Question

# Show that Lagrange's method automatically yields the correct equations of motion for a particle moving in a plane in a rotating coordinate system Oxy. (Hint: $T=\frac{1}{2} m \mathbf{v} \cdot \mathbf{v}$, where $\mathbf{v}=\mathbf{i}(\dot{x}-\omega y)+\mathbf{j}(\dot{y}+\omega x), \text { and } F_{x}=-\partial \mathrm{V} / \partial x, F_{y}=-\partial \mathrm{V} / \partial y$.)

Solution

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$\text{\underline{\textbf{To show:}}}$ Lagrange's Method leads to the correct equations of motion for a particle moving in a plane rotating coordinate system $O_{xy}$

$\text{\underline{\textbf{Proof}:}}$

$\text{\underline{Define the Kinetic energy and Lagrangian:}}$

$\begin{gather*} \mathbf{v} = (\dot{x} - \omega \cdot y) \mathbf{i} + (\dot{y} + \omega \cdot x) \mathbf{j} \\ T = \frac{1}{2} m \mathbf{v} \circ \mathbf{v} = \frac{1}{2} m [(\dot{x} - \omega \cdot y)^{2} + (\dot{y} + \omega \cdot x)^{2}] = T(\dot{x}, \dot{y}, y, x) \end{gather*}$

$\begin{gather*} L(\dot{x}, \dot{y}, y, x) = T(\dot{x}, \dot{y}, y, x) - V(x, y) = \frac{1}{2} m [(\dot{x} - \omega \cdot y)^{2} + (\dot{y} + \omega \cdot x)^{2}] - V(x, y) \end{gather*}$

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