Динамика

§2.

	1º.
	1.1.
	Oxyz –
	,
.	(
			A –
mQ –			VQ –
; Q
	1, N .
	,
			t .
	,
.
			M (e)
			A
A :

			A –				-
.			rA –			A	-
)							-
KA ,		&	&
KA
	Q	(rQ	rA ) mQ VQ ,
			PQ ,
			,			N	-
mQWQ FQ(e)	FQ(i) ,		Q		,
				1, N			(5.2.1)
KA
(5.2.1) WQ –						PQ , FQ(e) –	-
		, FQ(i) –					.
			,

–

M A(e)		&	&
	Q	(rQ	rA ) FQ(e) .

1.

			dKA	&
				M (e) ,		(5.2.2)
			dt	A
	. .					-
	KA ,				A ,	-
	M A(e)	,				A .
		.		,	1 §1.	:
x
		(5.2.1);			KA	-
x					;
x						KA
x	WQ	,				(5.2.1);
		1.
				- 171 -

	,		rQ rQ (t) , Q				—
						1, N
					KA (t)		&	&
						Q	(rQ (t)	rA ) mQVQ (t) ,
&
VQ (t)	drQ (t)		, Q 1, N .					t KA (t) .
		dt

dKA

dt (5.2.1)

,

	(5.2.1),	dVQ	&	—
			W
		dt	Q

&

¦drQ & mQVQ (t)

Q dt

		drQ		&
				VQ
		dt
mQ	dVQ
	dt
				PQ

&	&		dVQ
¦(rQ	rA )	mQ		.
			dt
Q

V mQVQ 0.

,

(5.2.1).

dKA

& &

&

(e)

& (i)

& &

& (e)

& &

dt

(5.2.1)

¦(rQ

rA )

(FQ

FQ

)

¦(rQ

rA )

FQ

¦(rQ

rA )

Q

Q

Q

(5.2.2),

.

(5.2.3)

-

mQWQ -

&

FQ(i) .

.

		&	&
	Q	(rQ	rA ) FQ(e) M A(e) .
					-
1	.
			.
1.2.	,
		A	C	.	-

				&	1		¦Q	&
				rC				mQrQ ,			(5.2.4)
						M
M —		, M	Q	mQ .
				rQ	rQ (t) ,			Q		,
									1, N		(5.2.5)
,	-	rC								t , . .	C
			,	.							, -
				MW			R(e) ,				(5.2.6)
					C
R(e) –				,
				&	&
			R(e)		¦FQ(e) ;
					Q
WC –		,

- 172 -

&

V

KC

KC

&				2 &
			dVC		d rC
	W						.
	C		dt		dt2
						VQ				WQ
1		&		&		&
	¦Q mQVQ ,				W			1	¦Q mQWQ .	(5.2.7)
M								M
&		&				&
Q (rQ		rC ) mQVQ					Q UQ mQVQ .			(5.2.8)

KC —

,	Q rQ rC	PQ	-
.
		KC ,	-
			-
.
2.

. .

MC(e)

(5.2.1).

MC(i) —

			dKC			&
						M (e)	,			(5.2.9)
				dt		C
										-
KC ,									,
								.
.				,					,
1.
								(5.2.5)	(5.2.8)	-
								,
	dKC	&		&		&		&	&
		¦(VQ		VC )		mQVQ		¦UQ	mQWQ .
	dt
		Q						Q

,

Q	VQ mQVQ 0			Q	VC mQVQ
			mQWQ
	dKC	&	& (e)		&	& (i)
		¦UQ	FQ	¦UQ		FQ
	dt
		Q			Q

VC MVC 0.

FQ(e) FQ(i)

-

&	&
M (e)	M (i) ,
C	C

C .

MC(i) 0.

2 .

- 173 -

&

VrQ

,

3

1.3.

,

KrC —

	~&		~
	dUQ	, Q			,	d
				1, N
	dt					dt

§2 4

,

.

3.

			. Э	-
		&
KrC	Q	UQ	mQVrQ ,	(5.2.10)
—			t .	,
			,	-
.			,
			KC и KrC .	-
VQ		VrQ	&
	VC		¦mQ UQ 0	(5.2.11)
			Q
KC	KrC .			(5.2.12)
				— -
			KrC ,	-
				-

~			&
	dKrC
			M (e) ,	(5.2.13)
		dt	C
. .				KrC ,	-
				,

.

(5.2.12)

KC

	dKrC				dKC
	dt				dt
a
&				~&
	da				Z
				da		&
	dt			dt

—

~

d . dt

ё

	.
	2
&
M (e) .	(5.2.14)
C

&	,		(5.2.15)
a
		,	-
	,	—	-
.		,	-

- 174 -

		0			ё			,		(5.2.15)
&		~&
	da		da	, . .						a		-
	dt
			dt
				—	.				a	KrC ,		-
								~
						dKrC		dKrC	.
						dt
								dt
		(5.2.14),			(5.2.13).			3		.
		(5.2.9)			(5.2.13)					(5.2.2)		1.
		(5.2.2)								A ,	(5.2.9)
(5.2.13) —					C .			2		3	,	-
		1						A ,			A	-
				C .
					A
		,			1,			,
			A . Э					.
		2º.
		2.1.								KA
									K rA

A — ,

O

—

-

rA (t) .

rA rA (t),

VA —

Ax y z

t

Oxyz . ,

PQ ,

O , .1,

Q .

										.	-
								t		rА ,
&			drA			&		dVA
V					,	W			,
	A						A
			dt					dt
A ; WA —										.
										,	-
						.				Ax , Ay , Az	-
										Ax y z
										t .
										,
		rQ ,									A —

					rQ	rА	Q ,
					VQ	VА VrcQ ,		(5.2.16)
					WQ	WА	WrcQ .	(5.2.17)
	&	~&		c
		d
			UQ
:	VrcQ				—		PQ , . .	PQ

dt

Ax y z ,

- 175 -

,

FQ(e)

&

~

~2 &

Q

UQ

—

Wrc

d

Q

dVrc

c

dt

dt

2

(5.2.16)

(5.2.17)

KA

KrA

&c

m Vrc

—

Q

UQ

Q Q

A

KA

.

2.2.

4.

A , . .

dKА

&

(e)

M

А

dt

KA

—

, VA

—

M (e)

A

PQ .

	,		Ax y z	-
&c		.
	m V	—		-
Q UQ	Q Q			-
		A
				-
		.		-
		,	K rA —	-

.

		-
		-
&	&
Q VA .		(5.2.18)
		A , M A(e) — -
	A ,	Q —
		A ,
Q	&c F(e) ,
	UQ	Q

—

,

.

.

.

KA

t

¦Q

&

dt

mQVQ

dtUQ

dKA

d c

&

&

c

~

&

c

&

d

d

UQ

UQ

(5.2.16)

Vrc

dt

dt

Q

d Q

VQ VА .

dt

(5.2.1).

mQWQ

U&c

dKA

&

&

&

(V VА ) m V

dt

¦

¦

Q

Q Q

Q

Q

Q

- 176 -

				PQ	-
	Uc				-
¦		m W .
	&		&
	Q		Q Q
Q	VrcQ
			VQ VА ,
				FQ(e)	FQ(i) -
&		¦ Q		&
Q				Q
F( )		Q	U&c	F(i) .	(5.2.19)

&	&	&	&
¦(VQ	VА ) mQVQ		¦(VQ
Q			Q

,

,

&	&	&
mQVQ )	¦(mQVQ	VА ) .
	Q

Q mQVQ

(5.2.19)

	&	&	&	& &
	¦(VQ	VА ) mQVQ		Q VА .
	Q			(5.2.19)
,	,

.

,(5.2.20) (5.2.19),

(5.2.18):			¦	U&c
& &					&		& &	&
	dKA	Q VА			F	( )	Q VА	А( ) .
	dt			Q	Q
			Q

4 .

1.

2.3.

5.

C — -

,

,

&

dKrA

dt

	dKA & ( )		&	&
		А	V	VА .
	dt
(5.2.21) VC —				.
			Q	MVC (5.2.18).

A

.

,

A ,

A

~

dKrA

dt

.

A ,

&	&	&
M A(e)	WA MUCc .

	K rA			&c	m Vrc				t
		Q UQ			Q	Q
~&		&			(5.2.1).
dKrA	,		Q		dVrcQ	WrcQ ,	UQ
		dVrc				&	d	c
dt		dt			dt		dt

A

M —

~&		c
d					,
	UQ Vrc
			&	Q
dt

- 177 -

Q .

(5.2.20)

-

,-

M A(e)

-

(5.2.21)

K rA ,

-

-

M C ,

.

(5.2.22)

-

	dt				~				~&
					dt		¦ dtUQ
	dKrA				dKrA			Q	d	c
WrcQ						WrcQ			WQ
	mQWQ
	~						U&c		&
		dKrA
								(F( )
		dt
					¦ Q				Q
					Q
MA(e)	MA(i) ,					&c
		M A(		)	Q			F	( )
						UQ		Q
—	,								,

A .

Q &cQ

m Vr

WА .

&

FQ(i) )

M A(i)

¦U&Qc

Q

.

¦mQU&Qc

Q

Q U&Qc

Q & cQ .

m Wr

(5.2.17),

(5.2.1).

&

WA .

,-

FQ(i)

Q mQU&Q MU&C ,

C —

-

A .

dKrA

~

dKrA

dt

dt

2.

~

dKA

dKrA

:

~

dt

dt

dKA

&

&

& &

dKrA

MUCc

WA MVC VА .

dt

dt

.

(5.2.22)

(5.2.21), (5.2.23)

(5.2.16) VC

, VrcC —

Ax y z ,

3º. З

(1)

(5.2.23).

VC

VА VrcC

	&	~
		dKrA
	M (e)
	A	dt
		2
	—

VC VА VrcC

VC

A

&	&
MUCc WA .

.

VА , (5.2.23)

.

A	M A(e) 0 .

dKA 0. dt

MA(i) 0,

(5.2.22).

(5.2.23)

. -

,

VrcC .

-

.

(5.2.2)

- 178 -

		KA const .	(5.2.24)
		,	,	-
			.
	6.		:
1)			;
2)			,
A	,	M A(e) 0 .
:				-
–
,			A ;
–			,	-
			A ,

.

.
(5.2.24)	,
M A(e) 0, . .
,
6,

A

,6

,

,

.

6

,

,

1.

,

.

6

.

В,

,

.

Э

, :

–

– KA

6);

–

A

C

2 3

:

~

dK dKr

dt dt

A ,

2)6.

	, . .	1)
.	M A(e)	0 .
	.	1),
		.
	A ,	,
	.
	A	KA ,
	.
		-
	KA ,	;
	(	-
	A .	.
	C
	.
	&
	M (e) .

- 179 -

,	MC(e)		0 ,
			KC K rC	const .		(5.2.25)
,	MC(e)	0 ,				-
В			.		,	-
,					.
				t .	ё	-
,			(			)
(5.2.25)	K rC		const ,	ё		-
				t .
	,			A	.	VA —
					.
7.
VA (t)VC	M A(e)	0 ,
KA			.
	,				4.
7	,		,	A		VA ,
				t	VC	,
				A		-

.

.

1.

,

)

(

,

&

J

mm

&

&

m

d 2r

0

r

F ,

dt

2

r

3

&

—

-

r OC

m —

, m0 —

,

—

:

&

mV .

KO

r

F

O

J

&

&

&

&

&

mm

&

MO (F )

r F

r

0

r

r

3

K

const , . .

,

KO ,

,

KO

r

V

t

- 180 -

.

		,
, O —		.
O (		)
.	,
0.
	O
	.
		,
	,	-