Question

# Consider a mass m moving in two dimensions with potential energy U (x, y) = 1/2 kr², where r² = x² + y². Write down the Lagrangian, using coordinates x and y, and find the two Lagrange equations of motion. Describe their solutions.

Solution

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Combined kinetic energy can be written like:

$T=\dfrac{m}{2}\vec v^2=\dfrac{m}{2}\left(\dot x^2 +\dot y^2 \right )$

While in the problem we have potential energy which we can rewrite using $x$ and $y$:

$U=\dfrac{1}{2}kr^2=\dfrac{1}{2}k\left(x^2+y^2\right)$

Now let's write $L$ and use $E-L$ theorem:

\begin{aligned} L=T-U\\ \dfrac{\partial L}{\partial x}=\dfrac{d}{dt}\dfrac{\partial L}{\partial \dot x}\\ \dfrac{\partial L}{\partial y}=\dfrac{d}{dt}\dfrac{\partial L}{\partial \dot y} \end{aligned}

This will give us following differential equations:

\begin{aligned} -kx=m\ddot x\\ -ky=m\ddot y \end{aligned}

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