MexLekcii2010prn

12

– , .

, .

.

) (

r), , ( r),

0. , .

,

F = G

Mm

r

;

r

- .

r 2 r

r

= F×dr = –F×(dr)×cosa = –F×dr×cos0o = –F×dr .

( )

r2

Mm

æ

Mm

Mm

ö

Mm

Mm

ç

÷

A = -òG

dr = - ç

(-G

) - (-G

) ÷

= -(U2

-U1) = G

- G

.

r2

r

r

r

r

r

è

2

1

ø

2

1

1

: ,

(r1>r2), , . ,

r1 r2 –

,

( ), .

U(r) = -G	Mm	+ C .	( j = -	U	= -	GM	- )
				m
	r					r

,

. . ,

,

, ,

= 0. U( )=0,

, ,

. =0 , ,

		U(r) = -G	Mm	.
	F(r)		r
		U(r) .

.

∞

U(B) = òF(r)dr = A (

B

= –DU

).

101

13

13.

. , ,

. .

§1. .

,

3n

, :

ìíx1, y1, z1; x2 , y2 , z2 ; ... xn , yn , zn . îx&1, y&1, z&1; x&2 , y&2 , z&2 ; ... x&n , y&n , z&n

n

3n

(1)

,

v

fα(r1, r2, … rn; v1, v2, … vn) = fα0 ,

(2)

fα0

( , fα0=mv2/2+mgx – ,

v).

( ).

. ,

, ,

( ).

( – ,

) (

).

,

,

, ,

:

1).

.

3), .

4),

, ,

( ).

, .

102

13

§2.

, ,

.

, .

, (

).

, .

n

1, 2, 3, ... n, . dt

dr1, dr2, dr3,…, drn ( . 1).

2- :

ìm

dv1

= F

+ F

+ ... + F

= F

dt

ï

1

12

13

1n

p1

ï

dv2

ïm

= F

+ F

+ ... + F

= F

ï

2

dt

21

23

2n

p2

dri ×

ï

M

M

M

M

M

(3)

í

ï

dvi

ïm

= F + F

+ ... + F

= F

dt

ï

i

i1

i2

in

pi

. 1

ïm

dvn

= F

+ F

+ ... + F

= F

ï

n

dt

n1

n2

n,n-1

pn

î

(3) .

:

dv

dr

v2

m v2

m

i

× dr

= m dv ×

i

= m v dv

= m d(

i

) = d(

i i

) = (dE )

.

(4)

dt

2

2

i dt

i

i i

i i

i

i

Fpi×dri – ,

, – . Fp1,Fp2,…,Fpi,…,Fpn –

( ), ,

,

,

–dAi = –Fpidri = dUi dAi = = –dUi . (5)

(4) (5), (3) :

	ì(dE )1			+ dU1 = 0
ï								n	n	(6)
+ í........................								å(dE )i	+ ådUi = 0 .
	ï(dE )		n	+ dU	n	= 0		i =1	i=1
	î

, U

(v , t),

dE i dU

( ):

103

				13
	n	n
	d(å(E )i	+ åUi ) = d(E + U) = 0 .		(7)
	i =1	i =1
,			– .	(8)
	+ U = = const

:

( )

( ), ( ) ,

.

, ( ) .

,

F , Fi, ( ,

). :

m	dvi	= F	+ F		;
i dt		pi,	pi,
(dE )i + dUi = F, dr = dAi,				dE = A E2 – E1 = A .

, ( )

0, ,

. , ,

,

, :

2 – 1 = ..

. ( , , ,

.): ( )

, .

§3. .

,

= v.

( ), n

,

, 2-

ìdp1

ïï dt ïdp2

ï

í dt

ïï M ïdpn

ï

î dt

= F12 + F13 + ... + F1n = Fp1
= F21 + F23		+ ... + F2n	= Fp2	.	(3 )
M	M	M	M

= Fn1 + Fn2 + ... + Fn(n-1) = Fpn

104

13

n (3). ( Fij)

n	dp		d	n	dp	n
å	i	=		åpi =		= åFij .
					dt
i =1	dt		dt i =1			i, j=1

i ¹ j

(3 ) ( . 3),

0, F12 = –F21 , F12+F21=0 –

.

	dp	= 0 ,		,	(9)
			= const

dt

n

p = åpi – ( ) –

i=1

. (9) :

( )

, :

,

. ( ).

– 3- .

,

.

!).

(≤ c), F12 i F21 ,

.

,

( ).

,

,

. ,

.

–

, ,

( , Z),

– . =const, a pz –

: .

§4. .

dN = M .

dt

n

1, 2, ... , ... n, r1, r2, … ri, … rn ,

Fp1, Fp2, … Fpi, …, Fpn,

105

13

, ( ) .

:

ìdN1 ïï dt

ïdN2

ïdt

+í

ïdN3

ïdt

ïdNn

ïî dt

n

= M1 = r1 ´ F12 + r1 ´ F13 + ... = år1 ´ F1j = r1 ´ Fp1

	j=2
		n
	= M2 = r2 ´ F21 + r2 ´ F23 + ... =	år2 ´ F2 j = r2 ´ Fp2
		j=1, j¹2
		n
	= M3 = r3 ´ F31 + r3 ´ F32 + ... =	år3 ´ F3 j = r3 ´ Fp3
		j=1, j¹3			F12=-F21 F12=F21
		n -1
					M1=|r1×F12|=r1×F12sina1=F12×l
	= Mn = rn ´ Fn1 + rn ´ Fn 2 + ... = årn ´ Fnj		= rn	´ Fpn
					M2=r2×F21sina2	=F21×r2×sinb

j=1

M=F12(r1×sina1-r2×sinb)=0

0 ,

r1×F12+ r2×F21 = r1×F12 – r2×F12 =Dr12×F12 = Dr12F12×sin0 = 0 –

( ) =0.

F23 F32 – .

d

(N

+ N

+ ... + N

) =

dN

= 0

,

(10)

2

N = const

dt

1

n

dt

:

; :

.

		dp	= F i	dN	= M
		dt		dt

.

, ,

.

§5. .

, ,

,

. , ,

.

, .

, ,

.

( , ),

.

,

106

13

: ,

, -

, .

1. –

, , .

, , ,

,

( ,

, ).

,

.

.

,

. 2-

3- ,

,

3- – .

3- ? – ,

.

n 1

2 , r,

,

: ( ),

– ( ). ,

A=(F12+F13+F14+…+F1n)×r. r 0, =0 ,

0, åFij = 0 .

i ¹ j

, , – , 3-

, Fij+Fji=0, ,

– .

2.

. , .

( , ,

).

– –

.

dj

.

0 ( ). ,

, , –

) , .

, 0. ,

107

13

dϕ . Fij

i Fji ri i rj. =ri×Fij + rj×Fji (=0).

dri,j=dϕ×ri,j.

:

′ = (ri +dri)×Fij + (rj +drj)×Fji = (ri×Fij+rj×Fji) + (dri×Fij+drj×Fji) =

=M + (dϕ×ri×Fij+dϕ×rj×Fji) = M + dϕ×(ri×Fij+rj×Fji) = M + dϕ×(ri×Fij-rj×Fij) =

=M + dϕ×Fij×(ri -rj) = M + dϕ×(Fij× rij) = M + dϕ×(0) = M.

:

, ,

.

3.–

. -

,

. , ,

, .

, .

. ( . „

”.

. , ,

, ), v=const,

, .

: v =const ,

,

.

,

, .

,

. ,

( ),

( ),

, ( ).

, .

. –

, ,

, , ,

.

108

14

14.

. .

.

. . (

. .)

§1.

, ,

.

.

, ,

( , ).

.

, (

, ): ,

, ,

– ,

,

, ( ).

:

.),

,

( .)

, ( . .)

(

).

, , .

. ,

,

,

, .

.

, ,

,

.

,

. , ,

, , ,

, .

,

.

109

14

, .

.

– ,

.

– ,

( ) .

,

. ( )

. ,

, ,

. ( . 1): 1)

2) .

, ,

(“ ”), .

. 1

–

:

				′	′
			k =	v2 − v1			,	(1)
	′	′		v2 − v1
			– 1- 2- , v1 i v2 –
	v1	i v2
	. ( )				k→1,			( )
	k→0.

§2.

(

=0 – . ), ,

, ( ′ =0).

2- . 1 2

v1 v2 .

v1′ v2′ .

( ) .

2

2

′2

′2

2

′2

′2

2

m1v1

m2v2

m1v1

m2v2

+

=

+

m1(v1

− v1

) = m2 (v2

− v2 )

→

2

2

2

2

→

m1(v1– v1′)(v1+ v1′) = m2(v2′– v2)(v2′+ v2)

(2)

110