How Do You Formulate the Lagrange Function for a Triatomic Molecule Model?

[Shafikae]

Consider the linear model of a molecule with three atoms. The outer atoms are of mass m and the atom in the molecules center is of mass M . The outer atoms are connected to the center atom through springs of a constant k.

(a) Find the Lagrange function of the system. Use as coordinates the deviations of the atoms from their equilibrium position.

(b) Find the eigenfrequencies of the system and corresponding eigenvectors.

(c) Write down the general solution of the equation of motion of the system.


I have not attempted this problem because I don't know how to do it at all. Any help will be appreciated. Thank you.

 

[Soveraign]

Consider the linear model of a molecule with three atoms. The outer atoms are of mass m and the atom in the molecules center is of mass M . The outer atoms are connected to the center atom through springs of a constant k.

(a) Find the Lagrange function of the system. Use as coordinates the deviations of the atoms from their equilibrium position.

(b) Find the eigenfrequencies of the system and corresponding eigenvectors.

(c) Write down the general solution of the equation of motion of the system.


I have not attempted this problem because I don't know how to do it at all. Any help will be appreciated. Thank you.

Tricky :) You may recall that in Newtonian mechanics, the Lagrangian L = T - V (potential energy - kinetic energy). The energy equations are in terms of position and velocity and are in "generalized coordinates". That is to say, you can use any coordinate system you like. A sometimes convenient system to use would be to first assign each object its own. So atom a has x_{a},y_{a}, b has x_{b},y_{b}, etc... Toss together an over "L" and then start connecting the relationships between the coordinates (add constraints). Once you have all constraints, the problem may become solvable.

Sometimes you can brute force this through, substituting where necessary. Another method is Lagrange multipliers.

Now, when a problem asks "what are the eigenfrequencies" it is typically hinting at a specific way to solve the problem. It is asking "what are the characteristic frequencies of the system?" You can arrive at this answer in more than one way.

Have you covered Lagrange multipliers or maybe some matrix method of solving systems recently? There are many ways to solve this.

 

[Shafikae]

Yes we have covered briefly a matrix method of solving this problem. But I have trouble with setting the Lagrangian and the whole process of understanding the coordinates etc. If you can help me with that would be like great. I just have a problem with setting up the equation. Thank you.