PHYS 705: Classical Mechanics. Small Oscillations: Example A Linear Triatomic Molecule

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1 PHYS 75: Clssicl echnics Sll Oscilltions: Exple A Liner Tritoic olecule

2 A Liner Tritoic olecule x b b x x3 x Experientlly, one ight be interested in the rdition resulted fro the intrinsic oscilltion odes fro these tritoic olecule. The potentil energy for the two springs is, V x xb x3 x b

3 3 A Liner Tritoic olecule x b b x x3 x Now, we will introduce generlized coordintes reltive to their x, x, x equilibriu positions, : 3 x x j j j j,, 3 Note: x x x3 x b

4 4 A Liner Tritoic olecule Expnding the potentil energy bout its equilibriu position, we hve: V 3 ultiplying the squres out, we hve: x x b x x x x x x xx V 3 3 V V V j j In trix for, this qudrtic for hs this for: V

5 5 A Liner Tritoic olecule Direct ethod to evlute using Recll, we hve V V j V j qjq V x xb x3 x b By ting the prtil derivtives directly nd evluting t, x j x j V V qq V : x xb nd q V : V q nd x x b x3 x b V qq V

6 6 A Liner Tritoic olecule The inetic energy for the three ss is given by: T x x x 3 Substituting our generlized coordintes or, x x j j j x j j T 3 T T j j In trix for, this qudrtic for hs this for: T

7 7 A Liner Tritoic olecule Cobining these two qudrtic fors into the chrcteristic eqution, VT V T Explicitly evluting this deterinnt, we hve the following eqution, 4

8 8 A Liner Tritoic olecule And, this hs three distinct solutions (eigenfrequencies): 3 Note: The solution ens tht the corresponding norl coordinte will hve the following trivil ODE: j or (unifor trnsltionl otion) The entire olecule will siply ove uniforly to the right or left; no oscilltions (not quite interesting otion by itself)

9 9 A Liner Tritoic olecule Now, we find the eigenvectors for ech solved eigenvlues using: V r T r r r r r r 3r Norl ode #: r 3 3

10 A Liner Tritoic olecule Norl ode #: r Solving st nd 3 rd equtions, we hve nd Putting the together, we hve 3 (Note tht the solution lso stisfies the nd eqution.) 3 So, the eigenvector for is:

11 A Liner Tritoic olecule Norl ode #: r The eigenvector needs to be norlized with respect to T: This gives: T Finlly, we hve the norlized eignvector for :

12 A Liner Tritoic olecule Norl ode #: r 3 3 3

13 3 A Liner Tritoic olecule Norl ode #: r 3 Agin, we need to norlized with respect to T: T This gives:

14 4 A Liner Tritoic olecule Norl ode #3: r First, let try to siply the trix eleents first: 3 3

15 5 A Liner Tritoic olecule Norl ode #3: r 3 Putting these vlues bc into the trix, we hve

16 6 A Liner Tritoic olecule Norl ode #3: r Agin, we need to norlized with respect to T: T 3 3 3

17 7 A Liner Tritoic olecule Norl ode #3: r This gives: 3 4 3

18 8 A Liner Tritoic olecule Then, our generl solution is given by: j jr r nd Ce Ce irt * irt r r r (the coplex coefficient will be deterined by IC) C r

19 9 Longitudinl Norl odes So, if single norl ode is ctive, the otion of the three generlized j r coordintes will loos lie the following,,, 3 :,, :,, 3: n,, 3 3 3

20 Longitudinl Norl odes nd Anitions for the two non-rigid-trnsltionl odes: 3 inole/linole.htl (fro Polytechnics Institute of NY Univ: K. ing Leung)

21 Sury. Pic generlized coordintes nd find nd. Expnd T nd V bout equilibriu q j Tq ( ) V( q j ) This gives two rel syetric qudrtic fors: with q q j j j j T ( ) V ( ) 3. Clculte eigenfrequencies fro chrcteristic eqution r det Vj r Tj 4. Clculte eigenvectors for ech eigenfrequencies using 5. Norlize eigenvectors with respect to T: 6. Generl solutions re in ters of the norl odes r * 7. Originl coordintes re relted bc through t C e C e r i t irt r r r Tjjrr j j j j V T j r j jr C deterined by IC r j jr r