Change of variables and gravity constants

[saybrook1]

Homework Statement

Hi guys, I'm struggling to figure out how the solution in the picture that I posted was able to get rid of their mg factors and then come up with a factor of k for x_1 in their eigenvalue equation. You can see that in the second equation of motion there is no k*x_1 but it shows up in the eigenvalue equation. I thought you could just drop those mg factors when you went to figure out eigenvalues since they only effect the equilibrium positions. Thanks a bunch of any help.

Homework Equations

Lagrangian for two springs and two hanging masses. Change of variables.

The Attempt at a Solution

I've tried to work out what the change of variables would be or how you would calculate it but not such luck yet.

Any help would be greatly appreciated. Thank you.

 

[TSny]

Homework Statement

You can see that in the second equation of motion there is no k*x_1 but it shows up in the eigenvalue equation.

Did you apply Lagrange's equations to the Lagrangian to check if the equations of motion are written correctly?

I thought you could just drop those mg factors when you went to figure out eigenvalues since they only effect the equilibrium positions.

I've tried to work out what the change of variables would be or how you would calculate it but not such luck yet.

Introduce new variables, $y_1$ and $y_2$, that are just shifted versions of the old variables: $y_1 = x_1 - a_1$ and $y_2 = x_2 - a_2$, where $a_1$ and $a_2$ are constants. Write the equations of motion in terms of the new variables and see if you can find values of the constants such that the mg terms disappear.

 

[saybrook1]

Did you apply Lagrange's equations to the Lagrangian to check if the equations of motion are written correctly?Introduce new variables, $y_1$ and $y_2$, that are just shifted versions of the old variables: $y_1 = x_1 - a_1$ and $y_2 = x_2 - a_2$, where $a_1$ and $a_2$ are constants. Write the equations of motion in terms of the new variables and see if you can find values of the constants such that the mg terms disappear.

Giving this a shot right now; Thank you very much. And yes the E.o.m.'s are correct.

 

[TSny]

Giving this a shot right now;

OK

Thank you very much. And yes the E.o.m.'s are correct.

The second equation (the one for $\ddot{x}_2$) is not correct. In particular, there appears to have been a mistake in calculating $\frac{\partial L}{\partial x_2}$.

 

[saybrook1]

Did you apply Lagrange's equations to the Lagrangian to check if the equations of motion are written correctly?Introduce new variables, $y_1$ and $y_2$, that are just shifted versions of the old variables: $y_1 = x_1 - a_1$ and $y_2 = x_2 - a_2$, where $a_1$ and $a_2$ are constants. Write the equations of motion in terms of the new variables and see if you can find values of the constants such that the mg terms disappear.

Okay, so I did come up with shifted variables that eliminated the mg's but I still can't figure out how they came up with that k*x_1 for the second line in the eigenvalue equation.

 

[saybrook1]

OK
The second equation (for $\ddot{x}_2$) is not correct. In particular, there appears to have been a mistake in calculating $\frac{\partial L}{\partial x_2}$.

Awesome! Thanks for pointing that out.

 

[saybrook1]

OK
The second equation (the one for $\ddot{x}_2$) is not correct. In particular, there appears to have been a mistake in calculating $\frac{\partial L}{\partial x_2}$.

Yeah, I just went through it and it's looking good now; I appreciate the help!

 

[TSny]

OK, great.