What system does this lagrangian represent?

[Jamoo]

Hey guys,

I am struggling with this tutorial problem: If you have the following lagrangian:

L = m/2(aX^2 + bXY + cY^2) - K/2(ax^2 + bxy + cy^2)

(Where captial letters indicate first derivative with respect to time, a,b,c constants)

What physical system does it represent?

It looks like a simple coupled oscillator, but I don't know what the extra velocity term (bXY) could represent? Any help would be much appreciated.

Thanks

 

[James R]

Does it matter what physical system it represents? What does the question ask?

It's really hard to work back to from the Lagrangian to a particular system, anyway, since the coordinates can be generalised coordinates rather than simple displacements, for example.

It would be possible to transform to different generalised coordinates (U,V) which would completely remove the XY terms in the expression, and give an equivalent Lagrangian with no terms of the form kUV.

For example, try:

U = X + Y
V = X - Y

 

[Jamoo]

The question is as follows:
1) Derive the Equations of motion for this sytem

Examine particularly cases a=c=0, and b=0,c=-a (I am not sure whther this applies to part 1 or 2/3)

2) What is the physical system described by the above Lagrangian
3) Write the natural form of the Lagrangian for this system

But thanks for the help so far.

 

[robphy]

You can probably write each term in terms of a symmetric matrix. Finding the eigenvectors and eigenvalues (i.e., "normal modes") should suggest a nicer choice of coordinates.

 

[sniffer]

for the first term, do you mean X or X dot?

do you mean \frac {m}{2} (a\dot{x}^2 . . .. ?

 

[Jamoo]

JamesR and robphy, thanks. Using normal coordiantes seems to be the way to go.

sniffer: yes, by X I meant x dot, I just haven't worked out how to use the equation editor on forum.

 

[sniffer]

sorry, i didn't read carefully.