PHYS 705: Classical Mechanics. Central Force Problems II

Transcription

1 PHYS 75: Cassica Mechanics Centa Foce Pobems II

2 Obits in Centa Foce Pobem Sppose we e inteested moe in the shape of the obit, (not necessay the time evotion) Then, a sotion fo = () o = () wod be moe sef! Fist, et ty to get = (): Stat with the EOM: dv m 3 m d We aso have the anga momentm eqation: dt both sides m m d dt dt m d d d dt m d (NOTE: switch to V notation since we wi be sing as invese adis ate)

3 3 Obits in Centa Foce Pobem Sbstitting this eation into o eqation, we have, m d d dv dt dt m 3 d m d d dv 3 m d m d m d d d dt m d d d dv 3 m d d m d To simpify this fthe, it wod be sef to intodce a cood tans: RHS: by chain e, we have bt, d d dv () d d dv d d So dv d V d d

4 Obits in Centa Foce Pobem 3 d d d m d m d m d d d m d d Ptting the two sides togethe, we have Now, the LHS: m d d d V d d md V d d ODE fo () o () 3 d d d d m m 4

5 5 Obits in Centa Foce Pobem In this fom, we can get a qaitative insight into the obit s symmety: Conside a tning point (apside) at with ICs: () and d d (tning pt) ( ) apside ( ) Integate the pevios eqation fowad et say in the + diection and we get ( ) Now, since d d with ' d ' d Integate the same ODE backwad in the - diection with the same ICs wi give the same vaes, i.e., ( ) ( ) Obit is symmetic abot the apsides!

6 6 Obits in Centa Foce Pobem So, we ony need to find the obit fom one apside to the NEXT. Then, to constct the f obit, one can efect this basic segment aong the axis connecting the apside and the oigin symmeticay. apside apside

7 7 Obits in Centa Foce Pobem If yo expess the ODE back in tems of and the foce F(), yo get, d m d m V d d This fom is sef if yo want to sove fo the foce aw fo a ( ) given known obit. (homewok) F Exampe: et ke ( k fo physica obits, needs to be +) (the obit is a spia: + ot and in ) () eca: dv () d dv F () d d d d V d Pg in st tem in ODE: d d e e d ke k d k d d ke k e

8 8 Obits in Centa Foce Pobem So, we have m F 3 m () F () ke So, fo the pescibed obit, the eqied foce aw is: F () m 3 (an invese cbic foce aw)

9 9 Obits in Centa Foce Pobem ( ) Instead of soving fo fom the pevios nd ode ODE Thee is aso an atenative way to get the invese obit eqation by soving a qadate. Reca the eqation obtained fom consevation of enegy eqation: d EV() dt m m d d d dt d dt () To eiminate t in the eqation, note that by chain e again, This can be ewitten sing the anga momentm eqation, d d d d dt dt d m d ( note : m ) d d dv 3 m d d m d

10 Obits in Centa Foce Pobem Sbstitting this into o eqation, we have d EV() m d m m Reaanging tems and integating both sides gives, m d m E V m The ight hand side can be integated by qadate.

11 Obits in Centa Foce Pobem Comments: () n n F,.., ie V(). If Then, the intega can be integated (in cosed fom) in sevea cases: n =, -, -3: can be soved in tems of tig fnctions. n = - is the Kepe s pobem n = is the hamonic osciato n 5,3,, 4, 5, 7 o n,,,, m d m E V m can be soved in tems of eiptic fnctions

12 Cosed Obits Comments: m d m E V m. This fom of the eqation is aso sef in detemining whethe o not obits ae cosed, i.e. if they eventay etn to whee it stated and etace the same path Reca we showed that the obit is symmetic abot its apsides min max So the anga change in in going fom min to max then max back to min is max d m E V m m min

13 3 Cosed Obits: Batad s Theoem Comments: a a b b If, whee is ationa, then the obit coses afte b cyces ( ). And, the obit wi have gone min max min aond the cente of foce a times. In this exampe,, the obit is a cosed eipse. a b min max

14 4 Cosed Obits: anothe exampe max Anothe exampe with : the obit is a cosed eipse. a b min Batad s Theoem (873) states that ony the invese sqae foce (n = -) and Hooke s aw (n = ) give ise to cosed obits. (We won t pove it bt we wi give a favo of it now.)

15 5 Stabiity of Cica Obits - Fo any attactive potentias, a (bonded) cica obit is aways possibe fo the ight choice of E and. This cica obit wi occ at vaes whee the effective potentia V () has its extema (eqiibia). V ' eqiibia detemines the shape of V () E Fo E = E,, obit wi be a E cica obit at =,. m (Note: The tem in V () is epsive so that an attactive potentia V() is needed to ceate a potentia we.)

16 6 Stabiity of Cica Obits - A given cica obit is stabe if: V ' E E dv d ' eqiibia, AND dv d ', So, fo this exampe, the cica obit at = wi be stabe and the one at = wi not be. - Conside a genea powe aw attactive centa foce: k k F () with n V() n n

17 7 Stabiity of Cica Obits and the effective potentia is: V '( ) - Appying the condition fo stabe cica obits: k n m n dv ' k d m n 3 mk n3 dv' nk 3 d m n 4 nk 3 m n3 n 3 Sbstitte fom the top into the bottom eqation, we have, nk 3 mk m 3n m

18 8 Stabiity of Cica Obits The ed boxed eqation impies that: n 3 is eqied fo stabe cica obits! - Now, we conside the sitation if the obit is sighty deviated fom the stabe cica obit. We want to anayze its osciations abot the cica obit Fo convenience, we escae the foce aw by m sch that: dv F () mg () d so that the eqation of motion is given by: m dv d m g ()

19 9 Stabiity of Cica Obits Sbstitting the constant anga momentm: m g() 3 - Now, we conside the sitation when the obit was initiay at and we appy a sma petbation x to it, i.e., m x 3 x x g( x) m Unde this sma petbation, we want to appoximate: (cick) ( note : x) x 3 x x

20 Sma Petbations of a Cica Obit x

21 Stabiity of Cica Obits dg g( x) g( ) x d - Ptting these two appoximations back into o ODE fo the petbation x, x g( x) m x x x 3 3 g( ) g '( ) x m x x x - Note, ON the cica obit ( ), we have 3 x 3 x 3 3 m 3 g( )

22 Stabiity of Cica Obits - Ptting this back into the petbation eqation, we have, x 3 ( ) x g( ) g g '( ) x 3 g ( ) x g'( ) x - This ooks ike the hamonic osciato eqation with nata feqency x x with 3 g ( ) g '( )

23 3 Stabiity of Cica Obits - If, this has the genea osciatoy sotion: it x() t Ae Be it - If, then is imaginay and the sotion wi no onge be osciatoy The petbation x wi exponentiay gow o decay in time cica obit wi not be stabe! - So, fo stabiity of the cica obit, we need : 3 g ( ) 3 g'( ) g( ) g'( ) we sed g ( ) 3 m 3 F'( ) F( )

24 4 Stabiity of Cica/Cosed Obits - Again, sing a powe aw foce aw: F() k n - We have, ( n) 3 nk n k 3 n (3 n) - Again, we have the condition n 3 needed fo stabe cica obit. - One step fthe, in ode fo s to have cosed (bt sighty off cica) obits, the anga speed of the deviation fom the cica obit, mst be commensate with the anga speed of the cica obit itsef,.

25 5 Sma Petbations of a Cica Obit x one osciation of the ippe one f cyce aond the cente The be obit osciate as it goes aond the cente of foce and it coses back onto itsef.

26 6 Stabiity of Cosed Obits - Let conside this fthe. On the cica obit, we have, ( const) Fom the eqation of motion, we can cacate this, m F( ) mg( ) g( ) m m F () - Fo cosed obits, we then need, / 3 g ( ) g'( ) g p g g q / '( ) / 3 ( ) ( ) p,q mst be integes

27 7 Batad s Theoem Again Again, conside a powe aw foce aw: F() k n / n / g '( ) nk 3 3 3n n g ( ) k / /? p q n, Check: Both give ationa sotions and they wi give cosed obits! Batad basicay epeated a simia anaysis by incding highe ode petbation tems to show that fo a obits (not necessay sma deviations fom a cica obit) to be cosed, n mst be - o -. F () k F() k