Virtual joy almashish
Analitik mexanikada amaliy matematika va fizikaning bir boʻlimi,
virtual siljish (yoki
cheksiz kichik oʻzgarish )
δ
γ
{\displaystyle \delta \gamma }
Mexanik tizimning trayektoriyasi gipotetik (shuning uchun virtual atamasi) haqiqiy traektoriyadan qanday qilib biroz chetga chiqishi mumkinligini koʻrsatadi.
γ
{\displaystyle \gamma }
tizimning cheklovlarini buzmasdan . Har lahza uchun
t
,
{\displaystyle t,}
δ
γ
(
t
)
{\displaystyle \delta \gamma (t)}
nuqtadagi konfiguratsiya fazosiga tangensial vektordir
γ
(
t
)
.
{\displaystyle \gamma (t).}
Vektorlar
δ
γ
(
t
)
{\displaystyle \delta \gamma (t)}
qaysi yoʻnalishlarni koʻrsating
γ
(
t
)
{\displaystyle \gamma (t)}
cheklovlarni buzmasdan „borish“ mumkin.
Misol uchun, ikki oʻlchovli sirtdagi bitta zarrachadan tashkil topgan tizimning virtual siljishlari, qoʻshimcha cheklovlar yoʻq deb hisoblab, butun tangens tekisligini toʻldiradi.
Biroq, cheklovlar barcha trayektoriyalarni talab qilsa
γ
{\displaystyle \gamma }
berilgan nuqtadan oʻting
q
{\displaystyle \mathbf {q} }
berilgan vaqtda
τ
,
{\displaystyle \tau ,}
yaʼni
γ
(
τ
)
=
q
,
{\displaystyle \gamma (\tau )=\mathbf {q} ,}
keyin
δ
γ
(
τ
)
=
0.
{\displaystyle \delta \gamma (\tau )=0.}
Belgilari
Mayli
M
{\displaystyle M}
mexanik tizimning konfiguratsiya maydoni boʻlsin,
t
0
,
t
1
∈
R
{\displaystyle t_{0},t_{1}\in \mathbb {R} }
vaqt lahzalari boʻlsin,
q
0
,
q
1
∈
M
,
{\displaystyle q_{0},q_{1}\in M,}
C
∞
[
t
0
,
t
1
]
{\displaystyle C^{\infty }[t_{0},t_{1}]}
ustidagi silliq funksiyalardan iborat.
Cheklovlar
γ
(
t
0
)
=
q
0
,
{\displaystyle \gamma (t_{0})=q_{0},}
γ
(
t
1
)
=
q
1
{\displaystyle \gamma (t_{1})=q_{1}}
Bu yerda faqat tasvir uchun. Amalda, har bir alohida tizim uchun individual cheklovlar toʻplami talab qilinadi.
Taʼrifi
Har bir yoʻl uchun
γ
∈
P
(
M
)
{\displaystyle \gamma \in P(M)}
va
ϵ
0
>
0
,
{\displaystyle \epsilon _{0}>0,}
ning oʻzgarishi
γ
{\displaystyle \gamma }
funksiya hisoblanadi
Γ
:
[
t
0
,
t
1
]
×
[
−
ϵ
0
,
ϵ
0
]
→
M
{\displaystyle \Gamma :[t_{0},t_{1}]\times [-\epsilon _{0},\epsilon _{0}]\to M}
shunday, har bir uchun
ϵ
∈
[
−
ϵ
0
,
ϵ
0
]
,
{\displaystyle \epsilon \in [-\epsilon _{0},\epsilon _{0}],}
Γ
(
⋅
,
ϵ
)
∈
P
(
M
)
{\displaystyle \Gamma (\cdot ,\epsilon )\in P(M)}
va
Γ
(
t
,
0
)
=
γ
(
t
)
.
{\displaystyle \Gamma (t,0)=\gamma (t).}
Virtual joy almashish
δ
γ
:
[
t
0
,
t
1
]
→
T
M
{\displaystyle \delta \gamma :[t_{0},t_{1}]\to TM}
(
T
M
{\displaystyle (TM}
ning tangens toʻplamidir
M
)
{\displaystyle M)}
oʻzgarishiga mos keladi
Γ
{\displaystyle \Gamma }
har biriga quyidagini belgilaydi
t
∈
[
t
0
,
t
1
]
{\displaystyle t\in [t_{0},t_{1}]}
tangens vektori
δ
γ
(
t
)
=
d
Γ
(
t
,
ϵ
)
d
ϵ
|
ϵ
=
0
∈
T
γ
(
t
)
M
.
{\displaystyle \delta \gamma (t)={\frac {d\Gamma (t,\epsilon )}{d\epsilon }}{\Biggl |}_{\epsilon =0}\in T_{\gamma (t)}M.}
Tangens xaritasi nuqtai nazaridan,
δ
γ
(
t
)
=
Γ
∗
t
(
d
d
ϵ
|
ϵ
=
0
)
.
{\displaystyle \delta \gamma (t)=\Gamma _{*}^{t}\left({\frac {d}{d\epsilon }}{\Biggl |}_{\epsilon =0}\right).}
Bu yerda
Γ
∗
t
:
T
0
[
−
ϵ
,
ϵ
]
→
T
Γ
(
t
,
0
)
M
=
T
γ
(
t
)
M
{\displaystyle \Gamma _{*}^{t}:T_{0}[-\epsilon ,\epsilon ]\to T_{\Gamma (t,0)}M=T_{\gamma (t)}M}
ning tangens xaritasi hisoblanadi
Γ
t
:
[
−
ϵ
,
ϵ
]
→
M
,
{\displaystyle \Gamma ^{t}:[-\epsilon ,\epsilon ]\to M,}
bu yerda
Γ
t
(
ϵ
)
=
Γ
(
t
,
ϵ
)
,
{\displaystyle \Gamma ^{t}(\epsilon )=\Gamma (t,\epsilon ),}
va
d
d
ϵ
|
ϵ
=
0
∈
T
0
[
−
ϵ
,
ϵ
]
.
{\displaystyle \textstyle {\frac {d}{d\epsilon }}{\Bigl |}_{\epsilon =0}\in T_{0}[-\epsilon ,\epsilon ].}
Xususiyatlari
δ
γ
(
t
)
=
∑
i
=
1
n
d
[
q
i
(
Γ
(
t
,
ϵ
)
)
]
d
ϵ
|
ϵ
=
0
⋅
d
d
q
i
|
γ
(
t
)
.
{\displaystyle \delta \gamma (t)=\sum _{i=1}^{n}{\frac {d[q_{i}(\Gamma (t,\epsilon ))]}{d\epsilon }}{\Biggl |}_{\epsilon =0}\cdot {\frac {d}{dq_{i}}}{\Biggl |}_{\gamma (t)}.}
Misollar
R 3 dagi erkin zarracha
Yagona zarracha erkin harakatlanadi
R
3
{\displaystyle \mathbb {R} ^{3}}
3 erkinlik darajasiga ega. Konfiguratsiya maydoni
M
=
R
3
,
{\displaystyle M=\mathbb {R} ^{3},}
va
P
(
M
)
=
C
∞
(
[
t
0
,
t
1
]
,
M
)
.
{\displaystyle P(M)=C^{\infty }([t_{0},t_{1}],M).}
Har bir yoʻl uchun
γ
∈
P
(
M
)
{\displaystyle \gamma \in P(M)}
va variatsiya
Γ
(
t
,
ϵ
)
{\displaystyle \Gamma (t,\epsilon )}
ning
γ
,
{\displaystyle \gamma ,}
noyobi mavjud
σ
∈
T
0
R
3
{\displaystyle \sigma \in T_{0}\mathbb {R} ^{3}}
shu kabi
Γ
(
t
,
ϵ
)
=
γ
(
t
)
+
σ
(
t
)
ϵ
+
o
(
ϵ
)
,
{\displaystyle \Gamma (t,\epsilon )=\gamma (t)+\sigma (t)\epsilon +o(\epsilon ),}
kabi
ϵ
→
0.
{\displaystyle \epsilon \to 0.}
Taʼrifga koʻra,
δ
γ
(
t
)
=
(
d
d
ϵ
(
γ
(
t
)
+
σ
(
t
)
ϵ
+
o
(
ϵ
)
)
)
|
ϵ
=
0
{\displaystyle \delta \gamma (t)=\left({\frac {d}{d\epsilon }}{\Bigl (}\gamma (t)+\sigma (t)\epsilon +o(\epsilon ){\Bigr )}\right){\Biggl |}_{\epsilon =0}}
olib keladi
δ
γ
(
t
)
=
σ
(
t
)
∈
T
γ
(
t
)
R
3
.
{\displaystyle \delta \gamma (t)=\sigma (t)\in T_{\gamma (t)}\mathbb {R} ^{3}.}
Sirtdagi erkin zarralar
N
{\displaystyle N}
ikki oʻlchovli sirtda erkin harakatlanadigan zarralar
S
⊂
R
3
{\displaystyle S\subset \mathbb {R} ^{3}}
bor
2
N
{\displaystyle 2N}
erkinlik darajasi. Bu erda konfiguratsiya maydoni
M
=
{
(
r
1
,
…
,
r
N
)
∈
R
3
N
∣
r
i
∈
R
3
;
r
i
≠
r
j
if
i
≠
j
}
,
{\displaystyle M=\{(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})\in \mathbb {R} ^{3\,N}\mid \mathbf {r} _{i}\in \mathbb {R} ^{3};\ \mathbf {r} _{i}\neq \mathbf {r} _{j}\ {\text{if}}\ i\neq j\},}
bu yerda
r
i
∈
R
3
{\displaystyle \mathbf {r} _{i}\in \mathbb {R} ^{3}}
ning radius vektori
i
th
{\displaystyle i^{\text{th}}}
zarracha. Bundan quyidagi kelib chiqadi
T
(
r
1
,
…
,
r
N
)
M
=
T
r
1
S
⊕
…
⊕
T
r
N
S
,
{\displaystyle T_{(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})}M=T_{\mathbf {r} _{1}}S\oplus \ldots \oplus T_{\mathbf {r} _{N}}S,}
va har bir yoʻl
γ
∈
P
(
M
)
{\displaystyle \gamma \in P(M)}
radius vektorlari yordamida tasvirlanishi mumkin
r
i
{\displaystyle \mathbf {r} _{i}}
har bir alohida zarrachaning, yaʼni
γ
(
t
)
=
(
r
1
(
t
)
,
…
,
r
N
(
t
)
)
.
{\displaystyle \gamma (t)=(\mathbf {r} _{1}(t),\ldots ,\mathbf {r} _{N}(t)).}
Bu shuni anglatadiki, har bir kishi uchun
δ
γ
(
t
)
∈
T
(
r
1
(
t
)
,
…
,
r
N
(
t
)
)
M
,
{\displaystyle \delta \gamma (t)\in T_{(\mathbf {r} _{1}(t),\ldots ,\mathbf {r} _{N}(t))}M,}
δ
γ
(
t
)
=
δ
r
1
(
t
)
⊕
…
⊕
δ
r
N
(
t
)
,
{\displaystyle \delta \gamma (t)=\delta \mathbf {r} _{1}(t)\oplus \ldots \oplus \delta \mathbf {r} _{N}(t),}
bu yerda
δ
r
i
(
t
)
∈
T
r
i
(
t
)
S
.
{\displaystyle \delta \mathbf {r} _{i}(t)\in T_{\mathbf {r} _{i}(t)}S.}
Baʼzi mualliflar buni shunday ifodalaydilar
δ
γ
=
(
δ
r
1
,
…
,
δ
r
N
)
.
{\displaystyle \delta \gamma =(\delta \mathbf {r} _{1},\ldots ,\delta \mathbf {r} _{N}).}
Ruxsat etilgan nuqta atrofida aylanadigan qattiq jism
Qoʻshimcha cheklovlarsiz qoʻzgʻalmas nuqta atrofida aylanadigan qattiq jism 3 daraja erkinlikka ega. Bu yerda konfiguratsiya maydoni
M
=
S
O
(
3
)
,
{\displaystyle M=SO(3),}
3 oʻlchovli maxsus ortogonal guruh (aks holda 3D aylanish guruhi deb nomlanadi) va
P
(
M
)
=
C
∞
(
[
t
0
,
t
1
]
,
M
)
.
{\displaystyle P(M)=C^{\infty }([t_{0},t_{1}],M).}
Biz standart belgidan foydalanamiz
s
o
(
3
)
{\displaystyle {\mathfrak {so}}(3)}
barcha egri-simmetrik uch oʻlchovli matritsalarning uch oʻlchovli chiziqli fazosiga murojaat qilish. Eksponensial xarita
exp
:
s
o
(
3
)
→
S
O
(
3
)
{\displaystyle \exp :{\mathfrak {so}}(3)\to SO(3)}
mavjudligini kafolatlaydi
ϵ
0
>
0
{\displaystyle \epsilon _{0}>0}
Shunday qilib, har bir yoʻl uchun
γ
∈
P
(
M
)
,
{\displaystyle \gamma \in P(M),}
uning oʻzgarishi
Γ
(
t
,
ϵ
)
,
{\displaystyle \Gamma (t,\epsilon ),}
va
t
∈
[
t
0
,
t
1
]
,
{\displaystyle t\in [t_{0},t_{1}],}
oʻziga xos yoʻl bor
Θ
t
∈
C
∞
(
[
−
ϵ
0
,
ϵ
0
]
,
s
o
(
3
)
)
{\displaystyle \Theta ^{t}\in C^{\infty }([-\epsilon _{0},\epsilon _{0}],{\mathfrak {so}}(3))}
shu kabi
Θ
t
(
0
)
=
0
{\displaystyle \Theta ^{t}(0)=0}
va har biri uchun
ϵ
∈
[
−
ϵ
0
,
ϵ
0
]
,
{\displaystyle \epsilon \in [-\epsilon _{0},\epsilon _{0}],}
Γ
(
t
,
ϵ
)
=
γ
(
t
)
exp
(
Θ
t
(
ϵ
)
)
.
{\displaystyle \Gamma (t,\epsilon )=\gamma (t)\exp(\Theta ^{t}(\epsilon )).}
Taʼrifga koʻra,
δ
γ
(
t
)
=
(
d
d
ϵ
(
γ
(
t
)
exp
(
Θ
t
(
ϵ
)
)
)
)
|
ϵ
=
0
=
γ
(
t
)
d
Θ
t
(
ϵ
)
d
ϵ
|
ϵ
=
0
.
{\displaystyle \delta \gamma (t)=\left({\frac {d}{d\epsilon }}{\Bigl (}\gamma (t)\exp(\Theta ^{t}(\epsilon )){\Bigr )}\right){\Biggl |}_{\epsilon =0}=\gamma (t){\frac {d\Theta ^{t}(\epsilon )}{d\epsilon }}{\Biggl |}_{\epsilon =0}.}
Chunki, baʼzi funksiyalar uchun
σ
:
[
t
0
,
t
1
]
→
s
o
(
3
)
,
{\displaystyle \sigma :[t_{0},t_{1}]\to {\mathfrak {so}}(3),}
Θ
t
(
ϵ
)
=
ϵ
σ
(
t
)
+
o
(
ϵ
)
{\displaystyle \Theta ^{t}(\epsilon )=\epsilon \sigma (t)+o(\epsilon )}
, kabi
ϵ
→
0
{\displaystyle \epsilon \to 0}
,
δ
γ
(
t
)
=
γ
(
t
)
σ
(
t
)
∈
T
γ
(
t
)
S
O
(
3
)
.
{\displaystyle \delta \gamma (t)=\gamma (t)\sigma (t)\in T_{\gamma (t)}SO(3).}
Manbalar
uz.wikipedia.org