# Homework Help: Small oscillations (normal modes)

1. Apr 16, 2006

### wormhole

Hi

see the attached picture...
2 coupled masses, each suspended from spring in gravitational field...
also entire construction can vibrate only vertically...

I need to write lagrangian for this system in the following form:

$$L=\frac{1}{2}\sum_{i=1,j=1}^2m_{ij}\dot{x}_i\dot{x}_j-\frac{1}{2}\sum_{i=1,j=1}^2k_{ij}x_ix_j$$

$x_i$ and $x_j$ are displacements of masses from equalibrium positions and springs are identical

the kinetic part of L is easy one but i stack with potential part...
i get to this expression(i always confused about right signs):

$$m_1g(l+x_1)+m_2g(l+x_1+l+x_2) + \frac{1}{2}m_1{x_1}^2+\frac{1}{2}m_2{x_2}^2$$

- positive direction taken along g direction
- l is length of unstreched spring

every potential term in L must in $k_{ij}x_ix_j$ form but i don't see a way how do i get it from my expression for potential energy because there are g force terms containing only $x_i$ or $x_j$

thanks

#### Attached Files:

• ###### 2springs.GIF
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Last edited: Apr 16, 2006