Васильев / Vasilev_mechanics_ru[1]

	d L	N
		=	M i & , 8 & = M & , 8 &
	d t
		i = 1

( , ' )

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. , '

( ),

' .

.

& ' ", '

"

' ' .

1.21 "

$' .(( - ' "

. , .((

" , .

" " . $ - , ,

, . $

, '

. 0 " II & ,

) . ( , )

" .

- , '

. , .(( " ,

" " , '

. "

: ϕ, ω, β.

( , .(( "

) " ' . $

" .(( "

.

+ , " "

ω,				mi. &
				.
	ω	z		. *
				vi . +			Li
		Ri	mi	vi ri. /			mi
Li					Ri = ri Cosϕi. $
		ϕi	vi	+( Li	=	mi [ri, vi]. !ri - -
ϕi
		ri		,			mi
	O			, vi - i -
				. + L
				' : L = Li = mi[ri, vi]. %
ω				, "
	L . $

" . 0 " , '

, . ! " )

	, z:		dLz		= Mz . &			.((
			dt
						L z. %
	z, )
	, Lzi = Li Cosϕi. $ " ri vi , Li						=	mi ri
	vi. *, Lzi = mi ri	vi Cosϕi = mi Ri vi, " Ri				- mi
	. !, vi = ω Ri , Lzi = ω Ri2				mi. $) Lz
	) Lzi :	Lz = Lzi = ω Ri2			mi = ω Ri2 mi. $

,

L. Iz = Ri2 mi '

' , )

. , ", ,

) , ,

", . () z

.(( Lz = Iz ω. $ , .((

,

" .((:

	I		dω z	= M		Iz βz = Mz.
		z	dt		z

& " . /

" ), - .

1.22 ( )

%, - ) . &

' ' , " " )

. & ' ,

.(( " . 0

' mi. / ,

" , L =

mi [ri,vi]. ( vi = [ω, ri], L = mi [ri, [ω, ri]] = mi ri2 ω - mi ri (ωri). $

" «) )» $ × × " = ($") - " ($ ). ( " L = ω mi ri2 - mi ri (ω,ri). 7

L ω , L = I ω. ( , ωr = ωx xi + ωy yi + ωz zi

ri2= xi2 + yi2 + zi2. ( " ) ( , x) "

: Lx = ωx mi (xi2 + yi2 + zi2 - xi2) - mi xi yi ωy - mi xi zi ωz .

:

Lx = Ixx ωx + Ixy ωy + Ixz ωz, "

Ixx = mi ( yi2 + zi2)

Ly = Iyx ωx + Iyy ωy + Iyz ωz

Ixy = - mi xi yi

Lz = Izx ωx + Izy ωy + Izz ωz

.

I

I

I

% '

, Iαβ = Iβα,

Ixx , Ixy ....

xx

xy

.

I ,

I

,

I

xz

xx

yy

zz

Iyx

Iyy

Iyz

),

Ixy = Iyx , Ixz

=

Izx

,

Iyz =

Izy

)

).

*

,

Izx

Izy

Izz

' , .

Ixx, Iyy, Izz

" , -

".

,

", . ( . $ - " ". * " ", - " ".

! " . 0 "

, "

. ( ", - "

. *, Ix = Ixx, Iy = Iyy , Iz = Izz "

). % '

) Izz = mi (ri2 - zi2) = mi Ri2, .

" .

1.23 ' ). ( 8 . $ " .((

+(. , ' ( ), '

" "

Izz = R 2 d m = ρ R 2 d V

m V

" ρ - , R - " 6 dV . ,

, " ' '

Izz = ρ R2 dV .

V

" " . /

' .

1). + ) ( m, - R).

" I+( = m R2.

2). + ) " (m, l, ' )

Z). * , dV = Sdx, R = x, ρ = m/V = m/(Sl), "

dx

m

l / 2

m 1

l

3

m l

2

I z z

= 2

x 2 d x = 2

=

.

l

l 3

1 2

0

2

3). + ) ) (m, r, )

% " 6 )

) dR(

), 2πRh, " h - ), R -

" 6 . ( "

dm = 2πρRh dR dIz = R2 dm =

2πρhR3 dR. !, m = ρ V = ρπr2 h, "

r

)

I z9%1 = 2 πρ R 3dR = mr 2 .

0

2

4). + ) (m, R , ' ) ).

& dV 6 ,

' R

R

" ϑ

: dV

= 2πr2Sinθdrdθ.

6

rSinθ rdθ×dr. (

θ

6

" dIz = ρR2 dV = 2πρr4

Sin3θ dr dθ. ) ,

R

π

4πR 3

2R 2

2

I z8 ./ = 2πρ r 4 dr Sin 3 ϑdϑ =

ρ

=

mR 2 .

3

5

5

0

0

	) (
' ) '							"). $	,
						) ' "
	y		y			, ) ,
				mi		a. %,
	Ri’					: ' )
			Ri			,	-		. +
	C					) ,			'
O	a			xi	x
						, I = Ri’2 Dmi = (xi’2 + yi’2)Dmi
	xi’					= [(xi + a)2 + yi2] Dmi = (xi2 + 2xia + a2 + yi2) Dmi =

Ri2 Dmi + 2a xi Dmi + a2 Dmi = Ic + m a2 + 2 a m xc = Ic + m a2. ( ' ' ) , xc = 0,

8 :

, ,

:

I = Ic + m a2.

1.24 # " " " .

, .(( " " ω,

Dmi, Ri,

vi = w Ri. *, " DEki = 1/2 Dmi vi2 = 1/2 Dmi w2 Ri2. * ' ' " " " Ek/=

1/2 w2 Ri2 Dmi = 1/2 I w2. " "

" , ),

- ". ( , ". &

	, .((.
ds	/ , "
	, .
R	F ½½ ds, dA = Fs ds
dj	F = Fs R dj = Mz dj. ,
O	,
	: F||(

), F (

) Fτ( ). (

, dA = Mω dj. 0 P = Mω w = Mω .

, , "

" " ,

, ' ) . ( " # "

:

E k	=	E k$ * ( +	E k / =	m v 29 +	+	I	c	w 2
				2				2

" , "

" '. w=v/R Ek = mv2/2 + m R2/2 (v/R)2 = m v2.

1.25 3. 3

3 , "
. 0 " " L
ω , " L = I ω. ( " , I , w.
		% ", ,
		' , ' " M = 0. "
dL		, L.								( .
F		$ " F,
L		" Dt. , "
		DD’.
		" . ( "
D’		, dL = M dt;
		L(t+dt) = L(t) + M dt
M		" , .
O			,	½L½>>½M dt½.			( ,
		" F,
D
		" ' .
). / ) " .
% ",
" b. & " mg, M
= [r, mg]. +"			M = mgbSinb, " b –
) ". 0
		O’	"			dL = Mdt
			" " dj. + "
A			½dL½= L Sinb dj.
			"							):
dL B
L			dj = mgbSinβdt ;w			= dϕ	= mgb		= mgb .
				LSinb		dt		L	Iw
			dt "
	β		" ,
mg	b		. )
	O		" w << w. $
		M	,		mgb	<< I	w2. $			"
			) " "
, - " ". %,								,
Ep << Ek.

1.26 * - ' ! ' , "

. II

&,

. $ - ,

.

1). ' :

!

,

! .

2). , ' ,

:

, ,

, .

3). % , ' ,

:

, ,

, .

& .

' ". % II

& A12 = Ek2 - Ek1. 0 , , 6,

, ) ",

) " - Ep(x,y,z,t). ( ,

. ( " ' ) "

∂E

p

∂E

p

∂E

p

A12

= −

dx +

dy

+

dz .!

) Ep

∂x

∂y

∂z

dEp

=

∂Ep

dx +

∂Ep

dy

+

∂Ep

dz +

∂Ep

dt , "

∂x

∂y

∂z

∂t

A12

= − dEp

+

∂Ep

dt

( ,k2 + Ep2 ) - ( Ek1 + Ep1 ) =

∂E p

dt . , ,

∂t

∂t

) "

. ( , ' " . $ ' . $ '

. F1, F2, ..., ,

, ' . "

, " ". +

, r1 , r2 , ... R - "

Ep (r1 , r2 , ...) = Ep (r1 + R , r2 + R , ...). R , ,

	R = dx ex : Ep (r1 + R,..) = Ep	( r1 ,.. ) +	∂Ep
				∂x1

+	∂E	p		.
			+...... dx
	∂x2

' , - ) ' , '

, ' ( ). R ,

' , ' , , ,

II & ' .

! ' , ' ,

' , . / '

" dϕ.

, dA = (M1 + M2 + ...) dϕ = 0 " '

, ' , .

1.27 *

$ -

. &, " - x,y,z,

- ρ, θ, ϕ. ' . $ 6 ",

. + , ",

, . ( " ", .

f(x,y,z)=0 - , "

. 3, .

, ". ",

	. $ "
	y	. $ ,
		(x,y). &
		(x2 + y2 = R2) ,
		.

x"

,, (0,R).

" L .

n

' . , " , " "

' 3n . 3n

. $ - . ( "

' f. 0 ' f q1 , q2 ,...., qf , ' n .

. $ '

. 2 ' '

.

.((. 2 '

: .(( (3 )

(3 ) . 0 " ' - " .(( 6

.

3 4. #

1.28. 3

/ , '

q. ( " , ) "

Ep(q) , . , , - ' " )

Ep

, )

'

'

'

q,

" ,p(q) ≤

E(

,

E

" ). $

q3

" ( ),

q1 q2 " (

). (

q

q0

"

q1

q0 q2

q3

. $ '

'

(

) ' ' ' " - ' . # ,

( . )). $ " . / )

" q0 - " :

E

(q) = E

(q

) +

dE p

(q − q

) +

d 2 E p

(q − q

)2

+......

p

p

0

dq

q=q0

0

dq2

0

q−q0

( q = q0 ) " ,

)	. ,
k/2,	) "			,
), q0
E p ( q ) = E p ( q 0 ) +	k	( q − q 0 ) 2 ; f ( q ) = − k ( q − q 0 )
	2

( , , " ' ), ,

, . %,

' ' -

( ) . #, ' ,

II &. 0 ) m

" mW = F

d2 q

m dt 2 = −k(q − q0 ) . x = q - q0, ).

, "

. ( ", ω2 = k / m, m x’’ = -k x x’’ + ω2 x = 0. $ : ,

, .

	" " '					) x (t) = A1 Sinωt + A2
	Cosωt = A Sin(ωt + ϕ0). tgϕ0		A 2
		=		;A = A12		+ A22 . * )
			A1

( ) :

x(t) = A Sin (ωt + ϕ0)

. , ω - , ϕ0 -

, ϕ(t) = ωt + ϕ0 - . ,

,

. " , ,

E =	mv 2	+	kx 2	=	mω 2 A 2 Cos2	ϕ	+	kA 2 Sin 2	ϕ	=	mω 2 A 2
	2		2		2			2			2

. ( , . 0

, ' .

1.29 * ( , " )

* '

' ' . 0 ' '

, , ' x << 1.

(+(

) ( ). ! II

& " , "

(" ):

m a = F

; OX:

ma = - mg Sinα

(

x << 1, "

g

x . 7, a = x’’,

Sinα α

= x/l. ( "

a = − l

T

g

" '

x"+ l x = 0 .

x

O

mg

ω0 =

g

l '

2π

l

" . *

T =

ω0

= 2π g

. %,

), "

x(t) = A Sin (ω t + ϕ0). 0 "

'

. , , :

) ":

1) +( '0

. ( "

x|t=0 = x0

"

v|t=0 = 0

x(t) = x0 Cos ω0t

2) +( v0 .

( "

x|t=0 = 0

ϕ0= 0

v|t=0 = v0

x(t) = v0/ω0 Sin ω0t

«" ». / ,

. $

" ",

' 3

m

F

= - k x ( - ),

. ( " II & ma = F + mg +

N ) ' : ma = - kx. *

x

" '

k

' ω0

=

m

T =

2

π m .

/ '

"

k

" .

«

». .((, "

"

.

2

ω 0 =

m g a , T

= 2 π

I

, " m - .((, - .

I

m g a

) .((, I - ) .(( ..

1.30 ! ' '

" ,

'. * ' " . # ", , ' '

, ' '.

« », , "

" ' 6 F = - βv = - βx’ . ! II

& ma = - kx - βv x’’ = - ω02 x - β/m x’. γ = β/2m;

ω02 = k/m, " : x’’ + 2 γ x’ + ω02 x = 0

/ " - ', ,

. %

x(t) = x0 e- α t Cos ωt

ω α. 0 '

'

. $

(α2 - ω2 + ω02 - 2αγ) x0 e- α t Cos ωt + (2αω - 2ωγ) x0 e- α t Sin ωt = 0

2 ',

" ( ) . $

α ω: α = γ; ω = ω20 − γ 2 . ( , ' '

" :

		− γ t
x ( t ) =	x 0 e			ω	2	− γ	2		t .
			C o s		0

γ '. & ' "

' ( ) '

( ) ' '.

x	x
O	t
	O	t

$ ,

"

1.31 '

,

' . '

, " . 7 II

& : ma = - kx - 2 γ m x’ + F0 Cosωt

x’’ + 2 γ x’ + ω02 x = (F0/m) Cos ωt.

$) ' . * "

τ ' (