PHYS 705: Classical Mechanics. Central Force Problems I

Transcription

1 1 PHYS 705: Cassica Mechanics Centa Foce Pobems I

2 Two-Body Centa Foce Pobem Histoica Backgound: Kepe s Laws on ceestia bodies (~1605) - Based his 3 aws on obsevationa data fom Tycho Bahe - Fomuate his famous 3 aws: - Obit of each panet is an eipse with sun at one of its foci - Equa aeas swept out in equa time by an obit R 3 - The atio is the same fo a panets, whee is the peiod and R is the semi-majo axis -A these esuts wee obtained though amazing shee mathematica effots. Befoe we get back to this impotant ceestia appication, we wi addess the genea pobem

3 3 Kepe s Notes

4 4 Two-Body Centa Foce Pobem Set up of the genea pobem: masses inteact via foces diected aong the ine that connects them (centa foce): stong fom of 3 d aw Fist Step: Centa foce pobems can be educed to an effective 1-body pobem: Change to geneaized coodinates: CM position R and eative position, CM ' ' 1 1 R 1 & R& ' ' 1 R ' 1 1 R ' Instead of using, use (CM & eative position) 1

5 5 Two-Body Centa Foce Pobem Fom def. of CM: MR m m m M i i i 1 1 ' ' R m R m 1 1 R m m m R 1 ' ' 11m 0 ' ' 1 Combining it with, we have m m ' ' ' 1 1 m1 m1 m 1 m m ' 1 1 m1 m ' 1 m1 m m m ' ' 1 1 and CM R ' ' ' 1 1 m1 m 1 m ' 1

6 6 Two-Body Centa Foce Pobem Now, fom the Lagangian: T T T CM about CM M m 1 ' m ' R 1 M 1 mm 1 mm 1 R m m m m M 1 mm 1 R m1 m 1 1 R& (expess in tems of ony) We have a centa foce so that U() depends on the eative distance ony. m ' 1 m1 m m ' 1 m1 m

7 7 Two-Body Centa Foce Pobem So, M 1 mm R m1 m 1 L T U U () Note that R does not appea in L (cycic) so that EL equation fo R wi ony give: MR const CM is eithe stationay o moving unifomy. Pick an inetia ef. fame (CM fame) in which CM is not R 0 moving and we can ignoe the 1 st tem in L. The esut is then: 1 L U( ), whee mm m m 1 1 Same as a singe patice with mass moving in U().

8 8 Refeence Fames fo Centa Foce Pobems Motion in the eative coodinate fame with (3 dofs) : R (fixed), m ' ' 1 m1 m m m ' 1 m1 m m 1 1 ' CM m 1 Effective one body pobem with educed mass and eative position (dist fom to ). m 1 m m 1 m Motion of and in CM fame

9 9 Refeence Fames fo Centa Foce Pobems, Motion in oigina space with 6 dofs (not a independent): 1 m 1 m and cice each m 1 CM R m othe in space as thei CM move with constant veocity aong a fixed diection. 1

10 10 Two-Body Centa Foce Pobem The educed pobem is a much simpe pobem with 3 dofs instead of 6. m Aso, we often have (e.g., sun-eath) m 1 Then, mm mm m m m m m the CM is so cose to and 1 '

11 11 Two-Body Centa Foce Pobem Futhe eduction of the two-body centa foce pobem: Since U() is centa, i.e., F aways diects towad CM aong, So, we have, dl N 0 and N 0 dt o L const (Atenativey, as we wi see ate, is cycic!) L wi points in a constant diection fixed by initia condition. Then, =const, and & p must be to L aways, i.e., motion has to be pana. L 0 If (tivia case), then the tajectoy is a 1D motion though the oigin. Lp //

12 1 Two-Body Centa Foce Pobem With a these consideations, it is natua to use spheica coodinates and to oient the poa axis with L. L Then, we can anayze the pobem entiey in a D poa pane. Lasty, using the fact that U()= U() depends ony on the magnitude of the distance, we can show that the Lagangian is effectivey one-dimensiona. To see that, stat with L in : m L U, () (we wi simpy ca m= fom now on) v ˆ θˆ (Reca, in poa cood.)

13 13 Two-Body Centa Foce Pobem Since U() is centa, is cycic (does not appea) in L. And, the geneaized momentum with espect to is conseved, L Repacing with the constant in the Lagangian, we then have an effectivey 1D system in ony: p m const m m m L U () m U() m Note: is ignoabe but it is NOT a constant. and change whie is fixed in time. Motion is sti in D in the CM fame*.

14 14 Two-Body Centa Foce Pobem Now, we cacuate the EOM fo both and : d L L dt 0 m L U () L 0 (L is cycic in ) L m This gives: m const I ( m ) [One can ecognize this as the constant ang momentum:.] (This is the equivaent statement about the consevation of L as befoe.)

15 15 Two-Body Centa Foce Pobem Connecting back to Kepe s nd aw d d Note that aea swept out by with an 1 infinitesima otation d is da d da 1 d 1 And, dt dt d dt Fom EL equation, we have m 0 d dt 1 0 (*) Thus, Eq (*) estabishes Kepe s nd Law!

16 16 Two-Body Centa Foce Pobem Now, we ae back to the EOM fo : L m L m du d m L U () d L L dt 0 gives m du d m 0 m Combining with the equation, we have, du m m m d m m 3 du d 0

17 17 Two-Body Centa Foce Pobem We have seen that the equation give one constant of motion. We can get anothe constant of motion E since: 0 L t so that h (Jacobi Intega/enegy function) is conseved. q j ( ) q j U doesn t depend on and does not depend on t expicity check: L h q j L q j so that h = E tota enegy is conseved. j 1 m m m U m m m U m U E

18 18 Two-Body Centa Foce Pobem Hee is anothe way to ague that E is conseved. Let stat with the EOM, du d m U d m d m Mutipying on both sides, 3 So, m d m d d U dt d m dt d m d U dt dt m d m U() 0 dt m E (evese chain ue) E is conseved!

19 19 Two-Body Centa Foce Pobem m L U () Summay: We get EOMs and integas of motion (, E) fo this pobem. m du m d m 3 m () m E U Note: The E equation effectivey is the 1 st intega of the equation. We can ewite it expicity as, EU() m m

20 0 Two-Body Centa Foce Pobem Integating once moe time gives: t 0 E m d ' U( ') m' Simia, we can integate the equation and get, With initia condition 0 at t=0 () t t 0 dt m() t 0 With initia condition 0 at t=0 So, the pobem is competey soved with t() and (t). The pobem has been educed to a quadatue (doing the integas).

21 1 Gaphica Anaysis of Centa Foce Pobem Using the concept of an effective potentia, one can get an usefu quaitative undestanding of the pobem without actuay integating! Let conside the equation: du m d m 3 The ast two tems combined can be consideed as an effective foce f '( ) This ooks ike a 1D pobem: a singe patice moving in 1 dimension unde the infuence of an effective foce, du f '( ) d m 3

22 Gaphica Anaysis of Centa Foce Pobem 0 Case 1 : du m d m 3 No angua momentum, then, it is eay a 1D pobem. (uninteesting case) Case 0 : Thee is now a effective exta foce tem that is not diecty associated with the two objects: m m v m m m m m whee v is the θˆ component of veocity v ˆ θˆ So, we ecognize this as the centifuga acc. tem in a co-otating ef. fame.

23 3 Gaphica Anaysis of Centa Foce Pobem 0 Fo the case, we can combine the RHS into an effective potentia tem: du d du '( ) m U 3 d m d m d whee U '( ) U( ) m We wi conside thee exampes: a. U k f k b. U a f 3a. / 3 4 c U k f k (gavitationa, EM) (Hooke s aw: HMO) Note: zeo point fo U() is diffeent in these thee cases.

24 4 Centa Foce Pobem: Invese Squae Foce Exampe a: invese-squae foce k U '( ) m Fo a fixed, we can pot: UU, ' m U '( ) Note the asymptotic: 1. UU, ' 0 as. UU, ' as dominates 1 as 4. 1 dominates 1 as 0 This gives, k U ' U k as U ' m as a we in the midde 0

25 5 Centa Foce Pobem: Invese Squae Foce Now, fo a fixed vaue of tota E > 0 & conside what kind of obit is possibe? Reca 1 1 k E m U '( ) m m Notice that 1 '( ) 0 m E U since the adia KE can t be negative. s.t. E U '( ), 0 min min The obit is to the ine fom the oigin to the obit: 0 hee v can t go beyond min, the tuning point [whee E U '( ) ] and since 0 min

26 6 Centa Foce Pobem: Invese Squae Foce U '( ) E > 0 (unbounded obit) 1 '( ) m E U E ( ) min tuning pt as U ' 0 E U ' E( const) const (cose by) At tuning pt, thee is no instantaneous adia motion. A motion is angua. (fa away) The system becomes ess infuence by U() and moe ike fee patices.

27 7 Centa Foce Pobem: Invese Squae Foce U ' E EU m U U m ' ' E (adia KE) (angua KE) U ' U U (cose by) min (fa away) gets big gets big A motion is angua hee 0 and 0 and A motion is adia hee const

28 8 Centa Foce Pobem: Invese Squae Foce In the eative position space, the obit wi ook ike: oigin (fa away) 0 and const as if it wee a fee patice min hee is the point of cosest appoach (tuning point) 0 is age

29 9 Centa Foce Pobem: Invese Squae Foce - As we have seen, if E > 0, fo age, the patices act ike fee patices with T = E. -Now, if E = 0, fo age, the patices wi be stopped with T = E = 0. - What happens to the obit when E < 0? obits wi be bounded E U ' 1 U tuning pts again 0 -Obit is bounded between 1 & -These distances ae caed apsides o apsida distances -Simia to pevious tuning points, obits ae tangent to cices: = 1 & = - is agest at inne adius (U -U is agest at 1 )

30 30 Centa Foce Pobem: Invese Squae Foce So, fo bounded obits (E < 0), the obits in space ook ike: (fo the invese-squae foce) 1 We wi tak moe about bounded obits ate. In paticua, cond fo cosed obits (fig. 3.7 is wong, cuvatue must be concave fo attactive foces) If E is at the minimum (= 0 ), the bounded obit wi have a constant adius 0 and it wi be a cicua obit. 0 E

31 31 Centa Foce Pobem: Invese Squae Foce U ' min tuning pts (apsides) 0 E E 0( unbounded) E 1 E 1 oigin E 0 U E E1 0( bounded) min 1 E E ( cicua) 0

32 3 Centa Foce Pobem: Stonge Attactive U Exampe b: Stonge Attactive Potentia This gives, m U '( ) a m 3 U a f 3a 3 4 E 1 1 dominates 1. E=E 1 : obit is unbounded. E=E : obit is bounded away E 3 1 fom o bounded within 1 patices can coide E 3 3. E=E 3 : obit is bounded U ' U 1 3 dominates within 3 and it can go thu oigin patices can coide

33 33 Centa Foce Pobem: Stonge Attactive U What is the condition fo the patice to go thu the oigin? Fom the tota enegy equation, we have, 1 m E U '( ) E U ( ) m U Let, () n Fo the oigin to be accessibe, nea the oigin. Thus, we must have, E n m 0 0 o E n m

34 34 Centa Foce Pobem: Stonge Attactive U E n m As the obit gets cose to the oigin, LHS 0 Fo this inequaity to be satisfied in the imit, RHS must be negative fo sma. Then, this is tue if 0 0 n This can cetainy be tue fo, attactive foce stonge than invese-squae type. E 0 m If n =, then must hod at the imit 0, ie.., m m

35 35 Centa Foce Pobem: Hooke s Law Exampe c: Hooke s Law/SHO U k f k (Homewok pobem) U '( ) k m 0: U ' U U is the geen ine E U ' U Motion is 1D on a ine and is simpe hamonic 0: Motion is on a D pane but m obit wi be bounded and typicay eiptic