Final Exam Classical Mechanics

Transcription

1 Final Ea Classical Mechanics. Consider the otion in one diension of a article subjected to otential V= (where =constant). Use action-angle variables to find the eriod of the otion as a function of energ. Solution: The kinetic energ of the article is given b: T The otential energ of the article is given b: V The Energ of the article is given b: E T V The Lagrangian of the sste is given b: L T V The conjugate oenta is given b: L The Hailtonian is given b: H L T V E The conjugate tie derivative of oenta and satial coordinates are: H H Algebraicall, solve for oentu in ters of E and. E

2 Final Ea Classical Mechanics The tie indeendent Hailton-Jacobi generating function as a function of is given b (eq..7): W d E d The action is given b (the factor of 4 ultilied b the integral of the os-,os- quadrant because of setr) (eq..8): a a J d E d E Ed Since the is onl in the ositive region: a J E Ed a E 3 E 3 E a J E E E E E Since the article is changing direction at a then it s kinetic energ becoes zero instantaneousl. Therefore a E 4 J E 3 3 Algebraicall eress E in ters of J: 4 3 J E E J 4 E J 4 The rescribed solution calls for finding the eriod fro the action angles (eqs.86,.87,.9). H w, w vt, J

3 Final Ea Classical Mechanics E J 3 4 J E H w J J E 4 E dw w dt E w vt t E v E 4 T E v E 4 E T The frequenc can also be found fro: J d E E E E de 3 3 The eriod is given b: T E 4 E 3

4 Final Ea Classical Mechanics. Deterine the Lagrangian densit for a three-diensional sound wave in air. Deterine the ressure, densit, velocit, and theral waves for a lane wave and a radiating oint source. Reference: Elore & Heald Phsics of Waves and htt://en.wikiversit.org/wiki/advanced_classical_mechanics/continuu_mechanics 3 Densit of air:.9 kg/ In considering volue changes in an elastic ediu, we can write Hooke s Law: PB P hdrostatic ressure strain B bulk odulus [force/area] The volue strain associated with increental ressure is related to the dislaceent (vector function of osition and tie): V V P B B dislaceent vector function P Pressure The increental changes in ressure cause sall changes V of volue V (equilibriu volue V) the work done is: W P V The otential energ densit is: W U P V P B P P V V B v Fro the net vector force on a cubical volue eleent a wave equation for ressure is found: df Pdddz dddz t ass of volue eleent acceleration of volue eleent P P P P P t t B t v t v B the velocit of the wave 4

5 Final Ea Classical Mechanics This is shown to result in a sinusoidal wave of the dislaceent vector: r, t P r e it P P Pr, t P r e v i k r t P r, t P e just set the direction to the ais a P, t P e a i k t it The kinetic energ densit of the wave based on the above is: T t Re Fro the kinetic energ densit and the otential energ densit the Lagrangian densit is found as: L Re t v If the gas is eanded adiabaticall then: B P B adiabatic bulk odulus [force/area] C C C P V P C V heat caacit at constant ressure heat caacit at constant volue So the state variables (ressure and theral roerties) of the air can be related to the velocit of sound in the air: v P Pressure T k B v k B Teerature ass of individual olecule Boltzann constant 5

6 Final Ea Classical Mechanics For a radiating oint source, the wave equation becoes (and solution for the ressure wave): P P r r r r v t A ikrt P e r 6

7 Final Ea Classical Mechanics 3. Take and * as indeendent field variables in the following Lagrangian densit and deterine the wave equation. What is it? What are the canonical oenta? (I hoe that s the sae thing as conjugate oenta!) LDensit * V * * * i Solution: Substitute into the Euler-Lagrange (for Lagrange Densit) equations: LDensit LDensit LDensit LDensit LDensit z t z t LDensit LDensit LDensit LDensit LDensit z t * * * * First do * * z t LDensit * V * * * V * * i i LDensit LDensit * * z z * V * * * * * i L Densit * LDensit LDensit z z * * Densit * * * * V * * * i L LDensit LDensit z z * * z z * V * * * z * z * i L Densit z z z z z * 7

8 Final Ea Classical Mechanics LDensit LDensit t t * * z z * V * * * t * t * i LDensit t t * i t i t Putting the all together: V z i t i With soe rearranging it becoes the ore failiar for of Schrodinger s Wave Equation for a article in an unsecified otential t V i The conjugate oenta are found fro: and * * * L LDensit * * z z * V * * * i i Densit * LDensit * LDensit * * z z * V * * * * i i 8

9 Final Ea Classical Mechanics 4. First show that the following transforation is canonical: sin, P Q P P Q Q P Q Q P Q P, sin A transforation is canonical if the following is an eact differential: I found this reall cool relationshi but it turned out to be not worth anthing it looks like angular oentu sin, P Q P P Q Q P Q Q, P sin Q P P Q P P Q P P Q P P Q Q P Q Q P Q Q P Q Q P sin Q P P Q P sin sin sin Back to roving it is an eact differential the first art is to eress d d in ters of P s and Q s P sin Q P d sin QdP P QdQ dp P P Q Q d P Q Q sin QdP P QdQ dp P d P Q sin QdP P Q P QdQ P QdP P sin P Q Q dp Q P Q dq Q dp d Q sin Q dp P Q dq P Q dp Q sin QdP Q P QdQ QdP P 9

10 Final Ea Classical Mechanics Now eress d in ters of P s and Q s P Q Q d QdP P sin QdQ dq P P sin Q P d P sin Q P QdP P sin QdQ dq P d P sin Q Q dp P sin Q P sin Q dq P sin Q dq P P QdP P P sin QdQ P dq P d sin Q Q dp P sin Q dq P sin Q dq P Q dp P P sin Q dq P dq P Now get an eression for d d in ters of P s and Q s: d d Q sin Q dp P Q dq P Q dp Q sin QdP Q P QdQ QdP P sin Q Q dp P sin Q dq P sin Q dq P Q dp P P sin Q dq P dq P d d PdQ P dq Q dp P Q dp P Q dp P P sin Q dq P Q sin Q dp Q P Q dq P sin Q dq P d d PdQ P dq Q dp d P P Q Q P sin Q d d PdQ P dq P dq Q dp d P P Q Q P sinq d d PdQ P dq d P P Q Q P sin Q P Q

11 Final Ea Classical Mechanics The final relationshi is what is needed to show the eact differential: d d PdQ P dq d P P Q Q P sin Q PQ So this is the final art of the roof to show that it is an eact differential: i? df dq P dq PdQ P dq d P P Q Q P sin Q P Q PdQ P dq d P P Q Q P sin Q P Q i dq i i j j j i i j j j P dq d d PdQ P dq df where F P P Q Q P sin Q P Q After construction the Hailtonian for a article of charge q oving in a lane that is erendicular to a constant agnetic field B, use the transforation to eress the Hailtonian in the (Q,P) and obtain the otion of the article as a function of tie. OK so the net thing to do is construct the Hailtonian. First eress the kinetic energ in and (assuing B is arallel to the z ais): T Now the otential energ due to the Magnetic field: B Bzˆ ˆ ˆ zˆ q q q V v A v B R v B c c c z Bq Bq V c c The Lagrangian is: ˆ ˆ ˆ ˆ Bq L T V c

12 Final Ea Classical Mechanics The conjugate oenta are: Bq L T V c L Bq c Bq c L Bq c Bq c Or ou can find the oenta fro the T- ethod: Bq L c L T a L T a (8.3 for) Find the T- atri: T T

13 Final Ea Classical Mechanics For via equation 8.7 H a T ( a) L H H H The Hailtonian is the sae as in equation 8.34 H Convert to new coordinates: sin, P Q P P Q Q P Q Q P Q P, sin H P Q Q P Q Q P sin Q P P sin Q P H P Q Q P Q Q P sin Q P P sin Q P sin H P Q P Q H 8P H P 3

14 Final Ea Classical Mechanics H P E H Q P Q t P E E sin t P E t Q LET t,, sin E P P E Q Q E Esin t E t E This is the equation of a circle. qb c E sin t E E t E r Note: the angular oentu is constant also: E E E P Q P 4