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== Formulimi matematik ==
Ekuacioni i Hamilton–Jakobit është një [[ekuacion differencial pjesor]] jolinear i rendit të parë , për një funksion <math>S(q_{1}q___L_CURLY__1__R_CURLY__,\dots,q_{N}q___L_CURLY__N__R_CURLY__; t)</math> i quajtur [[funksioni principal i Hamiltonit]]
:<math>
H\left(q_{1}q___L_CURLY__1__R_CURLY__,\dots,q_{N}q___L_CURLY__N__R_CURLY__;\frac{frac__L_CURLY__\partial S}{S__R_CURLY____L_CURLY__\partial q_{1q___L_CURLY__1}},\dots,\frac{frac__L_CURLY__\partial S}{S__R_CURLY____L_CURLY__\partial q_{Nq___L_CURLY__N}};t\right) + \frac{frac__L_CURLY__\partial S}{S__R_CURLY____L_CURLY__\partial t}t__R_CURLY__=0.
</math>
Siç përshkruhet më lart, ky ekuacion mund të derivohet nga [[mekanika e Hamiltonit]] duke trajtuar <math>S</math> (veprimin) si funksionin gjenerues për një [[transformim kanonik]] të [[mekanika e Hamiltonit|funksionit Hamiltonian klasik]] <math>H(q_{1}q___L_CURLY__1__R_CURLY__,\dots,q_{N}q___L_CURLY__N__R_CURLY__;p_{1}p___L_CURLY__1__R_CURLY__,\dots,p_{N}p___L_CURLY__N__R_CURLY__;t)</math>. Impulsi i konjuguar i korrespondon derivateve të para të <math>S</math> në lidhje me koordinata e përgjithshme
:<math>
p_{kp___L_CURLY__k} = \frac{frac__L_CURLY__\partial S}{S__R_CURLY____L_CURLY__\partial q_{kq___L_CURLY__k}}.</math>
të cilat mund të merren si më poshtë.<br />
Ndryshimi i veprimit nga një shteg tek një shteg fqinj jepet nga
:<math>\delta S=\sum_{ksum___L_CURLY__k=1}1__R_CURLY__^N\left[\frac{frac__L_CURLY__\partial L}{L__R_CURLY____L_CURLY__\partial \dot{q}_{kdot__L_CURLY__q__R_CURLY_____L_CURLY__k}}\delta q_k\right]_{t_1}___L_CURLY__t_1__R_CURLY__^{t_2}__L_CURLY__t_2__R_CURLY__+\sum_{ksum___L_CURLY__k=1}1__R_CURLY__^N\int_{t_1}int___L_CURLY__t_1__R_CURLY__^{t_2}__L_CURLY__t_2__R_CURLY__\left(\frac {__L_CURLY__\partial L}{L__R_CURLY____L_CURLY__\partial q_k} - \frac {d}{d__L_CURLY__d__R_CURLY____L_CURLY__d t} \frac {__L_CURLY__\partial L}{L__R_CURLY____L_CURLY__\partial \dot{q}_k}dot__L_CURLY__q__R_CURLY___k__R_CURLY__\right)\delta q_k \,dt.</math>
Meqenëse shtegjet e lëvizjes aktuale kënaqin [[Ekuacioni i Ojler-Lagranzhit|ekuacioni i Ojler–Lagranzhit]], integrali në <math>\delta S</math> është zero. Tek termi i parë vendosim <math>\delta q_k(t_1)=0</math>, dhe e quajmë vlerën e <math>\delta q_k(t_2)</math> me <math>\delta q_k</math>. Duke zëvendësuar <math>\partial L/\partial \dot{q}_{k}dot__L_CURLY__q__R_CURLY_____L_CURLY__k__R_CURLY__</math> me <math>p_k</math>, marrim
:<math>\delta S=\sum_{ksum___L_CURLY__k=1}1__R_CURLY__^N p_k \delta q_k</math>.
Nga ky relacion del se [[derivatet pjesore]] të [[Veprimi (fizikë)|veprimit]] në lidhje me koordinatat janë të barabarta me sasinë e lëvizjes (impulset) korrespondues.
== Notacioni ==
Për thjeshtësi përdorim variabla me tekst të trashë si <math>\mathbf{q}mathbf__L_CURLY__q__R_CURLY__</math> për të paraqitur listën e <math>N</math> [[Koordinatat e përgjithshme|koordinatave të përgjithshme]]
:<math>
\mathbf__L_CURLY__q}\ \stackrel__L_CURLY__\mathrm__L_CURLY__def}}__L_CURLY__=__R_CURLY__\ (q___L_CURLY__1__R_CURLY__, q___L_CURLY__2__R_CURLY__, \ldots, q___L_CURLY__N-1__R_CURLY__, q___L_CURLY__N__R_CURLY__)
\mathbf{q} \ \stackrel{\mathrm{def}}{=}\ (q_{1}, q_{2}, \ldots, q_{N-1}, q_{N})
</math>
:<math>
\mathbf__L_CURLY__p}\cdot \mathbf__L_CURLY__q}\ \stackrel__L_CURLY__\mathrm__L_CURLY__def}}__L_CURLY__=__R_CURLY__\ \sum___L_CURLY__k=1__R_CURLY__^__L_CURLY__N}p___L_CURLY__k}q___L_CURLY__k__R_CURLY__.
\mathbf{p} \cdot \mathbf{q} \ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N} p_{k} q_{k}.
</math>
== Derivimi ==
Çdo [[transformim kanonik]] që përfshin një funksion gjenerues të tipit të dytë <math>G_{2}G___L_CURLY__2__R_CURLY__(\mathbf{q}mathbf__L_CURLY__q__R_CURLY__,\mathbf{P}mathbf__L_CURLY__P__R_CURLY__,t)</math> na çon tek relacionet
:<math>
\qquad
{__L_CURLY__\partial G_{2G___L_CURLY__2} \over \partial \mathbf{qmathbf__L_CURLY__q}} = \mathbf{p}mathbf__L_CURLY__p__R_CURLY__, \qquad
{__L_CURLY__\partial G_{2G___L_CURLY__2} \over \partial \mathbf{Pmathbf__L_CURLY__P}} = \mathbf{Q}mathbf__L_CURLY__Q__R_CURLY__, \qquad
K = H + {__L_CURLY__\partial G_{2G___L_CURLY__2} \over \partial t}t__R_CURLY__
</math>
(Shikoni artikullin mbi [[transformimet kanonike]] për detaje të mëtejshme.)
Në mënyre që të derivojmë HJE, në zgjedhim një funksion gjenerues <math>S(\mathbf{q}mathbf__L_CURLY__q__R_CURLY__, \mathbf{P}mathbf__L_CURLY__P__R_CURLY__, t)</math> që prodhon funksion e ri Hamiltonian <math>K</math> i cili është identikisht zero. Pra, të gjitha derivatet e tija janë zero gjithashtu, dhe [[ekuacionet e Hamiltonit]] bëhen shumë të lehta
:<math>
{d__L_CURLY__d\mathbf{Pmathbf__L_CURLY__P} \over dt} = {d__L_CURLY__d\mathbf{Qmathbf__L_CURLY__Q} \over dt} = 0
</math>
pra, koordinatat dhe vrullet (vektorët e impulseve) të reja të përgjithshme janë [[konstante të lëvizjes|konstantet e lëvizjes]]. Vrullet e reja të përgjithshme <math>\mathbf{P}mathbf__L_CURLY__P__R_CURLY__</math> shpesh jepen nga <math>\alpha_{1}alpha___L_CURLY__1__R_CURLY__, \alpha_{2}alpha___L_CURLY__2__R_CURLY__, \ldots, \alpha_{Nalpha___L_CURLY__N-1}1__R_CURLY__, \alpha_{N}alpha___L_CURLY__N__R_CURLY__</math>, pra, <math>P_{mP___L_CURLY__m} = \alpha_{m}alpha___L_CURLY__m__R_CURLY__</math>.
HJE është një rezultat i transformimit të funksionit Hamiltonian <math>K</math>
:<math>
K(\mathbf{Q}mathbf__L_CURLY__Q__R_CURLY__,\mathbf{P}mathbf__L_CURLY__P__R_CURLY__,t) = H(\mathbf{q}mathbf__L_CURLY__q__R_CURLY__,\mathbf{p}mathbf__L_CURLY__p__R_CURLY__,t) + {__L_CURLY__\partial S \over \partial t} = 0.
</math>
:<math>
H\left(\mathbf{q}mathbf__L_CURLY__q__R_CURLY__,{__L_CURLY__\partial S \over \partial \mathbf{qmathbf__L_CURLY__q}},t\right) + {__L_CURLY__\partial S \over \partial t} = 0,
</math>
meqense <math>\mathbf{p}mathbf__L_CURLY__p__R_CURLY__=\partial S/\partial \mathbf{q}mathbf__L_CURLY__q__R_CURLY__</math>.
[[Koordinatat e përgjithshme|Koordinatat e reja të përgjithshme]] <math>\mathbf{Q}mathbf__L_CURLY__Q__R_CURLY__</math> janë gjithashtu konstante, ato tipikisht jepën nga <math>\beta_{1}beta___L_CURLY__1__R_CURLY__, \beta_{2}beta___L_CURLY__2__R_CURLY__, \ldots, \beta_{Nbeta___L_CURLY__N-1}1__R_CURLY__, \beta_{N}beta___L_CURLY__N__R_CURLY__</math>. Pasi zgjedhim për <math>S(\mathbf{q}mathbf__L_CURLY__q__R_CURLY__,\boldsymbol\alpha, t)</math>, marrim disa ekuacione shumë të dobishme
:<math>
\mathbf{Qmathbf__L_CURLY__Q} = \boldsymbol\beta =
{__L_CURLY__\partial S \over \partial \boldsymbol\alpha}alpha__R_CURLY__
</math>
:<math>
Q___L_CURLY__m}= \beta___L_CURLY__m}=
Q_{m} = \beta_{m} =
\frac{frac__L_CURLY__\partial S(\mathbf{q}mathbf__L_CURLY__q__R_CURLY__,\boldsymbol\alpha, t)}{__R_CURLY____L_CURLY__\partial \alpha_{malpha___L_CURLY__m}}
</math>
Idealisht, këto <math>N</math> ekuacione mund të invertohen për të gjetur [[koordinatat e përgjithshme]] origjinale <math>\mathbf{q}mathbf__L_CURLY__q__R_CURLY__</math> si një funksion i konstanteve <math>\boldsymbol\alpha</math> dhe <math>\boldsymbol\beta</math>, kështu që në këtë mënyre i japim fund zgjidhjes së problemit.
== Ndarja e variablave ==
:<math>
H = \frac{1frac__L_CURLY__1__R_CURLY____L_CURLY__2m}{2m} \left[ p_{r}p___L_CURLY__r__R_CURLY__^{2__L_CURLY__2} + \frac{p_{frac__L_CURLY__p___L_CURLY__\theta}theta__R_CURLY__^{2__L_CURLY__2}}{r__L_CURLY__r^{2__L_CURLY__2}} + \frac{p_{frac__L_CURLY__p___L_CURLY__\phi}phi__R_CURLY__^{2__L_CURLY__2}}{r__L_CURLY__r^{2__L_CURLY__2} \sin^{2__L_CURLY__2} \theta} \right] + U(r, \theta, \phi)
</math>
:<math>
U(r, \theta, \phi) = U_{r}U___L_CURLY__r__R_CURLY__(r) + \frac{U_{frac__L_CURLY__U___L_CURLY__\theta}theta__R_CURLY__(\theta)}{r__R_CURLY____L_CURLY__r^{2__L_CURLY__2}} + \frac{U_{frac__L_CURLY__U___L_CURLY__\phi}phi__R_CURLY__(\phi)}{r__R_CURLY____L_CURLY__r^{2}__L_CURLY__2__R_CURLY__\sin^{2}__L_CURLY__2__R_CURLY__\theta}theta__R_CURLY__
</math>
ku <math>U_{r}U___L_CURLY__r__R_CURLY__(r)</math>, <math>U_{U___L_CURLY__\theta}theta__R_CURLY__(\theta)</math> dhe <math>U_{U___L_CURLY__\phi}phi__R_CURLY__(\phi)</math> janë funksione arbitrare . Zëvendësimi i zgjedhjeve <math>S = S_{r}S___L_CURLY__r__R_CURLY__(r) + S_{S___L_CURLY__\theta}theta__R_CURLY__(\theta) + S_{S___L_CURLY__\phi}phi__R_CURLY__(\phi) - Et</math> tek ekuacioni i HJ jep
:<math>
\frac__L_CURLY__1__R_CURLY____L_CURLY__2m}\left( \frac__L_CURLY__dS___L_CURLY__r}}__L_CURLY__dr}\right)^__L_CURLY__2}+ U___L_CURLY__r__R_CURLY__(r) +
\frac{1}{2m} \left( \frac{dS_{r}}{dr} \right)^{2} + U_{r}(r) +
\frac{1}{2mfrac__L_CURLY__1__R_CURLY____L_CURLY__2m r^{2__L_CURLY__2}} \left[ \left( \frac{dS_{frac__L_CURLY__dS___L_CURLY__\theta}}{d__L_CURLY__d\theta} \right)^{2__L_CURLY__2} + 2m U_{U___L_CURLY__\theta}theta__R_CURLY__(\theta) \right] +
\frac{1}{2mfrac__L_CURLY__1__R_CURLY____L_CURLY__2m r^{2}__L_CURLY__2__R_CURLY__\sin^{2}__L_CURLY__2__R_CURLY__\theta} \left[ \left( \frac{dS_{frac__L_CURLY__dS___L_CURLY__\phi}}{d__L_CURLY__d\phi} \right)^{2__L_CURLY__2} + 2m U_{U___L_CURLY__\phi}phi__R_CURLY__(\phi) \right] = E
</math>
:<math>
\left( \frac{dS_{frac__L_CURLY__dS___L_CURLY__\phi}}{d__L_CURLY__d\phi} \right)^{2__L_CURLY__2} + 2m U_{U___L_CURLY__\phi}phi__R_CURLY__(\phi) = \Gamma_{Gamma___L_CURLY__\phi}phi__R_CURLY__
</math>
ku <math>\Gamma_{Gamma___L_CURLY__\phi}phi__R_CURLY__</math> është [[konstantja e lëvizjes]] e cila eliminon <math>\phi</math> varësinë nga ekuacioni i Hamilton–Jakobit
:<math>
\frac__L_CURLY__1__R_CURLY____L_CURLY__2m}\left( \frac__L_CURLY__dS___L_CURLY__r}}__L_CURLY__dr}\right)^__L_CURLY__2}+ U___L_CURLY__r__R_CURLY__(r) +
\frac{1}{2m} \left( \frac{dS_{r}}{dr} \right)^{2} + U_{r}(r) +
\frac{1}{2mfrac__L_CURLY__1__R_CURLY____L_CURLY__2m r^{2__L_CURLY__2}} \left[ \left( \frac{dS_{frac__L_CURLY__dS___L_CURLY__\theta}}{d__L_CURLY__d\theta} \right)^{2__L_CURLY__2} + 2m U_{U___L_CURLY__\theta}theta__R_CURLY__(\theta) + \frac{frac__L_CURLY__\Gamma_{Gamma___L_CURLY__\phi}}{__L_CURLY__\sin^{2}__L_CURLY__2__R_CURLY__\theta} \right] = E
</math>
:<math>
\left( \frac{dS_{frac__L_CURLY__dS___L_CURLY__\theta}}{d__L_CURLY__d\theta} \right)^{2__L_CURLY__2} + 2m U_{U___L_CURLY__\theta}theta__R_CURLY__(\theta) + \frac{frac__L_CURLY__\Gamma_{Gamma___L_CURLY__\phi}}{__L_CURLY__\sin^{2}__L_CURLY__2__R_CURLY__\theta} = \Gamma_{Gamma___L_CURLY__\theta}theta__R_CURLY__
</math>
ku <math>\Gamma_{Gamma___L_CURLY__\theta}theta__R_CURLY__</math> është prapë një [[konstante e lëvizjes]] e cila eliminon varësinë nga <math>\theta</math> dhe e redukton ekuacionin në [[ekuacionin diferencial ordiner]] final.
:<math>
\frac{1frac__L_CURLY__1__R_CURLY____L_CURLY__2m}{2m} \left( \frac{dS_{rfrac__L_CURLY__dS___L_CURLY__r}}{dr__L_CURLY__dr} \right)^{2__L_CURLY__2} + U_{r}U___L_CURLY__r__R_CURLY__(r) + \frac{frac__L_CURLY__\Gamma_{Gamma___L_CURLY__\theta}}{2m__L_CURLY__2m r^{2__L_CURLY__2}} = E
</math>
== Ekuacioni i Hamilton-Jakobit në një fushe gravitacionale ==
:<math>g^__L_CURLY__ik__R_CURLY__\frac__L_CURLY__\partial__L_CURLY__S}}__L_CURLY__\partial__L_CURLY__x^__L_CURLY__i}}}\frac__L_CURLY__\partial__L_CURLY__S}}__L_CURLY__\partial__L_CURLY__x^__L_CURLY__k}}} - m^__L_CURLY__2__R_CURLY__c^__L_CURLY__2}= 0</math>
:<math>g^{ik}\frac{\partial{S}}{\partial{x^{i}}}\frac{\partial{S}}{\partial{x^{k}}} - m^{2}c^{2} = 0</math>
ku <math>g^{ik}__L_CURLY__ik__R_CURLY__</math> janë komponentët [[kontravariant]] të [[tensorit të metrikës]], ''m'' është [[masa e prehjes]] e thërrmijës dhe ''c'' është [[shpejtësia e dritës]].
== Shiko gjithashtu ==
| 