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 ${\textstyle M\equiv \sum _{i}m_{i}}$ tō sī chit-liōng chóng-hô, ${\textstyle \sum _{i}m_{i}\mathbf {v} _{i}}$ tú-á-hó tō sī ${\textstyle \mathbf {P} }$, 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

Siông-sè chhiáⁿ khoàⁿ: ūn-tōng-liōng siú-hêng tēng-lí

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, ${\textstyle \gamma \equiv \left(1-u^{2}/c^{2}\right)^{-1/2}}$ 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 ${\textstyle \tau }$ 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

Siông-sè chhiáⁿ khoàⁿ: sì-ūn-tōng-liōng

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

$\mathbf {U} ={\frac {{\text{d}}\mathbf {X} }{{\text{d}}\tau }}={\frac {{\text{d}}\mathbf {X} }{{\text{d}}t}}{\frac {{\text{d}}t}{{\text{d}}\tau }}=\gamma \left(c,\mathbf {u} \right),$

${\textstyle \mathbf {u} ={\frac {{\text{d}}\mathbf {x} }{{\text{d}}t}}}$ 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 $E=\gamma mc^{2}$ 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:

$E^{2}=c^{2}\mathbf {p} \cdot \mathbf {p} +\left(mc^{2}\right)^{2}.$

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.