Lagrange Kinematic Formula

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Section 13.2 Lagrange Kinematic Formula

The following formula is pivotal to the Lagrange formulation.

Theorem 13.2.1. Lagrange Kinematic Formula.

Given a system \(\Sigma\) whose position is parametrized by \((n+1)\) variables \((\bq, t)\text{,}\) the acceleration of an arbitrary point \(P\) of the system satisfies

\begin{equation} \ba_P \cdot \frac{\partial \br_{OP}}{\partial q_i} = \left[\frac{d}{dt}\left(\frac{\partial}{\partial \dq_i}\right) - \frac{\partial}{\partial q_i} \right] \frac{\vel_P^2}{2}\tag{13.2.1} \end{equation}

where the variables \((\bq, \dbq, t)\) satisfy assumption (13.1.2).

Proof.

We seek to find the scalar quantity \(\ba_P \cdot \frac{\partial \br_{OP}}{\partial q_i}\) for a given coordinate \(q_i\text{.}\) Here \(P\) represents an arbitrary point of the system. By using (13.1.3) we find

\begin{equation*} \ba_P \cdot \frac{\partial \br_{OP}}{\partial q_i} =\ba_P \cdot \frac{\partial \vel_P}{\partial \dq_i} = \frac{d}{dt}\left(\vel_P \cdot \frac{\partial \vel_P}{\partial \dq_i} \right) - \vel_P \cdot\frac{d}{dt}\left( \frac{\partial \br_{OP}}{\partial q_i} \right) \end{equation*}

The first term can be written as

\begin{equation*} \frac{d}{dt}\left(\vel_P \cdot \frac{\partial \vel_P}{\partial \dq_i} \right) =\tfrac{1}{2}\frac{d}{dt}\left(\frac{\partial \vel_P^2}{\partial \dq_i}\right) \end{equation*}

As for the second term, we cannot permute the operators \(\frac{d}{dt}\) and \(\frac{\partial}{\partial q_i}\text{:}\) as we did for equation (13.1.1) we consider \(\frac{\partial \br_{OP}}{\partial q_i}\) as a function of \((\bq, t)\) to find

\begin{align*} \frac{d}{dt}\left( \frac{\partial \br_{OP}}{\partial q_i} \right) \amp = \sum_{j=1}^n \frac{\partial^2 \br_{OP}}{\partial q_j\partial q_i} \, \dq_j + \frac{\partial^2 \br_{OP}}{\partial t\partial q_i} = \frac{\partial}{\partial q_i}\left(\sum_{j=1}^n \frac{\partial \br_{OP}}{\partial q_j} \, \dq_j + \frac{\partial \br_{OP}}{\partial t} \right)\\ \amp =\frac{\partial \vel_P}{\partial q_i} \end{align*}

Now we can write \(\vel_P \cdot\frac{d}{dt}\left( \frac{\partial \br_{OP}}{\partial q_i} \right)\) as \(\tfrac{1}{2} \frac{\partial}{\partial q_i} \vel_P^2\text{.}\)