่ซพ็‰นๅฎš็†่ˆ‡ๅฐ็จฑๆ€ง-ๆ™‚็ฉบ่ˆ‡่ƒฝ้‡ๅ‹•้‡ๅฎˆๆ†

ๅœจ่จŽ่ซ–ๆ™‚็ฉบๅฐ็จฑๆ€งๆ™‚๏ผŒๆˆ‘ๅ€‘ๅ›บๅฎš็‰ฉ็†้‡\(\phi \)ไธๅ—่ฎŠๅˆ†ๅฝฑ้Ÿฟ๏ผŒ\(\Psi=0\)ๆ™‚๏ผŒๅชๅ‰ฉไธ‹ $$\partial _\mu \left[\left({\partial \mathcal{L}\over\partial \left(\partial _\mu \phi\right) } \partial _\alpha \phi - \delta ^\mu _\alpha L \right) ฮง^\alpha \right]=0$$ ๆˆ‘ๅ€‘ๅฎš็พฉ่ƒฝ้‡ๅ‹•้‡ๅผต้‡Energy momentum tensor \(T^{\mu\nu}\) $$T^{\mu\nu}\equiv {{\partial L}\over{\partial \left( \partial _\nu \phi \right)}} \partial ^\mu \phi -\eta ^{\mu\nu} L$$ ๅฎˆๆ†ๆต\(\partial _\nu T^{\mu\nu}=0\)ๅˆ†ๅˆฅไปฃ่กจ $$\partial _\nu T^{0\nu}=0่ƒฝ้‡ๅฎˆๆ†$$ $$\partial _\nu T^{i\nu}=0ๅ‹•้‡ๅฎˆๆ†$$ ๅœจ็›ธๅฐ่ซ–ไธญ๏ผŒไธ€ๅ€‹่‡ช็”ฑ็ฒ’ๅญ็š„่ƒฝ้‡ๅ‹•้‡ๅผต้‡็‚บ $$T^{\mu\nu}\equiv mU^\mu U^\nu$$ ๅ…ถไธญ ๅฐๅคๅ…ธ็ฒ’ๅญไพ†่ชช๏ผŒๅ›žๅˆฐ้ž็›ธๅฐ่ซ–ๆ€ง็š„่ฉฑ๏ผŒ็ฒ’ๅญ็š„่ปŒ่ทก\(q\)ๅช่ˆ‡\(t\)ๆœ‰้—œ๏ผŒ $$T^{\mu\nu}=T^{00}={\partial L\over \partial \dot{q} } \dot{q} -L=H$$ \(T^{00}\)ๅฐฑๆ˜ฏHamiltonian๏ผŒ่ƒฝ้‡ๅฎˆๆ†ๅพ‹ $$\partial _\nu T^{0\nu}=\partial _0 T^{00}={dH \over dt}=0$$ ่ƒฝ้‡ไธ้šจๆ™‚้–“ๆ”น่ฎŠ๏ผ
ๅคๅ…ธ็š„่ซพ็‰นๅฎš็†่กจ่ฟฐ็‚บ

$$ๅคๅ…ธ้ž็›ธๅฐ่ซ–ๆ€ง$$$$็›ธๅฐ่ซ–ๆ€ง$$
$$S=\int Ldt $$$$S=\int \mathcal{L}d^4 x$$
$$N\equiv {\partial L\over\partial \dot{q} _i } Q_i-\left({\partial L\over\partial \dot{q} _i } \dot{q} _i-L\right)T$$ $$N^\mu \equiv {\partial \mathcal{L}\over\partial (\partial _\mu \alpha^\nu ) } \Psi^\nu-\left({\partial \mathcal{L}\over\partial (\partial _\mu \alpha^\nu ) } \partial _\alpha \alpha^\nu- \delta ^\mu _\alpha L\right) ฮง^\alpha $$
ๅคๅ…ธไธญ๏ผŒๆ™‚้–“\(t\)ไฝ”็š„่ง’่‰ฒๅฐๆ‡‰ๆ–ผๆ™‚็ฉบ\(x^\mu\) ๏ผŒๆ‰€ไปฅๆ™‚้–“็š„ๅฐ็จฑๆ€งไปฃ่กจ่ƒฝ้‡ๅฎˆๆ†๏ผŒๅ…ถ้ค˜็š„ๅฐ็จฑๆ€ง\(Q_i\)ๅฏๆ˜ฏ็ฉบ้–“\(q_i\)ใ€่ง’ๅบฆ\(\theta_i\)๏ผŒๅˆ†ๅˆฅๆ˜ฏ $${\partial L\over\partial \dot{q} _i } ๅ‹•้‡ๅฎˆๆ†$$ $${\partial L\over \partial \dot{\theta}_i } ่ง’ๅ‹•้‡ๅฎˆๆ†$$

็Ž‹ๅŸนๅ„’ ่ซพ็‰นๅฎš็†