central force motion · !8! #o kepler's law!1! planets move in elliptical orbits about the sun...
TRANSCRIPT
Central Force Motion
Chapter 4
!1!
Central Force Motions
IO Reminder : Polar coordinates
. Equations of motion in some systems aren't separable in
Cartesian coordinates,
but can be # me simple and solvable in a
different choice of coordinates.
ryEo
'
| Polar coordinates ( in ZD )# EI cylindrical ( in 3D )
,5 '=reI= xetxtyej
To Recall (
darted) : n2=x 't
,'
)×
O= Arctaa
%-) endjetjieo' anddaoE= - ei
'± Elisa ⇒fdjfoitoeg
Spherical coordinates : more complicated but same logic .
Zn +
IE, F=sE '
\,e¢10Leg X= nsino cost1 y = n Sind Sino
•,
,
>Y Z
= ncoso cost
→1
¢ \ ! ei'
C ¥ +× Tinto
, .no e± '
+
=|5
'
' nicotine . rjzeat
a not Zio . ¢ }in¢ws•
niisinot Ziosiao + Znoiouoso
!2!
IIO Central Motion and potential
I particle : ¥75 '
In spherical coordinates' '
5'
a he' F= FGIE'
central Ei=Ipy
OEZ To force Hill1 -7 ×
g--
• Central forces are conservative : U= - f 'Fk'Ids '
.For central forces the motion takesplace on a plane :
I - Kei
'
→ Torque F= 5 'xF'
: 5 'since 5 '×ej=o '
=) 4¥ , o with F= 5 '×mF= & constant of motion+
I I'
E plane 1 to % =) motion is effectively ZD !
Moreover : F's (ngxmfnioo = mice E'
=) 726 = cst = § → if n small,
!6! largeConstant of motion
(angular momentum )
Equations of motion : mat . .m|5'
' 902
ziot.no. =/
- od%,
=) Zio + not =L dtp)= o @ we recover io
'
. est
Let Us get rid of o'
usingO' = £
MNZ
!3!
⇒ mti . no' ) = moi - nmmk÷
,
= - ¥⇐ min
'
= . day,
+ mb÷,
= - dodge
with Ueq ( n ) = UG ) tzbm÷ Effective potential
we've mapped the problem onto an effective one dimensional
system :
msi = -
d.Y÷A =) formally solvable !
This equation can be integrated once ( conservation of energy) :
tmn 't Uepp ( s ) = E= cst Id effectivemotion !
=) integrate
#O Motion in central potentials and Kepler 's Problem
So the first step is now to plot Uefp ( n ) :
E± : FG ) : - nk.( gravitation - Cotton ,
Coulomb Ke 9¥)2
UGK -
Kg =\ Ueq , - Kg t ¥mn
Note : E } Ueq for motion to take place
!4!
- Hui Ueed "
Kaska -
Fi€... ":YtIe⇒.
.=±1 Mr mk
!
bad . -- -
!• E70 : Unbounded motion
:Ueq ( nc ) TE ( o : bounded motion
. Uepplnd = E : circular motion : 5=0
Shape of the orbits : n ( tl,
OIH =) nkel trajectory( forget the dependence)
.
*i=¥od¥ . He ⇐ ⇒
t.FM#noGhf+lo1m__.n4mlE- U ) - lihir*
in'
= It ( gb•mln÷ ) =0
'
¥o(mP÷fFo) Formal solution an
any UGI
. fat ⇐ that # × TIED=mE÷¥÷iEnt¥5
so mini . ueeet± dado.
. ÷ftp.T.n-mgn.IFHwhere F=
- olddt
!5!
this equation is a bit nasty : non linear and second order !
To solve it,
we use the following trick :
U -
. lg Change of variable
¥o=¥Its = - theIfo :# '
adios = . EE.
+ ¥ GET
⇒ Etna . ÷ # l'
. n= . tho.tt#Eye2xYayii'
= - Il 's Eat ' 1 =E*Mkl⇒ dotohtu = - pnfqF( %) "
Harmonic oscillator"
with driving force-
EE.Flkl
Kepler problem : F=- khz = - Ku
'
⇒ dlyz,
+ u = my easy !
¢general solution ob homogeneous
⇒ uco ) = my + A cost . oo )equation
µ particular solution
f eccentricity
!6!
Let A=. mpgke ⇒ n(o1=GYmKl
- E cost - I. )
.
E is determined by the system 's energy and angular momentum:
E .
. Emnitueeohl =Emm?÷l¥ol'
. ¥tfn÷= . . .=mztg÷(e
'. i ) ⇒ e=TtE"
' ⇒ " e "
m KiEto : ET ,
E=o : E= 1
E- Em; :{ =o
Orbits :For the Kepler problem
,
the orbits are conic sections
( = hyperbola ,parabola
,ellipse ,
circle )
Ref : Given a point ( focus) and a line ( dinectnix ) :EIIT
PE conic section ⇐ ) d ( P,
focus ) = E d( P,
direct
.in/dinafnix#tP,
d( P, focus )= n
=d( P
,direct.ir/=dtncoso
d F( focus ) n= Ed + En cos 0(⇒ n= Edl- ECOSO
E== : Circle : d→ A,
E→ o,
Ed : R fixed
E=Emin= - M¥2 ,R= QYMK : nco1=R
circle !
!7!
EI : parabola ( E=o ) : d. =n( t use ) = n - ×
⇒ ( dtxl '=r '= city '
⇒
= × 't Zxdt de×= ¥ (y ?dY
anyNY c= fate
#.
=Ea
,ix
: k¥0Iii ×.
( : tic- minor
EIo.li Ellipse / hyperbolaa
axis-
( or { < 11 Ey ,major axis
Eto E > 0
Ed =n - n{ use = s - ex =) e%Hxt=i=x' tyi= Eydhtx't2dx|
⇒ × '( 1- e 4 - zdaixty' =eat
⇒ Keefe '÷r+t÷=i÷⇒ ( x .
ei÷t+y÷=e¥ot+÷t.
#-
Key"
ellipse14
- e'
oeeti: K÷+f÷= I E. × - Feet (closed)
at i÷ , ,ei=¥ :( e- . Feet~ hyperbolaE > l : ¥ -Ye÷= I
, t.EE (open
!8!
#O Kepler 's law
!1! Planets move in elliptical orbits about the Sun where the sun
atone focus ( Moss mpknef
,
: See next chapter )
!2! The area per unit time swept out by a radius vector fromthe son to a planet is constant :
to = est : DA :{ n' do so dff.
= est
nqI2*¥da
!3! The square of a planet 's period is proportional to the cube
of themajor axe ) of the planet 's orbit
.
Pnoob : daft =z£m =) f. It-
.T = 2¥ A over a period
A. tab for an ellipse
=I ,[d÷,g, ,
with ed= lilmk
= IT a a × # ⇒ TEEpm.la?4jIyt2a4(1- EY0
and {D= all - EY , lifnk =\ T '=4T,÷ma3
.a
>
!9!
Summary ( and different derivation of theeq . of )
the trajectory
!1! mini . no' )= - KG ' } mini = . dueeyd
,mrio = l
.
Uepp = - Ent IZmsh
!2! E = { most + Uepp ( r )
⇒ o¥t=±FmeF±FEm2n2
and o¥=odIo=f÷.
'Io
!3! This yields : f.to/mn# = ± @ - Oo )
- TFnEt2m±s - thin constant
!4! set
u=4n⇒ du
= - Ynzdn
The integral become $ : fbdu_ -
= 0 - do
5215 TEETHCan be computed : new - I
= - cost - Oo )Ftc
m-k~
@
Let e= them :
⇒ mf÷ f- = I - E cos Co - Oo )
# s= mkleil- E cos (O - oo )