se z l 2 fi l - Mark Wilde...l arc Cartesian coordinates se y z Z n a.jo dy a'z l Lds...

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