se z l 2 fi l - Mark Wilde...l arc Cartesian coordinates se y z Z n a.jo dy a'z l Lds dn2tdy2td2 fi...
Transcript of se z l 2 fi l - Mark Wilde...l arc Cartesian coordinates se y z Z n a.jo dy a'z l Lds dn2tdy2td2 fi...
Cartesian coordinates se y z
dyZ n a.joa'z
l l Lds dn2tdy2td22
fil l
I i t i s
Sn t dattdy2
Curvilinear coordinates cylindricalcoordinates Cr OrZ
first let's restrict ourselves to a y plane
Yordo ds
ds dr24 doPN
dS2 dv2 m2do2
arc length9hthis particularenample it wasverystraightforward to findds Ourgoal is todevelop aframework tofinddsfor anycoordinatesystemF cylindrical coordinatesystem one can write thefollowingrelations
n V cos0
Y r Sino2 Z
Furthermore an dr coso N Sinodo
dy dr sin0 t r CasOdOdz dZ
now ds2 dx2tdy2 dz2 drCoso rsino do drsino triosodddZ2
d P t n'd02 tdz 2
note nocrossterms survives in thecylindrical coordinatesystemwhen such a thinghappens we call thecoordinate systemorthogonalLet'sanalyze the coordinate surfaces
1 2 3Vs const Z Const O constant
2 tn r n n
t.fi If I il lis 3
i Y To i Ye I i
L n Ln
thesesurface are pendicularto eachother ateverypointtheyintersectrediom
now you can also talk about the intersectionof twosurfaces at a time n
ez
n
er
wewill call anycoordinate system curvilinear whenthecoordinatesurfaces are not planes orsomeofthem and the
coordinate lines are curves rather than straightline
Scalefactors and Basis vectorsIn rectangularsystem n ne du Y I constant
thenparticle moves a distancedm
however in a cylindrical coordinate systemif 0 changesby the amount do themCds ry
do
Scalefactorhowtofigureout scalefactorsIf the transformation is orthogonal as it was the casewith cylindrical coordinates then one can find thescalefactorsfrom ds2
ds d r t r'd 02t d z 2
Scalefactors 1 V l
Let's write a vector in cylindrical coordinate system
d5 Er d r t Io ndO t Ezd'td5 d5 d5 Ep TeoFez are orthonormalto each
other
wenowdiscuss an important distinction b w rectangularcoordinate system and curvilinear coordinatesystem
T j I are constant in magnitude and direction butthat's not the case with einIo
r feo
gin iz f fy
ie
ErL
these basis vectors alsochangeif yougofrom one pointtoanother
so thereshould be awelldefinedmethodtoconnectthese two coordinatesystems
ds e da i t dy I t dz kcosO d r nsins0do i t ChinOd v t n Coo do J dzkiiconof JimOd r t f In sin0 t j nano I t deka
ein i coso t j sirs0
nee Eirsince tiranowhite'umm
it is easyto see howtheychangedirectionsonevery point
Let'sseewhathappens whenwe take time derivativeofdisplacementincylindricalcoordinates
J r I n t 2 Ez
II offer t n dain t datiz
i In t n dinTt
t E iz
daff inCs'mOl ddi t J ar odddta nO eo
15 i Er t r ieIo t E Iz
Let'sjustfocus on themotion in a plane
if o noto
Jatinin
what if we differentiate twotimes
I I in Ent niEr t niOIo t n I Io t n ieIo t I Izri Ent ni oe'o i oEo no neo not Ep I Ez
a fi n02 int zirietnii To Inez1
even if n was notchanging withtime we know there
is a radialforcefarcelanation actingon particleair war Er
ta't aftab at
moreover if r t fLt then Flo NoiEon
do Nd lor
w
Example2 Am our enample we described a coordinatesystemwith variables n V az O Nz Z Is there anyformalwayofdescribing a coordinatesystemwitharbitrarymining
D8 da i t dy I t dza
IT dm it 3dmdan I t ftp.dknk
frozeit 37,5131.1 datfaze
it If It E I da i n
2 2 2
A dn t azdnzf9zdng
hhenea zm
Define gij ai.ajds2
dsdsd5ds aiadnftaz.az daft ajajdn20T 82dm dk 120T ajdmdnztzaz.az dnzdnz
da dnzdnz oi.az ai.azfan ai.a.ai.atlaj.aTaz.azaIa3
Eii l2d5 gij daidnj that'stheenprenion you
alsosee in relativitycomponentsofthemetric tensor
if Theban's vectors areorthogonal then Ai aj Ofori I
ds2 Cdnidmdn3 h
gdaffy
d5
hidnieithzdnzezthzdnzez.fmcylindrical h I hz M hz 1
for spherical h I her hz orsino