????? ??????? ?? ????? ?????? ????????? ??? ?????? ??????? ?????? ???? ????? ?? ??????? ?????? ?? ??????? ??????? . ?? ?? ????? ??????? ??????? ???? ???? ???? ??????? ??????? ??? ???? ???? ??????? ??????, ?? ???? ?? ???? ???????? ?????? ???? (???? ?????? ???????). ?????? ??????? ?? ??? ?????? ?????? ?????? ??????'????? ?????, ???? ?????? ?? ??????? ??????? ???? ???? ???? ?????? ???, ??????? ???? ????? ????? ??????? ????? ???????. ????? ? ???????? ????? ? ?????? ???????? ????? ????? ?? ????? ?????? ?????????.

???????? [ ????? ??? ???? | ????? ]

??????? ?? ? ??????'??? ?? ?????? ????? ???

$${\displaystyle \ L(x,{\dot {x}},t)=T-U=E_{k}(x,{\dot {x}},t)-E_{p}(x,{\dot {x}},t)}$$

????? ?? ??? ??? ?????? ???????????? ??? ??????? ???????'??? ???? ?? ?????? ?? ? ?????????? ?? ???

$${\displaystyle \ H=p{\dot {x}}-L}$$

???? ??????????? ????? ???? ?? ? ?????? ?????? ?? ??????.

? ????? , ???? ? ??????? ?? ???????'??? ????? ???? ?? ?????? ?????? ????? ?? ??? :

$${\displaystyle S=\int _{t_{1}}^{t_{2}}{L(x,{\dot {x}},t)\ dt}}$$

????? ??????? ???? ??????? ???? ?????? ?? ?????? ???? ?????????, ????? ?????? ????? ?

$${\displaystyle \ \delta S=0}$$

??????? ??????? ?? ???? ?? ??? ???? ????? ?? ${\displaystyle S}$ ??? ??????? ???? ?? ${\displaystyle x}$.

?? ??, ??????? ${\displaystyle \delta S=0}$ ????? ????? ???? ?? ???? ???????? ?? ??? ????? ?????? ??? ???? ????? ???? ${\displaystyle t_{1}}$ ????? ???? ???? ${\displaystyle t_{2}}$ ????? ??????? ????? ??? ?? ???? ????????? ???? ??? ?????? ???? ??????? ???????? ?????? ???????. ?????? ????? ??? ????? ????? ???? ???? ????? ??????? ???? ????????. ?? ???? ?????? ${\displaystyle t_{2}-t_{1}}$ ??? ????? ??? ?? ????? ?????? ??? ??????? ?????.

???? ?????? ????????? ?? ???? ??? ?????? (????? ?????? ????? ?? ?????? ??????) ???? ?????? ????? ????? ?? ????? ???????? ?????? ??? ??????? ???? ?????? ${\displaystyle x(t)}$ ??? ????????? ??"? ????? ?? ??????? ???????????? ????:

$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)-{\frac {\partial L}{\partial x}}=0}$$

?????? ?? ????? ? ?????? ????? ??????' , ????? ????? ?? n ????? ????, ?? ???? ???? ?????? ?? ??????? ??? (??? ?????? ????? ?? n ??????? ???????????? ???????).

????? - ???? ???? ???? ?? ?????? [ ????? ??? ???? | ????? ]

???? ???? ?????? ?? ??? ????? ?? ???????? ???? ??-????, ??? ??????:

$${\displaystyle \ F(x)=-{\vec {\nabla }}V(x)=-V'(x)}$$

???????'??? ??? ????:

$${\displaystyle \ L={\frac {1}{2}}m{\dot {x}}^{2}-V(x)}$$

???? ???????? ?????-??????':

$${\displaystyle \ {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)={\frac {\mathrm {d} }{\mathrm {d} t}}\left(m{\dot {x}}\right)=m{\ddot {x}}}$$

${\displaystyle \ {\frac {\partial L}{\partial x}}=-V'(x)}$ ( ??? ?????? )

????"? ??????

${\displaystyle \ m{\ddot {x}}=-V'(x)=F(x)}$

??

${\displaystyle \ F=ma}$

???? ???? ???? ?? ?????? . ???? ?? ???? ?? ??? ?????? ??????????? ???? ??? .

???? ?????? ??????? [ ????? ??? ???? | ????? ]

???? ???? ????????? ????? ?????? ??????? ???? ???? ????' ???? ?????? ^[1]

???? ???? ????? ?? ??? ?????? ????? ??? ?? ??? ?? ?????-???.

?????? ???? ??? ?????? ???? ?? ???? ?? ?? ???????? ????? ???? ???, ??? ????? ????? ??? ???? ??? ??? ????????? ????.

????? ????? ????? ${\displaystyle \sigma }$ ???? ????? ??????, (???? ??????, ?????? ?? ??? ????? ??????) ????????? ????? ?? ???????:

????? ${\displaystyle {\hat {u}}}$ ????? ????? ?????? ??????.

???? 2 ?????? ?? ??????:

????

????? ?? ???' ?????? ????????? ?????:

?????: ???? ?????? ???? ?????, ???? ??????? ???? ?????? ?????? ?????? ??? ? ???? ???? , ???????? ?? ????? (distance) ???? ????? ??????? ?? ??????'??? ??? ????? ??? 2 ??????? ????? ?????. ???? ??????? ????? ????? ???? ??? ??????? ?????, ?????? ?????, ????? ???? ??????? ????? ????? ?????? ??? ???? ??? ???, ??? ?? ?? ????? ???? ????? ?????? ?? ???? ??0 (?? ?? ? ??? ?????? ?? ?????? ).????? ?? ???? ??? ???? ??????? ????? ?????, ?? ??? ????? ??? ??????? ?? ??? ????: ??? ???? ??? ??? ????? ????? ???? ??? ??? ?????? ??????. ???? ????? ???? ??? ???? ???? ?????? ?? ???? ?????? ?? ?????? ??? ???? ???????, ???? ????? ?????? ??????? ?? ?????? ?????? ???? ?? ??? ????, ?? ?? ????? ????? ???? ?? ??????? ?? ???? ?????? ????? (????? ??????? ?? ?????? ????? ?????).

?? ??? ???? ?? ??? ????' ??"? ????? ????? ?????? ????? ?????' ?? ??? ???' ????? ????????? ???????:

${\displaystyle S[{\textbf {x}}(\sigma )]\equiv \int _{\sigma _{1}}^{\sigma _{2}}L\left({\textbf {x}}(\sigma ),{\frac {d{\textbf {x}}(\sigma )}{d\sigma }}\right)d\sigma }$

?S ???? ????? ????????.

???? ??? ???? ?? ? ???????? ?? ?????? ????? ??? ???', ?????, ?? ???? ???? ?????? ????? ????? ????? ???:

${\displaystyle \left.{\frac {d}{d{\varepsilon }}}\right|_{\varepsilon =0}S[{\textbf {x}}(\sigma )+\varepsilon h(\sigma )]=0}$

??? ?????? ?? ?????? ??????? ??????? ?????? ??????? ???? ?????? ??????.

??? ?? [ ????? ??? ???? | ????? ]

• ????? ????????
• ??????'???
• ????? ?'?????

??????? ???????? [ ????? ??? ???? | ????? ]

• ?????? ??????? ??????? ????? ?? ????' ?????? ?????? ?- MIT

????? ?????? [ ????? ??? ???? | ????? ]