Ūn-tōng-liōng

**Soàn-sêng ūn-tōng-liōng** (Hàn-jī: **線型運動量**; Eng-gí: **linear momentum**), ia̍h sī kán-chheng **ūn-tōng-liōng** (**運動量;** momentum), ê chi̍t chióng le̍k-ha̍k ê bu̍t-lí-liōng, i biâu-su̍t bu̍t-thé sóa-tāng ê chêng-hêng. 

It-poaⁿ só͘ kóng ê ūn-tōng-liōng sī hiòng-liōng, kòan-sì iōng \mathbf{p} lâi tāi-piáu, án-ne i ê tōa-sè tō sī p=\left|\mathbf{p}\right|, hong-hiòng tō sī \hat{\mathbf{p}}=\frac{\mathbf{p}}{p}. 

Newton tē-jī ūn-tōng tēng-lu̍t kóng, bu̍t-thé ūn-tōng-liōng ê piàn-hòa-lu̍t pêⁿ-pêⁿ i siū-tio̍h ê si tī i sin-khu-téng ê chēng-la̍t.

: \mathbf{F}= \frac{\text{d}\mathbf{p}}{\text{d}t} .

Ūn-tōng-liōng chīn-chāi bu̍t-thé ê chham-khó-hē, put-jî-kò, tùi jīm-hô ê chham-khó-hē lâi khòaⁿ, ūn-tōng-liōng sī **siú-hêng-liōng** (**守恆量**; conserved), che tō sī kóng, siat-sú chi̍t-ê ba̍t-hē-thóng bô khì hō͘ gōa-le̍k éng-hióng--tio̍h, nā-án-ne i ê chóng ūn-tōng-liōng lóng bē ū piàn-hòa. Tī te̍k-sû siong-tùi-lūn, tiān-tōng-le̍k-ha̍k, liōng-chú le̍k-ha̍k, liōng-chú tiûⁿ-lūn, it-poaⁿ siong-tùi-lūn tang-tiong, ūn-tōng-liōng sui-jiân lóng ū bô-kâng ê hêng-sek, in chóng--sī siú-hêng-liōng. Kî-si̍t, ūn-tōng-liōng sī chi̍t chióng khong-kan kap sî-kan ê ki-pún tùi-thīn: **sóa-ūi tùi-thīn** (**徙位對伨**; translational symmetry.)

## Newton Le̍k-ha̍k

### Chit-tiám
Tī Newton Le̍k-ha̍k tiong, ko͘-chi̍t-ê chit-tiám ê ūn-tōng-liōng chiū tēng-gī chòe i ê chit-liōng m kap sok-tō͘ \mathbf{v} ê hē-só͘-chek (係數積):

: \mathbf{p}= m\mathbf{v} .

Tī it-chhù-goân (1D) ê chêng-hêng tiong, \mathbf{p} kap \mathbf{v} iōng si̍t-sò͘ piáu-sī tō hó:

: p= mv,

in ê chèng-hū hû-hō tāi-piáu hong-hiòng. Ūn-tōng-liōng ê tan-ūi sī chit-liōng ê tan-ūi sêng sok-tō͘ ê tan-ūi. Phì-jû-kóng tī SI tan-ūi, chit-liōng ê tan-ūi sī kong-kin (kg), sok-tō͘--ê sī kong-chhioh múi bió (m/s), nā án-ne ūn-tōng-liōng--ê tō sī "kong-kin kong-chhioh múi bió (kg⋅m/s)." Tī cgs tan-ūi, chit-liōng ê tan-ūi sī kong-khek (g), sok-tō͘--ê sī kong-hun múi bió (cm/s), nā án-ne ūn-tōng-liōng--ê tō sī "kong-khek kong-hun múi bió (g⋅cm/s)." 

Siat-sióng chi̍t tâi chhia ē-ēng-tit khòaⁿ-chòe sī chit-liōng 2000 kong-kin ê chit-tiàm, taⁿ chit tâi chhia í 36 km/h, iā tō sī 10 m/s, ê sok-tō͘ ǹg sai sái. Kiám-chhái kā tang-hong tèng chòe +x hong-hiòng, chit tâi chhia ê ūn-tōng-liōng chiū-án-ne sī (2000 kg) × (−10 m/s) = 20000 kg⋅m/s, hū-hō tāi-piáu hong-hiòng ǹg sai.

Che Newton tē-jī ūn-tōng tēng-lu̍t

: \mathbf{F}= \frac{\text{d}\mathbf{p}}{\text{d}t} .

### Chit-tiám Hē-thóng
Chi̍t tīn chit-tiám ē-sái cho͘-sêng chit-tiám hē-thóng (質點系統). Chit-ê hē-thóng ê ūn-tiōng-liōng tō sī ta̍k-ê chit-tiám ūn-tiōng-liōng ê hiòng-liōng-hô (向量和):

: \mathbf{P} = \sum_i \mathbf{p}_i = \sum_i m_i\mathbf{v}_i = m_1\mathbf{v}_1+ m_2\mathbf{v}_2 + \cdots,

kî-tiong ē-thiam-jī (下添字) i sī chit-tiám ê phiau-thâu (標頭), i = 1,2,3,\ldots. 

Chit-tiám hē-thóng ū chi̍t-ē chit-sim. I ê ūi-tì sī \mathbf{r}_\text{cm} ta̍k-ê chi̍t-tiám ê ūi-tì \mathbf{r}_i í chit-liōng m_i ûi khoân-sò͘ ê ka-khoân-hô (加權和):

: \mathbf{r}_\text{cm} = \frac{\sum_i m_i\mathbf{r}_i}{\sum_i m_i} = \frac{m_1\mathbf{r}_1+ m_2\mathbf{r}_2 + \cdots}{m_1+m_2+\cdots},

chia ê ē-thiam-jī \text{cm} sī chit-sim (center-of-mass) ê ì-sù. Chit-sim sok-tō͘ \boldsymbol{v}_\text{cm} sī \boldsymbol{r}_\text{cm} ê bî-hun:

: \boldsymbol{v}_\text{cm} = \dot\boldsymbol{v}_\text{cm} = \frac{\sum_i m_i\dot\boldsymbol{r}_i}{\sum_i m_i} = \frac{\sum_i m_i\boldsymbol{v}_i}{\sum_i m_i},

tān-sī M \equiv \sum_i m_i tō sī chit-liōng chóng-hô, \sum_i m_i\mathbf{v}_i tú-á-hó tō sī \mathbf{P}, só͘-í,

: \mathbf{P} = M\mathbf{v}_\text{cm} .

Che hām ko͘-chi̍t-ê chit-tiàm ê ūn-tōng-liōng ū kāng-khoán ê hêng-sek. Che piáu-sī kóng, chí-iàu chiong chit-tiàm hē-thóng iōng i ê chit-sim lâi tāi-piáu, ko͘-chit-tiàm ê Newton le̍k-ha̍k tùi i mā ē ha̍h-ēng.

### Siú-hêng

### Sio-lòng

## Kóng-gī-hòa

### Lagrange Le̍k-ha̍k

### Hamilton Le̍k-ha̍k

## Liân-sio̍k-thè

## Siong-tùi-lūn
Ti te̍k-sû siong-tùi-lūn tiong, Lorentz piàn-ōaⁿ sī nn̄g-ê koàn-sèng chham-khó-hē chi-kan ê chò-piau piàn-ōaⁿ (pau-koat ūi-tì hām sî-kan.) Lorentz piàn-ōaⁿ chi-hā ê chit-liōng m̄-sī put-piān-liōng, tian-tò sī sûi chham-khó-hē chi-kan ê siong-tùi sok-tō͘ \mathbf{v} lâi piàn-hòa:

: \gamma m = \frac{m}{\sqrt{1-u^2/c^2}},

chia ê u=|\mathbf{u}|, c sī kng-sok, \gamma \equiv \left(1-u^2/c^2\right)^{-1/2} kiò-chòe Lorentz in-chú; m kiò-chòe put-piàn chit-liōng, i sī bu̍t-thé chit-liōng tiong bē siū-tio̍h Lorentz piàn-ōaⁿ éng-hióng ê pō͘-hūn.

Chi̍t-ê koan-chhat-chiá khiā tī koàn-sèng-hē, teh khòaⁿ lēng-gōa chi̍t-ê koàn-sèng-hē téng-koân ê chit-tiàm. Nā chit-tiàm ū put-piàn chit-liōng m_0, koan-chhat-chiá khòaⁿ-tio̍h ê **siong-lúi-lūn-sèng ūn-tōng-liōng** (相對論性運動量, relativistic momentum) tō sī

: \mathbf{p} = \gamma m\mathbf{u}=\gamma m\frac{\text{d}\mathbf{x}}{\text{d}\tau},

kî-tiong ê \tau
 sī kò͘-iú-sî-kan (固有時間, proper time), tō sī tòe-tio̍h hit-ê chit-tiàm ê sî-cheng sò͘ sǹg ê sî-kan.

### Sì-hiòng-liōng Hêng-sek

Sì-hiòng-liōng (四向量, four-vector) ê kài-liām tùi siong-tùi-lūn ê sì-chhù-goân sî-khong tiong kài hó-ēng. Chi̍t-ê sì-hiòng-liōng \mathbf{A} ū sì-ê hun-liōng, piáu-sī ê hoat-tō͘ ū chē-chē-chióng:

: \mathbf {A}=\left(A^{0},\,A^{1},\,A^{2},\,A^{3}\right)=A^{\mu},
tī chia ko͘-chi̍t-ê A^{\mu}tō ē-sái piáu-ta̍t sì ê hun-liōng, i ê téng-thiam-jī (頂添字) tio̍h ài iōng Hi-lia̍p-jī; Nā-sī \mu=0 tāi-piáu “ná chhiūⁿ sî-kan” ê hun-liōng, nā-sī \mu=1,2,3 tāi-piáu “ná chhiūⁿ khong-kan” ê hun-liōng. Siá-chòe A^{\mu} ê sî, lán tiāⁿ-tiāⁿ ē iōng Einstein koàn-lē.

Nn̄g-ê sì-hiòng-liōng \mathbf{A}=\left(A^0,A^1,A^2,A^3\right) kap \mathbf{B}=\left(B^0,B^1,B^2,B^3\right) tī Minkowski khong-kan tiong ê sûn-liōng-chek sī án-ne tēng-gī:

: \mathbf{A}\cdot\mathbf{B}=A^\mu B_\nu=A^\mu \eta_{\mu\nu}B^\nu=\left(\begin{matrix}A^0&A^1&A^2&A^3\end{matrix}\right)\left(\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}\right)\left(\begin{matrix}B^0\\B^1\\B^2\\B^3\end{matrix}\right),
\eta_{\mu\nu}  sī Minkowski metric tensor, siong-tong (−, +, +, +) ê metric signature.

Sì-ūi-tì sī 
: \mathbf{X}=(ct,\mathbf{x}),
c sī kng-sok, t sī sî-kan, \mathbf{x} sī sam-chhù-goân khong-kan ūi-tì hiòng-liōng. Sì-sok-tō͘ sī 
: \mathbf{U} = \frac{\text{d}\mathbf{X}}{\text{d}\tau} = \frac{\text{d}\mathbf{X}}{\text{d}t}\frac{\text{d}t}{\text{d}\tau}=\gamma\left(c,\mathbf{u}\right),
\mathbf{u}= \frac{\text{d}\mathbf{x}}{\text{d}t} sī sam-chhù-goân khong-kan sok-tō͘ hiòng-liōng.Só͘-í **sì-ūn-tōng-liōng** tō sī 
: \mathbf{P} = m\mathbf{U} = m\gamma\left(c,\mathbf{v}\right) = \left(\frac{E}{c},\mathbf{p}\right),
chia ê E=\gamma m c^2 sī chóng-lêng-liōng, \mathbf{p}=\gamma m\mathbf{u} sī siong-lúi-lūn-sèng ūn-tōng-liōng. Sì-ūn-tōng-liōng kap i ka-kī ê sûn-liōng-chek sī
: \|\mathbf {P} \|^{2}=P^{\mu }P_{\mu }=m^{2}U^{\mu }U_{\mu } = m^2 \gamma^2 \left(\|\mathbf{u}\|^2-c^2\right) = -mc^2=\mathbf {p} \cdot \mathbf {p}-{\frac {E^{2}}{c^{2}}},
che ē thàu-kàu lêng-liōng ūn-tōng-liōng koan-hē:
: E^2=c^2\mathbf{p}\cdot\mathbf{p}+\left(mc^2\right)^2.

## Liōng-chú Le̍k-ha̍k

## Tiān-tōng-le̍k-ha̍k

## Le̍k-sú
Tē-it ūi chèng-khak thê-chhut Ūn-tōng-liōng siú-hêng tēng-lí ê lâng sī Eng-kok sò͘-ha̍k-kā John Wallis. Tī 1670 ê chok-phín, *Mechanica sive De Motu, Tractatus Geometricus*, i kóng "m̄-koán mi̍h-kiāⁿ chi̍t-khai-sí sī khiā tiānn-tiānn ia̍h-sī teh tín-tāng, tāi-seng ê chōng-thài chhî-sio̍k chiâu-ûn," koh kóng "nā-sī le̍k-liōng khah tōa chó͘-le̍k, mi̍h-kiāⁿ tō ē khai-sí tín-tāng." Wallis kā ūn-tōng ê liōng hō-chòe *momentum*, le̍k-liōng hō-chòe *vis*. 

Newton ê *[[Chū-jiân Tiat-ha̍k ê Sò͘-ha̍k Goân-lí]]* tī 1687 chhut-pán ê sî, i ê piat-ta̍t hām Wallis chin sio-siāng ê tēng-gī. *Quantitas motus*, "ūn-tōng ê liōng", chiū I ê tēng-gī tō sī "iû sok-tō͘ kap bu̍t-chit ê liōng sò͘ lâi--ê", che ē-sái khòaⁿ-chòe sī ūn-tōng-liōng. Newton teh kái-soe Tē-jī Tēng-lu̍t ê sī sò͘ kóng ê *mutatio motus*, "ūn-tōng pia-hòa", sī chèng-pí sò͘ si ê la̍t. Só͘-í i tī hia kóng *motus* ê sî, ì-sù kî-si̍t i-pòaⁿ sī teh kóng ūn-tōng-liōng, m̄-sī ūn-tōng.