Small oscillations (normal modes)

[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

 

[Jhero]

for help



Thank you for sharing your question with us. I am happy to help you with writing the Lagrangian for the system you described. Let's first start by defining the variables and parameters of the system:

- x_1 and x_2 are the displacements of the two masses from their equilibrium positions.
- m_1 and m_2 are the masses of the two masses, respectively.
- k is the spring constant for both springs.
- l is the length of the unstretched springs.
- g is the acceleration due to gravity.

Now, let's take a closer look at the potential energy of the system. The potential energy of a mass suspended from a spring is given by the formula: U = 1/2 * k * x^2, where k is the spring constant and x is the displacement from the equilibrium position. In your system, both masses are suspended from springs, so the total potential energy can be written as:

U = 1/2 * k * (x_1)^2 + 1/2 * k * (x_2)^2

Next, we need to take into account the gravitational potential energy of the system. Since the masses are suspended vertically, the gravitational potential energy will depend on the height of the masses from the ground. We can calculate this potential energy as:

U = m_1 * g * (l + x_1) + m_2 * g * (l + x_1 + l + x_2)

Here, we have taken into account the fact that the first mass is at a height of l + x_1 from the ground, and the second mass is at a height of l + x_1 + l + x_2 from the ground.

Now, we can combine these two expressions to get the total potential energy of the system:

U = 1/2 * k * (x_1)^2 + 1/2 * k * (x_2)^2 + m_1 * g * (l + x_1) + m_2 * g * (l + x_1 + l + x_2)

Finally, we can write the Lagrangian for the system as:

L = 1/2 * (m_1 + m_2) * (\dot{x_1})^2 + 1/2 * (m_2) * (\dot{x_2})