- #1

brotof

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## Homework Statement

Consider the following Lagrangian:

\begin{equation} L = \frac{m}{2}(a\dot{x}^2 + 2b\dot{x}\dot{y} + c\dot{y}^2)- \frac{k}{2}(ax^2 + 2bxy + cy^2)\end{equation}

Assume that \begin{equation} b^2 - 4ac \ne 0 \end{equation}Find the equations of motion and examine the cases a=b=0 and b=c, c=-a. Which kind of physical system is described by this Lagrangian? What meaning does \begin{equation} b^2 - 4ac \ne 0 \end{equation} have?

## Homework Equations

\begin{equation} \frac{\partial L}{\partial q_i} -\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i} } = 0 \end{equation}

## The Attempt at a Solution

I'm able to solve the maths for this problem (assuming no errors in my calculations). In general for a, c not equal to 0 i get

\begin{equation} x(t) = \frac{1}{ac-b^2}(A_1 c \cos(\omega t + \phi_1) - A_2 b \cos(\omega t + \phi_2)) \end{equation}

\begin{equation} y(t) = \frac{1}{ac-b^2}(A_2 a \cos(\omega t + \phi_2) - A_1 b \cos(\omega t + \phi_1)) \end{equation}

While for the cases a=c=0 and b=0, c=-a i get

\begin{equation} x(t) = A_1 \cos(\omega t + \phi_1) \end{equation}

\begin{equation} y(t) = A_2 \cos(\omega t + \phi_2) \end{equation}

The problem is I can't exactly figure out what kind of system this could be. My first thoughts are some kind of 2D harmonic oscillator or some system of 2 masses coupled with springs. I'm having problem answering what \begin{equation} b^2 - 4ac \ne 0 \end{equation} "means" when I can't figure out what kind of system this is.

Thanks in advance,

Brotof