partial L}{\partial \dot{q}^i} + \lambda_i \right]. $$
This is the evolution equation for $\lambda_i$. We see from the last equation that a change in the definition of the derivative $\frac{\partial L}{\partial \dot{q}^i}$ leads to a corresponding change in $\lambda_i$, but this has no consequences for $q^i$, as expected.

