Васильев / Vasilev_mechanics_ru[1]

' )), )

, . /

),

x(t)= x0Cos( ωt + ϕ ) , , ' :

ϕ '0. 0 ), ' '

, ': [(ω02 - ω2)Cosϕ - 2γωSinϕ - F0/(mx0)]Cosωt + [(ω2 - ω02)Sinϕ - 2γωCosϕ]Sinωt = 0.

2 ',

( ) . $

'0 ϕ:

	=	F 0						1								; tg ϕ =		2 γ ω
x 0																				.
																		2	2
		m			(ω	2	2	)				2		2			ω
									2										− ω 0
							− ω 0			+ 4	γ		ω

$ " x0 =

x0

f(ω). $ ', '

, '

' .

,

- .

0.7xm

ω =

ω02 − 2γ 2 .

F0

'

Q.

'

mω02

ω

. 2

ω

,

.

Q =

ω

. $ Δω ,

ω

" 0.7 (

).

1.32 * . : / , ' "

( , , ' " ' )),

" ", "

. $ '

".

, ' ( "). %

, , , " α. ,

" ω0, ) )

' ' - , )

x = a Cos(ωt + α). !, "

, " ,

' ", .

( " ' " '

" .

	ω0	* ' " '1
		'2, : x1 = a1Cos(ω0t +
	a	α1); x2 = a2Cos(ω0t + α2). $
		1 2 , . $
	α	. 1 "
O	x	x , ) " '
		) " ' :
		' = '1 + '2 . *,

. " ω0,

1 2,

a

" ω0 ,

α. % ,

a2 = a12 + a22 + 2a1a2Cos(α2 - α1)

a2

a1

tgα =

a1Sinα1 + a2Sinα 2

. , '1

a1Cosα1 + a2 Cosα2

x '2 ,

1

2

O

x1

x2

.

x

).

-

"

,

" ' " '

" . /

' ' "

. ( . "

ω, " ω + Δω. $ Δω << ω.

. ' " . 0

, α1 = α2 = 0. ( " :

x1 = aCosωt , x2 = aCos(ω+ Δω)t . * "

,

	x = x 1 + x 2	=			ω		C o s ω t
				2 a C o s		t
					2

3 ) , . !

" , .

, - " ω, "

. "

, ', ' - 2 + 2 ,

- . .

Δω

: = 2aCos 2 t . - )

, , " ,

Δω. ( , ) - -

' .

1.33 & ) " ) " .

3 ) " ) " .

" " -

" ) . , " "

. 0 ' " ) " g = F/m . " ) " .

, ( ),

, . $

( "						" ). % "
", "			m M				M
	= -					= - G
F		G		e	r ; g				e r
			r 2				r 2
' ' + -		,			" ,			G - " )

. $) " ' ' + m

,

		m	M
E p = - F d r	+ C o n st = - G			+ C o n st
		r

, , ' ' ' " ",

, Const = 0. , , , - ) "

m ( , !), j = Ep = -G M

m r

) " ) " " ( ). , " ) ,

)

"

)

)

=

d F

j

=

d E

p

, " dF

g

dEp - ) " m

m

m

dM, . * )

, " ) " : F = − ÑEp g = − Ñj.

1.340 ' " ) . ! #.

/ " ( , !) " "

" ( , * )). $ " ),

, , !, ' ,

). 0 ) " '

: ) " . !,

! " * ) .

%, " ' , # ,

#:

I. # - ,

" ' * ).

II./ -

.

III. # '

' , " * ).

0 , ' " .

* , - . !

L = m r × v = m r × dr/dt. ( L = Const, -

,

.

II #. |L|= m|r × dr/dt|= m r dr/dt Sin

			(r, dr) = mrdh/dt = 2mdS/dt. ( "
O	dS		dS = dS n, II
			#
r		dr	dS	L
			dt =		= Const .
				2m
			.
I	#.		( , '

. %' ,

v

-

- r

ϕ.

vϕ

v = vr + vϕ , . ( ", dr = (dr)ϕ

vr

eϕ + (dr)r er = r dϕ eϕ + dr er,

v2

2 + vr2. $

v = dr/dt =

r ω e

+ dr/dt

e

= v

r

ϕ

r

ϕ

'

ϕ

"

L = m r × v = m r × (eϕ vϕ + er vr) = m r × vϕ eϕ = m ω r2 e

mv2

mM

m(v2r

+ v2

)

mM

− G

=

ϕ

− G

= Const .

2

r

2

r

+ L = m ω r2 = m r2 ddtϕ ddtϕ = L/(mr2). (

ρ = 1/r. ( "

dr

=

dr d ϕ

=

d

1

d ϕ

= −

1 d ρ d ϕ

= − r

2 L d ρ

= −

L d ρ

.

d ϕ dt

ρ 2 d ϕ dt

2 d ϕ

m d ϕ

dt

d ϕ

ρ

dt

mr

$ ' "

m

(v2

+ v2 )−G

mM

m

L 2

dρ 2

m

r2

. 2

L2

dρ 2

L2

ρ2 −GmMρ=Cons

=

+

ϕ −GmMρ=

+

2

r

ϕ

r

2 m

2

2m

2m

dϕ

dϕ

$ ) ϕ, , ,

2

d ρ

d 2 ρ

+ 2 ρ

d ρ

−

2 G m 2 M d ρ

=

0 ;

d 2 ρ

+ ρ =

G m 2 M

> 0 .

d ϕ

d ϕ 2

d ϕ

L 2

d ϕ

d ϕ 2

L 2

$ '

' ω = 1, " ρ = ASinϕ + BCosϕ + C

B

.

=

A 2 + B 2 C o s( ϕ − ϕ 0 ) + C ; tg ϕ 0 =

p

=

A

1

L2

A 2 + B 2

=

; e =

. 0 ρ ρ = 1 + Cos (ϕ - ϕ0), r

C

G m 2 M

C

p

r = 1 + eCos(ϕ − ϕ0 ) . "

. : < 1 - ; = 1 - ; > 1 - "; = 0 -

. + r * ) " ϕ = ϕ0. $ " . ,

- ". ( , "

	r =			p		.
		1		+ e C o s	ϕ
0 - .								b
:							L1	L2
1.	$ S = π a b.
2.	L1 + L2 = 2a ( ).
								O
3. = b2/a.							F1	F2 a

4.r = /(1+ ); r' = /(1- ).

III #. % II

	#
	S =	L	T . ( 1) S = π a b, " b -
		2m

. a b , )

p: r(0) + r (π) = 2a = 2p/(1- e2)

a = p/(1 - e2), ", 2 ,

, F2, b :

a2 = (a - r )2 + b2

b =

p

− p =

b 2

.

a

1 − e 2

2mS

2

=

4m

2

(πab)

2

=

4π

2

a3

T

2

= Const .

( " T2 =

m2 GMp

GM

L

a3