Mechanics Physics 151

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Milton Powell
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1 Mechanics Physics 5 Lectue 5 Centa Foce Pobem (Chapte 3) What We Did Last Time Intoduced Hamiton s Pincipe Action intega is stationay fo the actua path Deived Lagange s Equations Used cacuus of vaiation Discussed consevation aws Geneaized (conugate) momentum Symmety Invaiance Momentum consevation We ae amost done with the basic concepts One moe thing to cove Goas fo Today Enegy consevation Define enegy function Subte diffeence fom the Newtonian vesion Centa foce pobem Fist appication Motion of a patice unde a centa foce Simpify the pobem using angua momentum consevation Discuss quaitative behavio of the soution Use enegy consevation Distinguish bounded/unbounded obits Actua soution Thusday

2 Enegy Consevation Conside time deivative of Lagangian dl( q, q, t) L dq L dq L = + + dt q dt dt t L d L Using Lagange s equation = one can deive q dt q d L L q L + = dt q t Define this as enegy function hqqt (,, ) Conseved if Lagangian does not depend expicity on t Enegy Function? L hqqt (,, ) q L Does enegy function epesent the tota enegy? Let s ty an easy exampe fist Singe patice moving aong x axis V( x) mx L= h= mx L mx = + V( x) = T + V How genea is this? Tota enegy Enegy Function Suppose L can be witten as L( q, q, t) = L ( q, t) + L ( q, q, t) + L ( q, q, t) Tue in most cases of inteest Deivatives satisfy L L = q = L st ode in q L q = L L hqqt (,, ) q L= L L L hqqt (,, ) q L nd ode in q Eue s theoem

3 Enegy Function hqqt (,, ) = L L L= T V Enegy function equas to the tota enegy T + V if T L and = V = L st condition is satisfied if tansfomation fom i to q is time-independent nd condition hods if the potentia is veocity-independent No fictions Fiction woud dissipate enegy Let s ook into the st condition Kinetic Enegy mi T = i i Using the chain ue = ( q,..., q ) i i n This woudn t wok if = ( q,..., q, t) because di = dt di = dt i i n i q q i q q i + t Time-independent m m m = qq = qq i i i i i i i i k k i i k, q qk k, i q qk nd ode homogeneous No q Enegy Consevation L hqqt (,, ) q L Enegy function equas to the tota enegy if Constaints ae time-independent Kinetic enegy T is nd ode homogeneous function of the veocities Potentia V is veocity-independent Enegy function is conseved if Lagangian does not depend expicity on time These ae estatement of the enegy consevation theoem in a moe genea famewok Conditions ae ceay defined 3

4 Centa Foce Pobem Conside a patice unde a centa foce Foce F paae to Assume F is consevative F = V() V is function of if F is centa Such systems ae quite common Panet aound the Sun Sateite aound the Eath Eecton aound a nuceus These exampes assume the body at the cente is heavy and does not move O F m Two-Body Pobem Conside two patices without extena foce and eative to cente of mass m Lagangian is ( m+ m) mi i L = R + R V() O Motion of CoM m i= Motion of patices aound CoM m ( ) = = ( m+ m) m+ m CoM m Potentia is function of = Stong aw of action and eaction mi i mm = ( m + m ) i= Two-Body Centa Foce ( m+ m) R mm L = + V() ( m+ m) R is cycic CoM moves at a constant veocity Move O to CoM and foget about it mm L = V() ( m + m ) m O R CoM m Reative motion of two patices is identica to the motion of one patice in a centa-foce potentia mm Reduced mass µ = o ( m m ) µ = m + m + 4

5 Hydogen and Positonium Positonium is a bound state of a positon and an eecton Simia to hydogen except m(p) >> m(e + ) Potentia V() is identica Tun them into centa foce pobem µ = mm e e me positonium ( me + me) = µ = mm p e hydogen me ( m + m ) p Spectum of positonium identica to hydogen with m e m e / e e e + e p q V() = Spheica Symmety Centa-foce system is spheicay symmetic It can be otated aound any axis though the oigin Lagangian L= T( ) V( ) doesn t depend on the diection Angua momentum is conseved L= p=const Diection of L is fixed L by definition is aways in a pane Choose poa coodinates Poa axis = diection of L = (, θ, ψ) = (, θ) Azimuth Zenith = /π L O Moe Fomay Lagangian in poa coodinates = (, θ, ψ ) m L = T V = ( + sin ψθ + ψ ) V( ) θ is cycic, but ψ is not d L L m ( ψ sinψ cos ψθ ) dt ψ ψ = = We can choose the poa axis so that the initia condition is ψ = π, ψ = nd tem vanishes ψ = Now ψ is constant. We can foget about it 5

6 Angua Momentum m L = T V = + V θ is cycic. Conugate momentum p θ conseves L pθ = = m θ = const θ Atenativey Aea veocity ( θ ) ( ) Kepe s nd aw da dt = θ = Tue fo any centa foce const Magnitude of angua momentum d da Radia Motion m L = T V = ( + θ ) V( ) d V() Lagange s equation fo ( m ) m θ + = dt Deivative of V is the foce m = m θ + f () V() f() = Centifuga foce Centa foce Using the angua momentum = m θ m = + f () We know how to integate this. 3 m But we aso know what we get by integating this Enegy Consevation E = T + V = m + + V = m + + V = m ( θ ) () () const = E V() m m One can sove this (in pincipe) by t d t = dt = = t() E V() m m Then invet t() (t) Then cacuate θ(t) by integating θ = m st ode diffeentia equation of NB: This neve goes negative Done! (?) 6

7 Degees of Feedom A patice has 3 degees of feedom Eqn of motion is nd ode diffeentia 6 constants Each consevation aw educes one diffeentiation By saying time-deivative equas zeo We used L and E 4 conseved quantities Left with constants of integation = and θ We don t have to use consevation aws It s ust easie than soving a of Lagange s equations Quaitative Behavio Integating the adia motion = isn t aways easy E V() m m Moe often impossibe You can sti te genea behavio by ooking at V Quasi potentia incuding () V() + m the centifuga foce Enegy E is conseved, and E V must be positive m m E = + V () = E V () > E > V () Pot V () and see how it intesects with E Invese-Squae Foce Conside an attactive / foce k k f() = V() = Gavity o eectostatic foce k V () = + m / foce dominates at age Centifuga foce dominates at sma A dip foms in the midde V () m k 7

8 Unbounded Motion Take V simia to / case Ony genea featues ae eevant E = E > min E = V ( min) Patice can go infinitey fa E V () m Aive fom = E E 3 Tuning point E = V = Go towad = A / foce woud make a hypeboa Bounded Motion E = E min < < max Patice is confined between two cices Goes back and foth between two adii E E V () m E 3 Obit may o may not be cosed. (This one isn t) A / foce woud make an eipse Cicua Motion E = E 3 = (fixed) Ony one adius is aowed Stays on a cice E = V ( ) = = const = V () Cassification into unbounded, bounded and cicua motion depends on the genea shape of V Not on the detais (/ o othewise) E E E 3 8

9 Anothe Exampe V a 3a = f = 3 4 Attactive -4 foce V has a bump Patice with enegy E may be eithe bounded o unbounded, depending on the initia a V = + m 3 E V m V Stabe Cicua Obit Cicua obit occus at the bottom of a dip of V m dv = E V = m = = d = const Top of a bump woks in theoy, but it is unstabe Initia condition must be exacty = and = Stabe cicua obit equies dv > d E E stabe unstabe Powe Law Foce V () V() + m dv d = = f( ) = df d 3 m = Suppose the foce has a fom k > fo attactive foce Condition fo stabe cicua obit is n n kn < 3k n > 3 3 f ( ) < dv df 3 = + > d d m 4 = = f = k Powe-aw foces with n > 3 can make stabe cicua obit n 9

10 Summay Stated discussing Centa Foce Pobems Reduced -body pobem into centa foce pobem Pobem is educed to one equation m = + f () 3 Used angua momentum consevation m Quaitative behavio depends on V () V() + m Unbounded, bounded, and cicua obits Condition fo stabe cicua obits Next step: Can we actuay sove fo the obit?

Mechanics Physics 151

Mechanics Physics 151 Mechanics Physics 5 Lectue 5 Centa Foce Pobem (Chapte 3) What We Did Last Time Intoduced Hamiton s Pincipe Action intega is stationay fo the actua path Deived Lagange s Equations Used cacuus of vaiation