Set up the Lagrangian for a CO2 molecule

[FilipLand]

Homework Statement

The carbon dioxide molecule can be considered a linear molecule with a central carbon atom, bound

to two oxygen atoms with a pair of identical springs in opposing directions. Study the longitudinal

motion of the molecule. If three coordinates are used, one of the normal mode frequencies vanishes.

What does that represent physically? Calculate a numerical value for the ratio between the two

other (non-zero) normal mode frequencies of the molecule. (The exercise should be solved using Lagrangian mechanics.)

Homework Equations

The answer is
$$sqrt(\frac{2m_0 + m_c}{m_c})$$ or numerically 1.915

The Attempt at a Solution

I'm not sure how to set up the Lagrangian, or how to find the mode frequency once it done for that matter.

See my attempt of solution, which give me one manageable diff.equation but one very messy, since I can't express $\chi$ without x.

Any input on how to approach the exercise?

 

[drvrm]

looking at your Lagrangian i have a comment that it should be correctly expressed as difference of Kinetic and Potential energy.

the kinetic part is motion of two oxygen atoms added with the third middle atom KE and the potential part will be the energy of two springs. suppose you describe the coordinates as x1, X , x2 then the the lagrangian will be simpler and the characteristic equation can be written..which can give you the modes pf oscillation. see small oscillations in any textbook of mechanics.

 

[FilipLand]

looking at your Lagrangian i have a comment that it should be correctly expressed as difference of Kinetic and Potential energy.

the kinetic part is motion of two oxygen atoms added with the third middle atom KE and the potential part will be the energy of two springs. suppose you describe the coordinates as x1, X , x2 then the the lagrangian will be simpler and the characteristic equation can be written..which can give you the modes pf oscillation. see small oscillations in any textbook of mechanics.

Thanks.

Yes my solution is a bit messy, but I tried to put up kinetic-potential. The first term is KE of the two edge-balls and the second term is KE for the middle one. x1, x2 and X is also used, any input on my choice of coordinates?
Thanks again.

 

[drvrm]

any input on my choice of coordinates?
Thanks again.

if you consider the small disturbance of the atoms from their equilibrium positions the the lagrangian can be related to the motion characteristics...for small oscillations the lagrangian can be expanded in taylor series around the equilibrium condition and the depending on degrees of freedom the characteristic equation can be set up.
you can see details in the chapter on small oscillation to get to understand the methodology...e.g.
<http://www.physics.usu.edu/torre/6010_Fall_2010/Lectures/08.pdf>
i do not know whether it is allowed but unless you do the theoretical exercise it will be difficult tp proceed. which textbook you are using?

 

[FilipLand]

if you consider the small disturbance of the atoms from their equilibrium positions the the lagrangian can be related to the motion characteristics...for small oscillations the lagrangian can be expanded in taylor series around the equilibrium condition and the depending on degrees of freedom the characteristic equation can be set up.
you can see details in the chapter on small oscillation to get to understand the methodology...e.g.
<http://www.physics.usu.edu/torre/6010_Fall_2010/Lectures/08.pdf>
i do not know whether it is allowed but unless you do the theoretical exercise it will be difficult tp proceed. which textbook you are using?

I'm using Classical mechanics by John Taylor, its ok buth rather simple.

 

[drvrm]

I'm using Classical mechanics by John Taylor, its ok buth rather simple.

The lagrangian will look symmetrical if you take the three coordinates as displacements from their equilibrium condition.
Then one can expand the L as Taylor series and minimize it to get the characteristic equations involving the KE and potential energy.Then one can get the frequencies of normal modes of oscillations. take the help of that lecture quoted by me.if possible.
i think physically the three modes are when the middle atom remains stationary and the atom 1 and 3 oscillate ; the 2nd and third mode will involve movement of the middle one towards left and towards right.

 

[FilipLand]

Thanks!