由哈密頓原理得到哈密頓正則方程

哈密頓原理即:

哈密頓原理即保守的、完整的力學體系在相同時間內, 由某一初位形轉移到另一已知位形的一切可能運動中, 真實運動的主函數具有穩定值, 即對於真實運動來講, 主函數的變分等於零. 哈密頓原理與牛頓運動定律是等價的原理.

(這裡直接給出了,可以考慮改天試從E-L equation推導出來)

#其中作用量被定義為 $[S=int_{{{t}_{1}}}^{{{t}_{2}}}{Ldt}]$

已知哈密頓量: $[H=sumlimits_{alpha }{frac{partial L}{partial {{{dot{q}}}_{alpha }}}{{{dot{q}}}_{alpha }}}-L]$ 且記 $[frac{partial L}{partial {{{dot{q}}}_{alpha }}}={{p}_{alpha }}]$

(#注意上面這個式子成立的前提是在保守系下)

則哈密頓原理可寫為: $[delta S=delta int_{{{t}_{1}}}^{{{t}_{2}}}{Ldt}=int_{{{t}_{1}}}^{{{t}_{2}}}{delta left[ sumlimits_{alpha }{{{p}_{alpha }}{{{dot{q}}}_{alpha }}}-H ight]dt=0}]$

$[Rightarrow int_{{{t}_{1}}}^{{{t}_{2}}}{left[ sumlimits_{alpha }{left( {{{dot{q}}}_{alpha }}delta {{p}_{alpha }}+{{p}_{alpha }}delta {{{dot{q}}}_{alpha }} ight)}-delta H ight]dt}=0]$

$[Rightarrow int_{{{t}_{1}}}^{{{t}_{2}}}{left[ sumlimits_{alpha }{left( {{{dot{q}}}_{alpha }}delta {{p}_{alpha }}-{{{dot{p}}}_{alpha }}delta {{q}_{alpha }} ight)}+sumlimits_{alpha }{frac{d}{dt}left( {{p}_{alpha }}delta {{q}_{alpha }} ight)}-delta H ight]dt}=0]$ $[Rightarrow sumlimits_{alpha }{left. {{p}_{alpha }}delta {{q}_{alpha }} ight|_{{{t}_{1}}}^{{{t}_{2}}}}+int_{{{t}_{1}}}^{{{t}_{2}}}{left[ sumlimits_{alpha }{left( {{{dot{q}}}_{alpha }}delta {{p}_{alpha }}-{{{dot{p}}}_{alpha }}delta {{q}_{alpha }} ight)}-delta H ight]dt}=0]$

其中 $[sumlimits_{alpha }{left. {{p}_{alpha }}delta {{q}_{alpha }} ight|_{{{t}_{1}}}^{{{t}_{2}}}}=0]$ 是顯然的,因為系統在初末時刻的狀態是給定的,所以變分為0.

$[Rightarrow int_{{{t}_{1}}}^{{{t}_{2}}}{left[ sumlimits_{alpha }{left( {{{dot{q}}}_{alpha }}delta {{p}_{alpha }}-{{{dot{p}}}_{alpha }}delta {{q}_{alpha }} ight)}-sumlimits_{alpha }{left( frac{partial H}{partial {{p}_{alpha }}}delta {{p}_{alpha }}+frac{partial H}{partial {{q}_{alpha }}}delta {{q}_{alpha }} ight)} ight]dt}=0]$

$[Rightarrow int_{{{t}_{1}}}^{{{t}_{2}}}{sumlimits_{alpha }{left[ left( {{{dot{q}}}_{alpha }}-frac{partial H}{partial {{p}_{alpha }}} ight)delta {{p}_{alpha }}-left( {{{dot{p}}}_{alpha }}+frac{partial H}{partial {{q}_{alpha }}} ight)delta {{q}_{alpha }} ight]}dt}=0]$

其中 $[delta {{p}_{alpha }}]$ $[delta {{q}_{alpha }}]$ 都是獨立變數的變分,所以只能是有下面關係式成立:

$[left{ egin{align} & frac{partial H}{partial {{p}_{alpha }}}={{{dot{q}}}_{alpha }} \ & frac{partial H}{partial {{q}_{alpha }}}=-{{{dot{p}}}_{alpha }} \ end{align} ight.]$ #又見到了哈密頓正則方程