Mechanics Physics 151
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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
2 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 Enegy Consevation Conside time deivative of Lagangian dl( q, q, t) L dq L dq L = + + dt q dt q dt t Using Lagange s equation one can deive d L L q L + = 0 dt q t Conseved if Lagangian does not depend expicity on t L d L = q dt q Define this as enegy function hqqt (,, )
3 Enegy Function? Does enegy function epesent the tota enegy? Let s ty an easy exampe fist Singe patice moving aong x axis mx L = V( x) h= mx L mx = + V( x) = T + V How genea is this? L hqqt (,, ) q L q Tota enegy Enegy Function Suppose L can be witten as L( qqt,, ) = L( qt, ) + L( qqt,, ) + L( qqt,, ) Tue in most cases of inteest Deivatives satisfy L0 L = 0 q q q 0 = L st ode in q q L q nd ode in q = L L hqqt (,, ) q L= L L q L hqqt (,, ) q L q 0 Eue s theoem 3
4 Enegy Function hqqt (,, ) = L L L = T V 0 Enegy function equas to the tota enegy T + V if T L and = V = L0 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 di = dt i q q 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 This woudn t wok if nd ode homogeneous di = dt = ( q,..., q, t) i i n i q q i + t because No q 4
5 Enegy Consevation L hqqt (,, ) q L q 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 Centa Foce Pobem Conside a patice unde a centa foce Foce F paae to Assume F is consevative 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 F = V() These exampes assume the body at the cente is heavy and does not move O F m 5
6 Two-Body Pobem Conside two patices without extena foce and eative to cente of mass Lagangian is ( m+ m) mi i L= R + V() Motion of CoM i= Motion of patices aound CoM m O R CoM m Potentia is function of = Stong aw of action and eaction m m ( ) = = ( m+ m) m+ m ( ) mi i mm = i= m+ m Two-Body Centa Foce L ( m + m ) R mm ( m + m ) = + V() R is cycic CoM moves at a constant veocity Move O to CoM and foget about it L mm ( m + m ) = V() m O R CoM m Reative motion of two patices is identica to the motion of one patice in a centa-foce potentia Reduced mass mm µ = ( m ) o µ = + + m m m 6
7 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 7
8 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 dt ψ ψ ( sin cos ) 0 = m ψ ψ ψθ = We can choose the poa axis so that the initia condition is ψ = π, ψ = 0 nd tem vanishes ψ = 0 Now ψ is constant. We can foget about it 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 8
9 Radia Motion m L= T V = ( + θ ) V( ) Lagange s equation fo Deivative of V is the foce m = m θ + f () Centifuga foce Using the angua momentum m = + f () 3 m d V() ( m ) m θ + = 0 dt Centa foce V() f() = = We know how to integate this. But we aso know what we get by integating this m θ 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 = 0 = t() 0 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! (?) 9
10 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 = 0 and θ 0 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 E = + V () m Pot V () and see how it intesects with E = E V () > 0 E > V () 0
11 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 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 Tuning point E = V = 0 Go towad = E 3 A / foce woud make a hypeboa
12 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 = 0 (fixed) Ony one adius is aowed Stays on a cice E = V ( ) = 0 V () Cassification into unbounded, bounded and cicua motion depends on the genea shape of V Not on the detais (/ o othewise) 0 = const = 0 E E E 3 0
13 Anothe Exampe V a = f = 3 4 Attactive -4 foce V has a bump 3a 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 = E V = Top of a bump woks in theoy, but it is unstabe Initia condition must be exacty 0 = const = 0 and = Stabe cicua obit equies dv m = = 0 d 0 > d dv 0 E E 0 stabe unstabe 3
14 Powe Law Foce V () V() + m dv d = 0 = f( ) = 0 df d 0 3 m0 = 0 Suppose the foce has a fom k > 0 fo attactive foce Condition fo stabe cicua obit is n n kn < 3k n > f ( 0 ) < 0 dv df 3 = + > 0 d d m 4 = = f = k Powe-aw foces with n > 3 can make stabe cicua obit n 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? 4
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