User:EditingPencil/sandbox/action

Let $C=\; \backslash \{\; q(t)\backslash ,|\backslash ,t\_1\backslash leq\; t\; \backslash leq\; t\_2\; \backslash \}$ be a trajectory for a particle. Then, the action is given by:

$\backslash Phi\; [C]\; =\; \backslash text\{Action\; of\; trajectory\; C\}\; =\; \backslash int\_\{t\_1\}^\{t\_2\}\; L(q(t),\backslash dot\; q(t),t)\backslash ,\; dt$

Let a variation of the trajectory be given as: $C\text{'}\; =\; \backslash \{\; q\text{'}(t)=q(t)+\backslash delta\; q(t)\backslash ,\; |\backslash ,t\_1+\backslash delta\; t\_1\backslash leq\; t\; \backslash leq\; t\_2\; +\backslash delta\; t\_2\backslash \}$, then the change in action is given by:

$\backslash begin\{align\}$

\delta\Phi[C]&=\Phi[C^{\prime}]-\Phi[C]\\

&=\int_{t_{1}+\delta t_1}^{t_{2}+\delta t_2}d t L(q(t)+\delta q(t),\dot{q}(t)+\delta\dot{q}(t))-\int_{t_{1}}^{t_{2}}d t L(q(t),\dot{q}(t),t)\\

&= \int_{t_{1}}^{t_{2}} [ L(q(t)+\delta q(t),\dot{q}(t)+\delta \dot q(t),t)- L(q(t),\dot{q}(t),t) ] \, dt +\,\int_{t_{2}}^{t_{2}}L(q_{s}^{\prime}\,\dot{q}_{s}^{\prime}\,t)\,d t\,-\,\int_{t_{1}}^{t_{1}}L(q_{s}^{\prime}\,\dot{q}_{s}^{\prime},t)\,d t\\

&=\int_{t_{1}}^{t_{2}} \left(\frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot q} \delta \dot q\right) \, dt +\,\int_{t_{2}}^{t_{2}+\delta t_2}L(q^{\prime},\dot{q}^{\prime},t)\,d t\,-\,\int_{t_{1}}^{t_{1}+\delta t_1}L(q^{\prime},\dot{q}^{\prime},t) \,d t\\

&=\int_{t_{1}}^{t_{2}}d t\left[\frac{\partial L}{\partial q}\,\delta q(t)+\frac{\partial L}{\partial\dot{q}}\ \frac{d}{d t}\,\delta q(t)\right]+\bigg[L\Delta t\bigg]_{t_{1}}^{t_{2}}\\

&=\int_{t_{1}}^{t_{2}}d t\left[\frac{\partial L}{\partial q}-\frac{d}{d t}\left(\frac{\partial L}{\partial\dot{q}}\right)\right]\delta q(t)+\frac{\partial L}{\partial\dot{q}}\delta q(t)\biggr|_{t_{1}}^{t_{2}}+L\Delta t\biggr|_{t_{1}}^{t_{2}}

\end{align}

Defining the total variation of path as $\backslash Delta\; q\; =\; q\text{'}(t\text{'})-q(t)=\; \backslash delta\; q(t)\; +\; \backslash dot\; q\; (t)\; \backslash delta\; t$ and momentum as $\backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q\}=p$, we get:

$\backslash begin\{align\}$

\Delta\Phi&=\int_{t_{1}}^{t_{2}} d t \left[\frac{\partial L}{\partial q}-\frac{d}{d t}\,\frac{\partial L}{\partial\dot{q}}\right]\delta q(t)+\bigg[p\Delta q-(p\dot q - L)\Delta t\bigg]_{t_{1}}^{t_{2}}\\

&=\int_{t_{1}}^{t_{2}} d t \left[\frac{\partial L}{\partial q}-\frac{d}{d t}\,\frac{\partial L}{\partial\dot{q}}\right]\delta q(t)+\bigg[p\Delta q-H \Delta t\bigg]_{t_{1}}^{t_{2}}\\

\end{align}