1-darajali erkinlik.2-darajali erkinlik.Egri chiziq bilan chegaralangan *m* massali zarra uchun **C** cheklash kuchi va virtual siljish *δ***r**. Natijada cheklanmagan kuch **N** dir. Virtual siljishning komponentlari cheklash tenglamasi bilan bog'langan.
Analitik mexanikada amaliy matematika va fizikaning bir boʻlimi, **virtual siljish** (yoki **cheksiz kichik oʻzgarish**) {\displaystyle \delta \gamma } Mexanik tizimning trayektoriyasi *gipotetik* (shuning uchun *virtual* atamasi) haqiqiy traektoriyadan qanday qilib biroz chetga chiqishi mumkinligini koʻrsatadi. {\displaystyle \gamma } tizimning cheklovlarini buzmasdan . Har lahza uchun {\displaystyle t,}{\displaystyle \delta \gamma (t)} nuqtadagi konfiguratsiya fazosiga tangensial vektordir {\displaystyle \gamma (t).} Vektorlar {\displaystyle \delta \gamma (t)} qaysi yoʻnalishlarni koʻrsating {\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 {\displaystyle \gamma } berilgan nuqtadan oʻting {\displaystyle \mathbf {q} } berilgan vaqtda {\displaystyle \tau ,} yaʼni {\displaystyle \gamma (\tau )=\mathbf {q} ,} keyin {\displaystyle \delta \gamma (\tau )=0.}

Belgilari
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Mayli {\displaystyle M} mexanik tizimning konfiguratsiya maydoni boʻlsin, {\displaystyle t\_{0},t\_{1}\in \mathbb {R} } vaqt lahzalari boʻlsin, {\displaystyle q\_{0},q\_{1}\in M,}{\displaystyle C^{\infty }[t\_{0},t\_{1}]} ustidagi silliq funksiyalardan iborat.

Cheklovlar {\displaystyle \gamma (t\_{0})=q\_{0},}{\displaystyle \gamma (t\_{1})=q\_{1}} Bu yerda faqat tasvir uchun. Amalda, har bir alohida tizim uchun individual cheklovlar toʻplami talab qilinadi.

Taʼrifi
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Har bir yoʻl uchun {\displaystyle \gamma \in P(M)} va {\displaystyle \epsilon \_{0}>0,} *ning oʻzgarishi* {\displaystyle \gamma } funksiya hisoblanadi {\displaystyle \Gamma :[t\_{0},t\_{1}]\times [-\epsilon \_{0},\epsilon \_{0}]\to M} shunday, har bir uchun {\displaystyle \epsilon \in [-\epsilon \_{0},\epsilon \_{0}],}{\displaystyle \Gamma (\cdot ,\epsilon )\in P(M)} va {\displaystyle \Gamma (t,0)=\gamma (t).} *Virtual joy almashish* {\displaystyle \delta \gamma :[t\_{0},t\_{1}]\to TM}{\displaystyle (TM} ning tangens toʻplamidir {\displaystyle M)} oʻzgarishiga mos keladi {\displaystyle \Gamma } har biriga quyidagini belgilaydi {\displaystyle t\in [t\_{0},t\_{1}]} tangens vektori

{\displaystyle \delta \gamma (t)={\frac {d\Gamma (t,\epsilon )}{d\epsilon }}{\Biggl |}\_{\epsilon =0}\in T\_{\gamma (t)}M.}

Tangens xaritasi nuqtai nazaridan,

{\displaystyle \delta \gamma (t)=\Gamma \_{\*}^{t}\left({\frac {d}{d\epsilon }}{\Biggl |}\_{\epsilon =0}\right).}

Bu yerda {\displaystyle \Gamma \_{\*}^{t}:T\_{0}[-\epsilon ,\epsilon ]\to T\_{\Gamma (t,0)}M=T\_{\gamma (t)}M} ning tangens xaritasi hisoblanadi {\displaystyle \Gamma ^{t}:[-\epsilon ,\epsilon ]\to M,} bu yerda {\displaystyle \Gamma ^{t}(\epsilon )=\Gamma (t,\epsilon ),} va {\displaystyle \textstyle {\frac {d}{d\epsilon }}{\Bigl |}\_{\epsilon =0}\in T\_{0}[-\epsilon ,\epsilon ].}

Xususiyatlari
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* *Koordinatali vakillik.* Agar {\displaystyle \{q\_{i}\}\_{i=1}^{n}} ixtiyoriy diagrammadagi koordinatalar {\displaystyle M} va {\displaystyle n=\mathop {\rm {dim}} M,} keyin

{\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)}.}
* Agar, bir zumda {\displaystyle \tau } va har bir {\displaystyle \gamma \in P(M),}{\displaystyle \gamma (\tau )={\text{const}},} keyin, har biri uchun {\displaystyle \gamma \in P(M),}{\displaystyle \delta \gamma (\tau )=0.}

* Agar {\displaystyle \textstyle \gamma ,{\frac {d\gamma }{dt}}\in P(M),} keyin {\displaystyle \delta {\frac {d\gamma }{dt}}={\frac {d}{dt}}\delta \gamma .}

Misollar
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### R 3 dagi erkin zarracha

Yagona zarracha erkin harakatlanadi {\displaystyle \mathbb {R} ^{3}} 3 erkinlik darajasiga ega. Konfiguratsiya maydoni {\displaystyle M=\mathbb {R} ^{3},} va {\displaystyle P(M)=C^{\infty }([t\_{0},t\_{1}],M).} Har bir yoʻl uchun {\displaystyle \gamma \in P(M)} va variatsiya {\displaystyle \Gamma (t,\epsilon )} ning {\displaystyle \gamma ,} noyobi mavjud {\displaystyle \sigma \in T\_{0}\mathbb {R} ^{3}} shu kabi {\displaystyle \Gamma (t,\epsilon )=\gamma (t)+\sigma (t)\epsilon +o(\epsilon ),} kabi {\displaystyle \epsilon \to 0.} Taʼrifga koʻra,

{\displaystyle \delta \gamma (t)=\left({\frac {d}{d\epsilon }}{\Bigl (}\gamma (t)+\sigma (t)\epsilon +o(\epsilon ){\Bigr )}\right){\Biggl |}\_{\epsilon =0}}

olib keladi

{\displaystyle \delta \gamma (t)=\sigma (t)\in T\_{\gamma (t)}\mathbb {R} ^{3}.}

### Sirtdagi erkin zarralar

{\displaystyle N} ikki oʻlchovli sirtda erkin harakatlanadigan zarralar {\displaystyle S\subset \mathbb {R} ^{3}} bor {\displaystyle 2N} erkinlik darajasi. Bu erda konfiguratsiya maydoni

{\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 {\displaystyle \mathbf {r} \_{i}\in \mathbb {R} ^{3}} ning radius vektori {\displaystyle i^{\text{th}}} zarracha. Bundan quyidagi kelib chiqadi

{\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 {\displaystyle \gamma \in P(M)} radius vektorlari yordamida tasvirlanishi mumkin {\displaystyle \mathbf {r} \_{i}} har bir alohida zarrachaning, yaʼni

{\displaystyle \gamma (t)=(\mathbf {r} \_{1}(t),\ldots ,\mathbf {r} \_{N}(t)).}

Bu shuni anglatadiki, har bir kishi uchun {\displaystyle \delta \gamma (t)\in T\_{(\mathbf {r} \_{1}(t),\ldots ,\mathbf {r} \_{N}(t))}M,}

{\displaystyle \delta \gamma (t)=\delta \mathbf {r} \_{1}(t)\oplus \ldots \oplus \delta \mathbf {r} \_{N}(t),}

bu yerda {\displaystyle \delta \mathbf {r} \_{i}(t)\in T\_{\mathbf {r} \_{i}(t)}S.} Baʼzi mualliflar buni shunday ifodalaydilar

{\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 {\displaystyle M=SO(3),} 3 oʻlchovli maxsus ortogonal guruh (aks holda 3D aylanish guruhi deb nomlanadi) va {\displaystyle P(M)=C^{\infty }([t\_{0},t\_{1}],M).} Biz standart belgidan foydalanamiz {\displaystyle {\mathfrak {so}}(3)} barcha egri-simmetrik uch oʻlchovli matritsalarning uch oʻlchovli chiziqli fazosiga murojaat qilish. Eksponensial xarita {\displaystyle \exp :{\mathfrak {so}}(3)\to SO(3)} mavjudligini kafolatlaydi {\displaystyle \epsilon \_{0}>0} Shunday qilib, har bir yoʻl uchun {\displaystyle \gamma \in P(M),} uning oʻzgarishi {\displaystyle \Gamma (t,\epsilon ),} va {\displaystyle t\in [t\_{0},t\_{1}],} oʻziga xos yoʻl bor {\displaystyle \Theta ^{t}\in C^{\infty }([-\epsilon \_{0},\epsilon \_{0}],{\mathfrak {so}}(3))} shu kabi {\displaystyle \Theta ^{t}(0)=0} va har biri uchun {\displaystyle \epsilon \in [-\epsilon \_{0},\epsilon \_{0}],}{\displaystyle \Gamma (t,\epsilon )=\gamma (t)\exp(\Theta ^{t}(\epsilon )).} Taʼrifga koʻra,

{\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 {\displaystyle \sigma :[t\_{0},t\_{1}]\to {\mathfrak {so}}(3),}{\displaystyle \Theta ^{t}(\epsilon )=\epsilon \sigma (t)+o(\epsilon )}, kabi {\displaystyle \epsilon \to 0} ,

{\displaystyle \delta \gamma (t)=\gamma (t)\sigma (t)\in T\_{\gamma (t)}SO(3).}

Manbalar
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1. ↑  Takhtajan, Leon A. „Part 1. Classical Mechanics",*. Classical Field Theory* (PDF), Department of Mathematics, Stony Brook University, Stony Brook, NY, 2017.
2. ↑  Goldstein, H.*. Classical Mechanics*, 3rd, Addison-Wesley, 2001 — 16 bet. ISBN 978-0-201-65702-9.
3. ↑  Torby, Bruce „Energy Methods",*. Advanced Dynamics for Engineers*, HRW Series in Mechanical Engineering. United States of America: CBS College Publishing, 1984. ISBN 0-03-063366-4.