Краткий курс теоретической механики. Яковенко Г.Н. - 96 стр.

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r = a + r
e
r
∗
= r
e
+ c = r
e
+ a cos ϕ + b sin ϕ.
r
∗
= Λ ◦ r ◦
e
Λ
(27.4)
=
³
r ◦ Λ + 2[e sin
ϕ
2
, r]
´
◦
e
Λ = r ◦
1
z}|{
Λ ◦
e
Λ +2b ◦
e
Λ sin
ϕ
2
=
= r + 2b ◦
³
cos
ϕ
2
− e sin
ϕ
2
´
sin
ϕ
2
= r + b sin ϕ − a(1 − cos ϕ) =
= r − a
|{z}
r
e
+ a cos ϕ) + b sin ϕ
| {z }
c
.
b ◦ e = −(b, e) + [b, e] = −([e, r], e) + [b, e] = [b, e] = a.
¥
r
Λ
−→ r
∗
M
−→ r
∗∗
,
r
N=?
−−→ r
∗∗
.
Λ r r
∗
M r
∗
r
∗∗
N
r r
∗∗
r
∗
= Λ ◦r ◦
e
Λ
r
∗∗
= M ◦r
∗
◦
f
M = M ◦Λ ◦ r ◦
e
Λ ◦
f
M
(27.11)
= M ◦Λ ◦ r ◦
^
M ◦Λ = N ◦r ◦
e
N,
N = M ◦ Λ
Λ M
N = M ◦ Λ
n
r
Λ
1
−→ ···
Λ
n
−→ r
∗
r
N
−→ r
∗
,
N = Λ
n
◦ ··· ◦ Λ
1
. (28.2)
r = a + re (ðèñ. 28.1). Óòâåðæäåíèå, êîòîðîå íóæíî äîêàçàòü, ïðè ïîìîùè ââå-
ä¼ííûõ âåêòîðîâ ôîðìóëèðóåòñÿ ñëåäóþùèì îáðàçîì (ðèñ. 28.1):

                            r∗ = re + c = re + a cos ϕ + b sin ϕ.

Ïðîäåëàåì íåîáõîäèìûå äëÿ äîêàçàòåëüñòâà âû÷èñëåíèÿ
                                                                1
                            ³                            z }| {
      ∗          e (27.4)              ϕ ´ e                         e sin ϕ =
                                                             e +2b ◦ Λ
     r = Λ ◦ r ◦ Λ = r ◦ Λ + 2[e sin , r] ◦ Λ = r ◦ Λ ◦ Λ
                                       2                                   2
                   ³  ϕ          ϕ´      ϕ
        = r + 2b ◦ cos − e sin       sin = r + b sin ϕ − a(1 − cos ϕ) =
                      2          2       2
                        = |r {z
                             − a} + a cos ϕ) + b sin ϕ .
                                    |       {z       }
                                    re                    c

Êðîìå ââåä¼ííûõ ðàíåå îáîçíà÷åíèé ó÷òåíî, ÷òî (ñì. (27.6))

           b ◦ e = −(b, e) + [b, e] = −([e, r], e) + [b, e] = [b, e] = a.

¥
    Ðàññìîòðèì ñèòóàöèþ äâóõ ïîñëåäîâàòåëüíûõ ïîâîðîòîâ.
                                           Λ          M
                                     r −→ r∗ −→ r∗∗ ,
                                               N =?
                                           r −−→ r∗∗ .
Êâàòåðíèîí Λ ïðè ïîìîùè (28.1) ïåðåâîäèò ïðîèçâîëüíûé âåêòîð r â âåêòîð r∗ ,
Êâàòåðíèîí M ïåðåâîäèò âåêòîð r∗ â âåêòîð r∗∗ . Íàéòè êâàòåðíèîí N , ïåðåâî-
                                                                                e,
äÿùèé ïðè ïîìîùè (28.1) âåêòîð r â âåêòîð r∗∗ . Èç (28.1) ñëåäóåò: r∗ = Λ ◦ r ◦ Λ

                   f=M ◦Λ◦r◦Λ
    r∗∗ = M ◦ r∗ ◦ M          f (27.11)
                            e◦M            ^
                                  = M ◦Λ◦r◦M         e,
                                             ◦Λ=N ◦r◦N

ãäå îáîçíà÷åíî N = M ◦ Λ. Òàêèì îáðàçîì, äâà ïîñëåäîâàòåëüíûõ ïîâîðîòà 
ïåðâûé, çàäàííûé Λ, âòîðîé, çàäàííûé M ,  ìîæíî çàìåíèòü îäíèì, çàäàí-
íûì ïðè ïîìîùè (28.1) êâàòåðíèîíîì N = M ◦ Λ. Ïî èíäóêöèè äîêàçûâàåòñÿ
ôîðìóëà â îáùåì ñëó÷àå: n ïîñëåäîâàòåëüíûõ ïîâîðîòîâ
                                           Λ          Λ
                                            1
                                         r −→        n
                                              · · · −→ r∗

çàìåíÿþòñÿ îäíèì
                                                N
                                            r −→ r∗ ,
çàäàííûì ïðè ïîìîùè (28.1) êâàòåðíèîíîì

                                     N = Λn ◦ · · · ◦ Λ1 .                  (28.2)

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