Answer to Question #253453 in Classical Mechanics for Alwande

Question

Let r= r(q1 ... qn, t). The generalized coordinates qi are not necessarily independent. Show that

a) ar/(agi)=ai/(aqi)

b)ai/(aqi) =d/(dt) ar/(aqr)

Answer 1

Given that,

$r= r(q_1,q_2,q_3,q_4 ....... q_n, t)$

The terms $q_i$ is not necessarily independent.

a) Now, a system of the degree of freedom with Lagrangian $L(q,q,t)$

$\frac{\partial}{\partial t}(\frac{\partial L}{\partial s})-\frac{\partial L}{\partial s_i}=0$

Now, as per the chain rule,

$\frac{\partial L}{\partial S_i}=\Sigma_k \frac{\partial L}{\partial q_k}\frac{\partial q_n}{\partial_i}+\frac{\partial L \partial q_k}{\partial q_k \partial Si}$

Now, we can say that the first function will be vanishes because $q_k$ is depending on the $S_k$ co-ordinates.

$q_i= \Sigma_i \frac{\partial q_i}{\partial S_i} S_i +\frac{\partial q_i}{\partial t}$