coordinate system 1

7/23/2019 Coordinate System 1

1/11

r

r0

0

x

MATEMATICAL PHYSICSDra. Suparmi, M.A., Ph.D

COORDINATE SYSTEM

POLAR COORDINATE

Polar coordinat ! o"#ct can mo$ on a lin, a plan, or %pac.Polar coordinat i% u%d to d%cri" th motion o& an o"#ctmo$in' on a plan. Exampl uni&orm circular motion (ucm)

0 unit $ctor in dirction

r ($ctor) i% po%ition $ctor o& th o"#ct.r0ia unit $ctor o& rdirction

r* ix +jy * ir co% +jr %in

r0* ico% +j%in

0* i%in +jco%

r * -r - * ma'nitud o& $ctor r-r0- * -0- *

r0x* -r0- co% * co%

r0y* -r0- %in * %in

0x* -0- %in * %in

0y* -0- co% * co%

I& / ha$ a particl /hich i% po%ition i% at r&rom ori'in,s* r* ror

v*d sdt

*ddt

(ror) * rd r0

d t+ r0

drdt

d r0

dt*

d

dt(i co% +j%in ) * i(%in ) +j(co% ) * (i%in +j

co% ) * 0

dr

dt* r

v* r

0 + r0 r

* 0 r + r0 r * 0 r+ r0 $r

v* 0 $+ r0 $r

a*d v

d t*

d

dt(0r ) +

d

dt(r0 r )

*r d

0

dt+ 0

dr

dt + 0 r

d dt

+ r dr

0

dt+ r0

drdt

d0dt* ddt (

ico% +j%in ) * i(%in ) j(co% ) * ( i%in +j

co% ) * r0

a* r ( r0) + 0 r + 0 r + r 0+ r0 r a * r0( r r 2 ) + 0( r +1 r ) * r0ar+ 0a

1203211405%t.l'iyo

0y0

r0y r0

0x r0x


2/11

6or 7ni&orm Circular Motion (7CM),r * 0r * 0

a * r0( r 2 ) * r0r 2 * r0r 1* acp

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3/11

r0

0r

r %in co%

8

0

x

y

r0(xy)

r0

r0(8)

r0(8)

0

y

x 0y * co%

0x

SPHERICAL COORDINATESphrical Coordinat i% th ad$anc o& polar coordinat

r = r0 rr* i r %inco%+jr %in%in+ kr co% *r(i %in

co%+j %in%in+ k co% )

r0* i %inco%+j%in%in+ kco%

0* i co%co%+jco%%in k%in

0* i%in +jco%

Th dirction o& 0(xy)* r0(xy)0(8)* -0- %in * %in

0(xy)* -0- co% * co%

0(y)* -0- co%%in* co% %in

0(x)* -0- co% co% * co% co%

0y* - 0|co% * co%

0x* - 0|%in * %in

ProblemI& a particl mo$ on %phrical o"#ct and th in%tantanou% po%ition o& particl i% at $ctorr, dtrmin th $locity and acclration o& th particl.

Solution

r0* i %inco%+j%in%in+ kco% s* ror

0* i co%co%+jco%%in k%in d

0

dt * i %in co% ico%%in j %in %in+jco%

co% kco% * (i %inco%+j %in%in+ kco%) + (jco%co% i

co%%in)

* r0+ co%(j co% i%in )

* r0+ co% 0 0* i%in +jco%

ddt 0* i co% j %in

* i co%(%in1+co%1) j %in(%in1+co%1) + k %inco%

k %inco%

* ( i co%%in1j %in%in1 k %inco%) + ( i co%co%1j

%inco%1+ k %inco%)

* %in( i co%%in+j %in%in+ kco%) co%(i co%co%

+j %inco% k %in)

* %inr0 co% 0* (%inr0 + co% 0)

v*d s

dt*

d

dt(ror) * r

d r0

dt+ r0

dr

dt

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d r0

dt *

d

dt(i

%inco%+j%in%in+ kco% )

* i:(co% ) co% + %in (%in) ; +j:(co% ) %in+

%in(co%) ; k %in )

* i co% co% i %in %in+j co% %in +j

%inco% k %in

* (ico% co% +jco% %in k%in ) + %in( i%in +j

co%)

* 0+ %in 0dr

dt * r

v* r ( 0 + %in 0) + r0 r

* 0 r + 0r %in + r0 r

* v+ v+vr

v* v+ v+vr

a*d v

dt*

d

dt(r0 r + 0 r + 0r %in ) *

d

dt(r0 r )+

d

dt

(0 r ) +d

dt( 0r %in )

()d

dt(r0 r ) * r0

d

dt r + r

d

dtr0 * r0 r + r ( 0+ %in

0) * r0 r + r 0+ r %in 0

(1)d

dt(0 r ) * 0r

d

dt + 0

d

dtr + r

d

dt 0* 0r + 0

r + r ( r0+ co% 0)* 0r + 0 r r 2 r0+ r co% 0* 0(r + r ) r 2 r0+ r co% 0

(9) ddt (

0r %in ) * 0r %in ddt +0r ddt sin +

0%in ddt

r + r %in d

dt 0

* 0r %in + 0r co% + 0%in r + r %in ( %inr0 co% 0)

* 0r %in + 0r co% + 0%in r r %in1

2 r0 r %in

2 co% 0

* 0(r %in + r co% + %in r ) r %in1 2 r0 r %inco%

2 0

a * () + (1) + (9)a * r0 r + r 0+ r %in 0+ 0(r + r ) r 2 r0+

r co% 0+ 0(r %in + r co% + %in r ) r %in1 2 r0 r %inco%

2 0

* r0 r r %in1 2 r0 r 2 r0+ r 0+ 0(r + r ) r %inco%

2 0 + r %in 0+ r co% 0+ 0(r %in + r co% +

%in r )

1203211405%t.l'iyo


5/11

r0

0r

r %in co%

80

x

y

* ( r r 2 r %in1 2 ) r0 + (r + 1 r r %inco%

2 ) 0+ (r %in

+ 1 %in r +1 r co% ) 0

a * arr0+ a 0+ a 0

Application of spherical coordinat on Laplacian

x * r %in co% y * r %in %in

8 * r co%

r * x2+y2+z2 dr *1

2(2x dx+2y dy+2z dz )

x2+y2+z2

tan *y

x %c1d*

x dyy dx

x2

*dy

x

y dx

x2

co% *z

r

z

x

2

+y

2

+z

2 %in d

* dz

r

z1

2(2x dx+2ydy+2z dz)

r3

*r

2

dzz (x dx+y dy+zdz )

r3

dr *

1

2(2x dx+2y dy+2z dz )

x2+y2+z2

dr

dx*

1

2 (2x+0+0)

x2+y2+z2

* xr

* %in co%

dr

dy*

1

2(0+2y dy+0)

x2+y2+z2

*y

r* %in %in

dr

dz*

1

2( 0+0+2z dz )

x2+y2+z2 *

z

r* co%

d

dx=

y cos2 x

2*

r sin cos

r sin sin cos2

*sin r sin

d

dy=

cos2

r sin cos *

cos

r sin

d

dx=

zx

r

3

sin

*r2cossin cos

r

3

sin

*coscos

r

d

dy=

zy

r3

sin*

r2

cos sin sin

r3

sin*

cos sin

r

d

dz=

r2z2

r3 sin*

r2(rcos)2

r3 sin*

r2(1cos2)

r3 sin*

sinr

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7/11

sin cos

2

r sin

r+

cos cos2

r2

sin

cos sin

r2

sin2

+

cos2

r2

sin2

( )2

(

z )

2

* (co%

r

sin

r

) (co%

r

sin

r

)

* co%1 ( r )2

+ cos sin

r2

+

cos2

r

r+

sincos

r2

+

sin2

r2

( )2

2

x2

+

2

y2

+

2

z2

*

( %in1

co%1

+

%in1%in1+co%1)

2

r2

+ (cos

2

cos2

r2

+cos

2

sin2

r2

+sin

2

r2

)

2

2

+ (

sin2

r2

sin2

+

cos2

r2

sin2

)

2

2

+ (cos

2

cos2

r+

cos2

sin2

r+

sin2

r

+sin

2

r+

cos2

r)

r+ (

sin cos cos2 r

2+

sin cos cos2 r

2+

sin cossin 2r2

+sin cossin 2

r2

+sin

2

cos

r2

sin+

cos2

cos

r2

sin+

sincos

r2 +

sin cos

r2 )

*( %in1+ co%1)(co%1+%in1)

2

r2

+(sin2+cos2)(cos2 +sin2 )

r2

2

2

+

sin2

+cos2 r

2sin

2

2

2

+ ( (sin2+cos2)(cos2 +sin2 )

r +

sin2

+sin2 r

r+

(1sincos cos

2

r2

1sin cos sin

2

r2

+

sin

(2+cos2)cos

r2

sin

+ 1

sincosr

2 )

*

2

r2

+1

r2

2

2

+1

r2

sin2

2

2

+1

r

r+ (1

sincos cos2

r2

1sin cos sin

2

r2

+

sin

(2 +cos2)cos

r2

sin

+ 1sincos

r2 )

*

2

r2

+1

r2

2

2

+1

r2

sin2

2

2

+ (1

r+

1

r

r (1

sincos (cos2 +sin2 )

r2

sin

(2 +cos2)cos

r2

sin

1sincos

r2 )

*

2

r2

+1

r2

2

2

+1

r2

sin2

2

2

+2

r

r+

cot

r2

1203211405%t.l'iyo >


8/11

*

2

r2

+2

r

r+

1

r2

2

2

+cot

r2

+

1

r2

sin2

2

2

?hr

1

r2

r

r2

r

) *

2

r2

+2

r

r

1

r2

sin

sin

) *1

r2

2

2

+cot

r2

2

x2

+

2

y2

+

2

z2

*1

r2

r

r2

r

) +1

r2

sin

sin

) +1

r2

sin2

2

2

Laplacian * 2 * *

(i x+j

y+k

z ) (i

x+j

y+k

z ) =

2

x2+

2

y2+

2

z2

2 *

1

r

2

r

r

2

r

) +1

r

2

sin

sin

) +1

r

2

sin2

2

2

Application of spherical coordinat on Laplacian dan Angular Momentum

Elctron i% rotatin' on %phrical path or lliptical path. Th an'ular momntumo& lctron i% xpr%%d a%

L* rx p

?hr ri% po%ition o& lctron &rom ori'in

pi% th linar momntum

I& pi% d@nd in dirntial oprator,

p* iB * iB(i

x+ j

y+ k

z)

thn L* rx p

*

i j x y i

x i

y

| i j k

x y z

i

x i

y i

z|

1203211405%t.l'iyo 3


9/11

* i(iBy

z+ iB8

y) + j(iB8

x+ iBx

z) + k(iBx

y+

iBy

x)=

iLx+ jLy+ kL8

Lx* (iBy

z

+ iB8

y

)

* iB:r %in %in (co%

r

sin

r

) r co% ( %in %in

r+

cossin

r

+cos r sin

);

= iB(r %in %in co%

r r sin sin

sin

r

r co% %in %in

r r cos

cossin

r

r cos

cos

r sin

)

= iB: sin2

+ cos2 ) sin )

+ cotcos

;

= iB( sin

+ cot cos

)

Ly* (iB8

x+ iBx

z)

* iB:r co% (%in co%

r

coscos

r

sin r sin

) r %in co%

(co%

r

sin

r

);

* iB:rco%%inco%

r+ cos co%1

sin cot

r %in

co% co% r+ %in1co%

;

* jiB:( cos co%1

+ cos %in1

) sin cot

* iB:(co%1+%in1) cos

sin cot

;

* iB( cos

sin cot

)

L8* (iBx

y+ iBy

x)

* iB: r %in co% (%in %in r

cossinr

cos r sin

) +

r %in %in (%in co%

r+

coscos

r

sin r sin

)

* iB:r%in co% %in %in

r

cossin

rr %in co%

cos r sin

r sin cos

+ r %in %in %in co%

r+

coscos

rr %in

%in

sin r sin

r sin sin

;

* iB: %inco%%in co%

+cos2

%inco%%in

co%

+sin

2

;

* iB

1203211405%t.l'iyo 2


10/11

L1* L L* (Lx)1+ (Ly)1+ (L8)1

(Lx)1

* iB( sin

+ cot cos

) iB( sin

+ cotcos

)= B1( sin

2

2

2

c%c1 sin cos

+ cotco%1

cot

2

cossin

+

cot2

cos2

2

2

)

= B1: sin2

2

2

+ cot2

cos2

2

2

(c%c1+ cot1) sin cos

+ cotco%1

;

(Ly)1* (iB( cos

sin cot

) (iB( cos

sin cot

)

= B1( cos

sin cot

) ( cos

sin cot

)

= B1: cos2

2

2

+ c%c1 sin cos

+ cot%in1

+ cot

2

cos sin

+

cot2sin

2

2

2

;

= B1: cos2

2

2

+ cot2sin

2

2

2

+ ( cot2 + c%c1) sin cos

+ cot

%in1

;

(L8)1* (iB

) (iB

) * B1

2

2

L1* L L* (Lx)1+ (Ly)1+ (L8)1= B1: sin

2

2

2

+ cot2

cos2

2

2


+ cotco%1

;

B1: cos2

2

2

+ cot2

sin2

2

2


+ cot

%in1

;

B1

2

2

* B1: sin2

2

2

+ cot2

cos2

2

2


+ cotco%1

1203211405%t.l'iyo 0


11/11

+ cos2

2

2

+ cot2

sin2

2

2


+ cot%in1

+

2

2

* B1:( sin2

2

2

+ cos2

2

2

)+( cot2

cos2

2

2

+ cot2

sin2

2

2

)

+ (cotco%1

+ cot%in1

) +

2

2

;

* B1:

2

2

+ cot2

2

2

+ cot

+

2

2

;

* B1:(

2

2

+cos

sin

) + ( cot

2 +1

2

2

;

* B1( 1

sin

(%in

+ csc2

2

2

)

mm"r!1

f(x )

x&(x)

x*

2

x2

+1

f(x )

f(x)

L1* L L* (Lx)1+ (Ly)1+ (L8)1 * B1( 1

sin

(%in

)+ csc

2

2

2

)

And 2 *

r

1r

2

r

) +1

r

2:

sin

1

sin

) +1

sin

2

2

2

;

So

2 *

r

1

r2

r

) L

2

2

r2

1203211405%t.l'iyo

coordinate system 1

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