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I have derive the Euler-Lagrange equation which is equation (2) for a condition in which generalised velocity is independent on the generalised coordinate but when generalised velocity is dependent on the generalised coordinate then we get a familiar Euler-Lagrange equation which is equation (3).

Is equation (2) a valid Euler-Lagrange equation?

Qmechanic
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Keshav Shrestha
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  • Possible duplicates: https://physics.stackexchange.com/q/885/2451 and links therein. – Qmechanic Aug 31 '22 at 13:19
  • " when generalized velocity is dependent on the generalized coordinate " then you get non holonomic constraint equations. – Eli Aug 31 '22 at 14:36
  • @Eli so equation (2) is non-holonomic constraint equation? – Keshav Shrestha Aug 31 '22 at 15:39
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    $q$ and $\dot{q}$ are independent variables, so the relation $\frac{\partial L}{\partial q}= \frac{\partial L}{\partial q}\frac{\partial q}{\partial q}+\frac{\partial L}{\partial \dot{q}}\frac{\partial \dot{q}}{\partial q}$ is meaningless. One doesn't substitute equation of motion to derive EL eqn of motion, rather $\dot{q}$ as function of $q$ is derived from solving EL eqn – KP99 Aug 31 '22 at 16:33

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