frenet eq in physics

7/31/2019 Frenet Eq in Physics

1/21

x = x(s) s

e1, e2 e3

de1ds

= (s)e2 ,

de2ds

= (s)e1 + (s)e3 ,

de3ds

= (s)e2 ,

(s) (s)

(s) (s)e1

dx

ds= e1 ,

dxds = 1


2/21

e2 e3

e3 = [e1e2].

s

d2x

ds2= (s)e2 ,

d3x

ds3=

d(s)

dse2

2(s)e1 + (s)(s)e3 ,

,,e1

dx

ds= cos cos sin sin cos,

dy

ds= sin cos + cos sin cos,

dz

ds= sin sin.


3/21

d

ds=

sin

sin,

d

ds= sin ctg,

dds

= cos.

(s) (s)

x1, x2, x3, 1, 2, 3

x1, x2, x3 xa a = 1, 2, 3

eA, A = 1, 2, 3,

A

d

O

d = de,

ee O

e d

deA = [deA].

ds

deAds

= [d

dseA] = [, eA],

= d/ds


4/21

a) eAeB =

AB =

1 A = B0 A = B

,

b) eA

e

A

=

[] = 1 = 0 =

,

A,B... = 1, 2, 3, , , = 1, 2, 3,

,,... A,B...

deA = de

A or d

= T

dx

,

deAds

=d

dseA or

deAds

= Tdx

dseA,

T = eAe

A, = e

Ae

A,, , =

x.

TAB = eAT

e

B ,

deAds

= TA Bdx

dseB .

e(1), e(2)

e(3)

e(1)

(s) = T(1) (2)dx

ds, (s) = T(2) (3)

dx

ds.

dx

ds= e(1) and e

(1)e

(1) = 1 .


5/21

T[] = ..,

.. = 1

2eA(e

A, e

A,),

0g= ABe

Ae

b, AB =

AB = diag(1, 1, 1),

ds2 =0g dx

dx = ABeAe

bdx

dx .

=

+ T

= eA

eA,

,

=1

2

0g (

0g, +

0g,

0g,)

T = ..+

0g (

0g

..+

0g

..)

..

1, 2, 3

d = dx

,

= eAe

A, = e

Ae

A,.


6/21

=2x

xxx

x+

x

xx

xx

x.

S = 2[,] + 2[||] = 0.

d = 0, dx = 0

d2x

ds2=

dx

ds

dx

ds T

dx

ds

dx

ds.

a)deAds

= dx

dseA + T

dx

dseA, or b)

deAds

= dx

dseA T

dx

dseA,

e (1) = dx/ds,

x1, . . . , xn


7/21

ja = ja(x

k),

a , b , c . . . = 1 . . . n

i , j , k . . . = 1 . . . n

det(ja) = 0

ja

jakb,j

ibka,j = C

..fab

kf,

C..fab

C..fab = C..fba ,

C..afb C..fcd + C

..afc C

..fdb + C

..afdC

..fbc = 0.

n TnC..fab

jbeja

det(eja) = 0.

eja(xk0) =

ja(x

k0),

xk0eja(x

k0)

ejaekb,j e

jbeka,j = C

..fab e

kf.

eaieja =

ji , e

aieib =

ab

C..ijk = 2eiae

a[k,j] = e

iaC

..abce

bje

ck.

..ijk =1

2C..ijk ,


8/21

k[ij] = ..kij = T

k[ij] =

1

2C..ijk .

Sijk

ijk= ijk + (Sijk Sjki Skij),

ijk Sijk

Sijk..ijk

..ijk = ..ikj , S

..ijk = S

..ikj .

..i

jk = ..ijk

xj

xjxk

xkxi

xi, S..i

jk = S..ijk

xj

xjxk

xkxi

xi.

Tijk = ..ijk + g

im(gjs..smk + gks

..smj), K

ijk = S

..ijk + g

im(gjsS..smk + gksS

..smj)

Tijk = Tjik , Kijk = Kjik .

i.jk =2

3i[kj] +

1

3njks

s + i

.jk , Si.jk =

2

3i[kSj] +

1

3njks S

s + Si

.jk ,

i.jk = gimgks

..smj, S

i.jk = g

imgksS..smj


9/21

j = i.ji , Sj = S

i.ji ,

j =1

2jins

ins, Sj =1

2jinsS

ins,

s

.js = 0, ijs + jsi + sij = 0, Ss

.js = 0, Sijs + Sjsi + Ssij = 0.

..ijk

..ijk = 1

2eia(e

aj,k e

ak,j), i, j, i... = 0, 1, 2, 3, a, b, c... = 0, 1, 2, 3,

S..ijk xi

..ijkS..ijk

..ijk

d2 = TabkTbandx

kdxn,

S..ijk

..ijk..ijk

..ijk

..ijk S

..ijk


10/21

x = x(s).

= 0 = 0 = 0 = 0

= 0 = 0

a)de1ds

= (s)e2 , b)de2ds

= (s)e1 , c)de3ds

= 0 ,

a)d2x

ds2= (s)e2 , b)

d3x

ds3=

(s)

dse2

2(s)e1 .

dx

dt= v,

t

v =dx

dt=

dx

ds

ds

dt= e1

ds

dt.

e1

|v| =ds

dt= v.

w = d2x/dt2

w =d2x

ds2

ds

dt2

+ e1d2s

dt2

.

w = e2

ds

dt

2+ e1

d2s

dt2= e2v

2 + e1dv

dt.

m J


11/21

w = e1dv

dt,

wn = e2v2

= 0 = 0

de1ds

= 0 ,de2ds

= (s)e3 ,de3ds

= (s)e2 ,

d2x

ds2= 0 ,

d3x

ds3= 0 .

e1e2

e3 e1ds/dt = v

de1dt

= 0 , de2dt

= e3 , de3

dt= e2 ,

= v

p = mdx

dt,

= v


12/21

= 0 = 0

m

Finer = m(W + [r] + [[r]] + 2[v]) .

F1 = mW

F2 = m[r]

F3 = m[[r]]

F4 = 2m[

v]

r = 0

Finer = mW 2m[v]) .

= v = 0

Finer = mW.


13/21

W = e2v2 + e1

dv

dt.

dv/dt = 0

xi

1, 2 and 3

De(0)

kds =

1e(1)k ,

De(1)k

ds= 1e

(0)k + 2e

(2)k ,

De(2)k

ds= 2e

(1)k + 3e

(3)k ,

De(3)k

ds= 3e

(2)k .

e(0)k , e

(1)k , e

(2)k and e

(3)k D

ijk =1

2gim(gjm , k + gkm, j gjk , m)

e(a)k

d2

xi

ds2= ijk dx

j

dsdx

k

ds Tijk dx

j

dsdx

k

ds.

ijk = ijk + T

ijk = e

iae

aj,k = e

aje

ia,k.

Tijk = eiake

aj = e

ajke

ia,


14/21

keai (e

ja)

eaieja =

ji , e

aieib =

ab ,

ji

a) keaj = T

abke

bj or b) ke

ia = T

ijke

ja.

dxk/ds

Deajds

= Tabkebj

dxk

ds.

Deiads

= Tijkeja

dxk

ds.

ei (0) = dxi/ds

Deakds

= Tabjebk

dxj

ds.

De(0)k

ds= T

(0)(1)je

(1)k

dxj

ds,

De(1)k

ds= T

(1)(0)je

(0)k

dxj

ds+ T

(1)(2)je

(2)k

dxj

ds,

De(2)k

ds= T

(2)(1)je

(1)k

dxj

ds+ T

(2)(3)je

(3)k

dxj

ds,

De(3)k

ds= T

(3)(2)je

(2)k

dxj

ds.

1 = T(0)(1)j

dxj

ds, 2 = T

(1)(2)j

dxj

ds, 3 = T

(2)(3)j

dxj

ds.

Tikj


15/21

[keam] e

b[kT

a|b|m] = 0, (A)

Rabkm + 2[kTa|b|m] + 2T

ac[kT

c|b|m] = 0, (B)

[keaj] + T

i[kj]e

ai = 0, (A)

Rjm 1

2gjmR = Tjm, (B.1)

Cijkm + 2[kTi,|j|m] + 2T

is[kT

s,|j|m] = J

ijkm , (B.2)

Tjm = 2

{([iT

i|j|m] + T

is[iT

s|j|m])

1

2gjmgpn

([iTi

|p|n] + Ti

s[iTs

|p|n])}

Jijkm = 2g[k(iTj)m] 1

3T gi[mgk]j .

(A) (B)

Rik = 0

(B.1)

(A) (B)

CDiAB AB

iCD = (TCD)

PA

iPB +

iAR(T

+DC

) RB

(TAB)P

C iPD

iCR(T

+BA

) RD , (+

A s)

2ABCD + ABCD = TACBD , (+B s+.1)

CABCD CDTAB + ABTCD + (TCD)F

A TFB + (T+DC

) FBTAF

(TAB)F

C TFD (T+BA

) FDTCF [TAB , TCD] = JABCD , (+

B s+.2)

A, C... = 0, 1 , B, D... = 0, 1 .


16/21

ds2 =

1

20

r

c2dt2

1

20

r

1dr2 r2(d2 + sin2 d2),

0 = MG/c2,

u, r, x2 and x3

i00 = (0, 1, 0, 0), i11 = (1, U, 0, 0),

i01 = (0, 0,P,iP),

00i = (1, 0, 0, 0), 11i = (U, 1, 0, 0),

01i =

1

2P(0, 0, 1, i),

U = 1/2 + 0

/r, P = (2)1/2

(1 + /4), = x2

+ ix3

,0 = .

= 1/r, = = 0/r, = 0/2r,

= 0/r + 20/r2, = /4.

= 0/r3.

(ijk)(Tijk)

ds2 = (1 2MG

rc2)c2dt2 (1 +

2MG

rc2)(dx2 + dy2 + dz2) .

eai

e(0)

0 = (1 +2

c2)1/2 ,

e(1)

1 = e(2)

2 = e(3)

3 = (1 2

c2)1/2 ,

= MG/r


17/21

gik = abeaiebk , ab =

ab = diag(1 1 1 1).

2

c2


18/21

mx = eE+e

c[xH] +

2e2

3c3...x ,

2e2

3c3...x

t

x = ve1, x = ae1 + kv2e2

...x= (a 2v3)e1 + (3va + v

2)e2 + v3e3,

x = dx/dt, ds = vdt, a = v, a = d3/dt3 = d/dt.

Frad =2e2

3c3{(a 2v3)e1 + (3va + v

2)e2 + v3e3}.

=h

mc= 3, 6 1011sm.

s = J = Jv =h

2,


19/21

J

J =2

5m2,

= v

1021rad/s.

(E 108 V/sm) Fe Fg

|Fe| = eE = mihte = mv

2e =e2

r20 4, 8 102 ,

|Fg| = mG = mintg = mv

2g =mMn

r

2

0

0, 6 1042 din.

|F| =2e2

3c3int.

int = v2 1025 sm/s2

F 2, 9 104 din.

(A) (B)

[iTi|j|m] + T

is[iT

s|j|m] = 0,


20/21

(A) (B)

E =

Tjmgjm(g)1

2 dV 0.

Tijk

d2xi

ds2+ ijk

dxj

ds

dxk

ds+ Tijk

dxj

ds

dxk

ds= 0,

Rijkm

M

s = MGrr2 + r2scos

2,

rsr rs

r rs

(s) = T(2) (3)dx

ds


21/21

frenet eq in physics

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