Lagrangian of a system of several masses and springs

[fluidistic]

Homework Statement

I challenged myself with a problem I invented, but I'm stuck.
Consider a 1 dimensional problem consisting of 3 masses, each one separated by a spring. So that from the left to the right of my sketch we have $$m_1$$, a spring ($$k_1$$ with natural length $$l_1$$), $$m_2$$, another spring ($$k_2$$, $$l_2$$) and $$m_3$$. Find the equations of motion of the system.

Homework Equations

$$L=L_1+L_2+L_3$$ where $$L_i=T_i-V_i$$.
After this, Euler-Lagrange equations.

The Attempt at a Solution

For the first mass I reached $$L_1=\frac{m_1 \dot x ^2}{2}-\frac{k_1 (\Delta x_1 )^2}{2}$$ though this $$\Delta x_1$$ really bothers me.
Now to find $$L_2$$, $$V_2$$ is a real headache. Because this mass is connected to 2 springs, I'm not sure at all how to calculate the potential energy of it. Maybe adding both springs' extensions? I mean $$V_2=\frac{k_1 (\Delta x_1)^2 + k_2 (\Delta x_2)^2}{2}$$?
Its kinetic energy would be $$T_2=\dot x ^2 + 2 \dot x \Delta \dot x_1 + (\Delta x_1)^2$$. Am I in the right direction?

 

[zachzach]

Homework Statement

I challenged myself with a problem I invented, but I'm stuck.
Consider a 1 dimensional problem consisting of 3 masses, each one separated by a spring. So that from the left to the right of my sketch we have $$m_1$$, a spring ($$k_1$$ with natural length $$l_1$$), $$m_2$$, another spring ($$k_2$$, $$l_2$$) and $$m_3$$. Find the equations of motion of the system.

Homework Equations

$$L=L_1+L_2+L_3$$ where $$L_i=T_i-V_i$$.
After this, Euler-Lagrange equations.

The Attempt at a Solution

For the first mass I reached $$L_1=\frac{m_1 \dot x ^2}{2}-\frac{k_1 (\Delta x_1 )^2}{2}$$ though this $$\Delta x_1$$ really bothers me.
Now to find $$L_2$$, $$V_2$$ is a real headache. Because this mass is connected to 2 springs, I'm not sure at all how to calculate the potential energy of it. Maybe adding both springs' extensions? I mean $$V_2=\frac{k_1 (\Delta x_1)^2 + k_2 (\Delta x_2)^2}{2}$$?
Its kinetic energy would be $$T_2=\dot x ^2 + 2 \dot x \Delta \dot x_1 + (\Delta x_1)^2$$. Am I in the right direction?

I think you need three different x positions, one for each mass. That way your answer will be three equations of motion, one for each mass. This way you could also write $$\Delta x_1$$ = x_2 - x_1

 

[vela]

Breaking the Lagrangian up by masses isn't a good idea because the potential energies aren't quantities associated with any single mass, but with each spring. In terms of the x_i's, what is the distance between the ends of spring 1? Once you have that, it's easy to write down what the potential energy stored by spring 1 is in terms of your variables.

 

[fluidistic]

I think you need three different x positions, one for each mass. That way your answer will be three equations of motion, one for each mass. This way you could also write $$\Delta x_1$$ = x_2 - x_1

Good observation. Actually that's what I partly did. Hence my equation for $$T_2$$. I took $$\vec r _2$$ or your $$x_2$$ as $$x+l_1+\Delta x_1$$.

Breaking the Lagrangian up by masses isn't a good idea because the potential energies aren't quantities associated with any single mass, but with each spring. In terms of the x_i's, what is the distance between the ends of spring 1? Once you have that, it's easy to write down what the potential energy stored by spring 1 is in terms of your variables.

Oh... hmm. Looking at zach's reply, x_2-x_1 is the distance between the ends of spring 1. So the potential energy of these 2 masses (or not?) would be $$\frac{k_1 (l_1 +x_2 - x_1 )^2}{2}$$? I'm not sure whether it's a "+x_2-x_1" or "-x_2+x_1" and also the second's mass is also attached to the other spring...

 

[vela]

The masses don't have potential energy; the springs do.

Your expression for the potential energy of spring 1 is almost correct, but not quite.

 

[fluidistic]

The masses don't have potential energy; the springs do.

Your expression for the potential energy of spring 1 is almost correct, but not quite.

Oh you're right.
Correct me if I'm wrong. I assumed 5 Lagrangians. 3 for the masses and 2 for the springs. The springs only carry a potential energy while the masses only a kinetic energy. Is this right?
If so, I reach the following Lagrangian: $$L=\frac{m_1\dot x_1 ^2}{2}+\frac{k_1}{2}(l_1 - x_2 ^2 + 2 x_1 x_2 - x_1 ^2)+\frac{m_2 \dot x_2 ^2}{2}+\frac{k_2}{2}(l_2-x_3 ^2 + 2 x_2 x_3 - x_2 ^2)+\frac{m_3 \dot x_3 ^2}{2}$$.

As for the motion equations, I get $$m_1 \ddot x_1 + k_1 (x_1-x_2)=0$$.
$$k_1 (x_1-x_2)+k_2 (x_3-x_2)-m_2 \ddot x_2 =0$$.
And $$k_2(x_2-x_3)-m_3 \ddot x_3=0$$. They look too simple in my opinion... What do you think?

 

[vela]

The equations of motion look okay. They're just what you'd get if you applied F=ma to each mass, right?

Your Lagrangian, as you typed it, is obviously wrong. The units don't work out in the spring terms. You can't combine l₁ with x₁² for instance.

 

[fluidistic]

The equations of motion look okay. They're just what you'd get if you applied F=ma to each mass, right?

I guess so. But I applied Euler-Lagrange's equations to my wrong Lagrangian to get them.

Your Lagrangian, as you typed it, is obviously wrong. The units don't work out in the spring terms. You can't combine l₁ with x₁² for instance.

Ah right... Err... But is the following true:

Correct me if I'm wrong. I assumed 5 Lagrangians. 3 for the masses and 2 for the springs. The springs only carry a potential energy while the masses only a kinetic energy. Is this right?

.

 

[fluidistic]

I see my error. My Lagrangian was $$L=\frac{m_1 \dot x_1 ^2}{2}+\frac{k_1}{2}[l_1- (x_2 -x_1)]^2 + \frac{m_2 \dot x_2 ^2}{2}+\frac{k_2}{2}[l_2 - (x_3-x_2)]^2+\frac{m_3 \dot x_3 ^2}{2}$$.
It gives me the equations $$m_1 \ddot x_1 +k_1 [(x_2 - x_1)-l_1]=0$$.
$$m_2 \ddot x_2 + k_1 [l_1 +(x_2 - x_1)]-k_2 [l_2 -(x_3-x_2)]=0$$ and $$m_3 \ddot x_3 +k_2 [l_2 - (x_3 - x_2)]=0$$.
Does this looks nicer?

 

[vela]

Looks good.

 

[fluidistic]

Looks good.

Ok thank you. :)