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[SOLVED] Springs, Normal Modes, and Center of Mass coordinates
1.) Problem
The problem of the linear triatomic molecule can be reduced to one of two degrees of freedom by introducing coordinates [tex]y_{1} = x_{2}  x_{1}, y_{2} = x_{3}  x_{2}[/tex], and eliminating [tex]x_{2}[/tex] by requiring that the center of mass remain at rest. Obtain the frequencies of the normal modes in these coordinates and show that they agree with the results of Section 6.4.
Classical Mechanics, Goldstein, 3rd Edition, pg 272
2.) Useful Formulae and Context
I've attached a picture I drew of the setup.
In section 6.4, the normal modes of the triatomic molecule are found by
1. writing out the potential and kinetic energy,
2. converting to coordinates relative to the equilibrium positions,
3. expressing them (V and T) as tensors, and
4. solving the characteristic equation [tex]V  \omega^{2} T = 0[/tex]
Explicitly,
[tex]V = \frac{k}{2} (x_{2}  x_{1}  b)^{2} + \frac{k}{2} (x_{3}  x_{2}  b)^{2} [/tex]
The coordinates relative to the equilibrium positions are introduced:
[tex]\eta_{i} = x_{i}  x_{0i}[/tex]
where
[tex]x_{02}  x_{01} = b = x_{03}  x_{02}[/tex]
So the potential energy becomes,
[tex]V = \frac{k}{2} (\eta_{2}  \eta_{1})^{2} + \frac{k}{2} (\eta_{3}  \eta_{2})^{2}[/tex]
[tex]V = \frac{k}{2} (\eta_{1}^{2} + 2\eta_{2}^{2} + \eta_{3}^{2}  2\eta_{1}\eta_{2}  2\eta_{2}\eta_{3})[/tex]
which can pretty easily be written in tensor form. A similar thing is done with kinetic energy.
3.) Attempt at the Solution
Goldstein writes that we should impose the constraint that "the center of mass remain stationary at the origin:"
[tex]m(x_{1} + x_{3}) + M x_{2} = 0[/tex]
and that this equation should be used to eliminate one of the coordinates from V and T.
Clearly, this coordinate should be [tex]x_{2}[/tex], since it appears in both [tex]y_{1}, y_{2}[/tex], right? I've scribbled and rearranged these equations over and over, and can't figure out how express V and T only in terms of [tex]y_{1}, y_{2}[/tex].
The "Center of Mass" is described by:
[tex]R = \frac{\sum m_{i} x_{i}}{\sum m_{i}} = \frac{m(x_{1} + x_{3}) + M x_{2}}{2m + M}[/tex]
How does knowing this help me? If someone could just point the way, or give me the smallest hint, I'm sure I could push this through  I'm just having a block on this. Thanks!
1.) Problem
The problem of the linear triatomic molecule can be reduced to one of two degrees of freedom by introducing coordinates [tex]y_{1} = x_{2}  x_{1}, y_{2} = x_{3}  x_{2}[/tex], and eliminating [tex]x_{2}[/tex] by requiring that the center of mass remain at rest. Obtain the frequencies of the normal modes in these coordinates and show that they agree with the results of Section 6.4.
Classical Mechanics, Goldstein, 3rd Edition, pg 272
2.) Useful Formulae and Context
I've attached a picture I drew of the setup.
In section 6.4, the normal modes of the triatomic molecule are found by
1. writing out the potential and kinetic energy,
2. converting to coordinates relative to the equilibrium positions,
3. expressing them (V and T) as tensors, and
4. solving the characteristic equation [tex]V  \omega^{2} T = 0[/tex]
Explicitly,
[tex]V = \frac{k}{2} (x_{2}  x_{1}  b)^{2} + \frac{k}{2} (x_{3}  x_{2}  b)^{2} [/tex]
The coordinates relative to the equilibrium positions are introduced:
[tex]\eta_{i} = x_{i}  x_{0i}[/tex]
where
[tex]x_{02}  x_{01} = b = x_{03}  x_{02}[/tex]
So the potential energy becomes,
[tex]V = \frac{k}{2} (\eta_{2}  \eta_{1})^{2} + \frac{k}{2} (\eta_{3}  \eta_{2})^{2}[/tex]
[tex]V = \frac{k}{2} (\eta_{1}^{2} + 2\eta_{2}^{2} + \eta_{3}^{2}  2\eta_{1}\eta_{2}  2\eta_{2}\eta_{3})[/tex]
which can pretty easily be written in tensor form. A similar thing is done with kinetic energy.
3.) Attempt at the Solution
Goldstein writes that we should impose the constraint that "the center of mass remain stationary at the origin:"
[tex]m(x_{1} + x_{3}) + M x_{2} = 0[/tex]
and that this equation should be used to eliminate one of the coordinates from V and T.
Clearly, this coordinate should be [tex]x_{2}[/tex], since it appears in both [tex]y_{1}, y_{2}[/tex], right? I've scribbled and rearranged these equations over and over, and can't figure out how express V and T only in terms of [tex]y_{1}, y_{2}[/tex].
The "Center of Mass" is described by:
[tex]R = \frac{\sum m_{i} x_{i}}{\sum m_{i}} = \frac{m(x_{1} + x_{3}) + M x_{2}}{2m + M}[/tex]
How does knowing this help me? If someone could just point the way, or give me the smallest hint, I'm sure I could push this through  I'm just having a block on this. Thanks!
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