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Let $C=\{q(t)\,|\,t_{1}\leq t\leq t_{2}\}$ be a trajectory for a particle. Then, the action is given by:

$\Phi [C]={\text{Action of trajectory C}}=\int _{t_{1}}^{t_{2}}L(q(t),{\dot {q}}(t),t)\,dt$

Let a variation of the trajectory be given as: $C'=\{q'(t)=q(t)+\delta q(t)\,|\,t_{1}+\delta t_{1}\leq t\leq t_{2}+\delta t_{2}\}$, then the change in action is given by:

${\begin{aligned}\delta \Phi [C]&=\Phi [C^{\prime }]-\Phi [C]\\&=\int _{t_{1}+\delta t_{1}}^{t_{2}+\delta t_{2}}dtL(q(t)+\delta q(t),{\dot {q}}(t)+\delta {\dot {q}}(t))-\int _{t_{1}}^{t_{2}}dtL(q(t),{\dot {q}}(t),t)\\&=\int _{t_{1}}^{t_{2}}[L(q(t)+\delta q(t),{\dot {q}}(t)+\delta {\dot {q}}(t),t)-L(q(t),{\dot {q}}(t),t)]\,dt+\,\int _{t_{2}}^{t_{2}}L(q_{s}^{\prime }\,{\dot {q}}_{s}^{\prime }\,t)\,dt\,-\,\int _{t_{1}}^{t_{1}}L(q_{s}^{\prime }\,{\dot {q}}_{s}^{\prime },t)\,dt\\&=\int _{t_{1}}^{t_{2}}\left({\frac {\partial L}{\partial q}}\delta q+{\frac {\partial L}{\partial {\dot {q}}}}\delta {\dot {q}}\right)\,dt+\,\int _{t_{2}}^{t_{2}+\delta t_{2}}L(q^{\prime },{\dot {q}}^{\prime },t)\,dt\,-\,\int _{t_{1}}^{t_{1}+\delta t_{1}}L(q^{\prime },{\dot {q}}^{\prime },t)\,dt\\&=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}\,\delta q(t)+{\frac {\partial L}{\partial {\dot {q}}}}\ {\frac {d}{dt}}\,\delta q(t)\right]+{\bigg [}L\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\&=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}}}\right)\right]\delta q(t)+{\frac {\partial L}{\partial {\dot {q}}}}\delta q(t){\biggr |}_{t_{1}}^{t_{2}}+L\Delta t{\biggr |}_{t_{1}}^{t_{2}}\end{aligned}}$

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

${\begin{aligned}\Delta \Phi &=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\,{\frac {\partial L}{\partial {\dot {q}}}}\right]\delta q(t)+{\bigg [}p\Delta q-(p{\dot {q}}-L)\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\&=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\,{\frac {\partial L}{\partial {\dot {q}}}}\right]\delta q(t)+{\bigg [}p\Delta q-H\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\\end{aligned}}$

Hamilton's action principle

Considering a special class of variation of path that leaves the end-points and terminal times unchanged, ie. $\Delta t_{i}=\Delta q(t_{i})=0.$ For such actions, the change in action functional is given by:

${\begin{aligned}\Delta _{\sigma }\Phi &=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\,{\frac {\partial L}{\partial {\dot {q}}}}\right]\delta q(t)\end{aligned}}$

From Lagrange's equation of motion, it follows that the infinitesimal change in action functional vanishes if the given trajectory is a solution for trajectory of the particle.

Weiss action principle

Using Lagrange's equations of motion, we have the following value for the change in action functional:

${\begin{aligned}\Delta \Phi &=\int _{t_{1}}^{t_{2}}dt\left[{\frac {\partial L}{\partial q}}-{\frac {d}{dt}}\,{\frac {\partial L}{\partial {\dot {q}}}}\right]\delta q(t)+{\bigg [}p\Delta q-H\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\&={\bigg [}p\Delta q-H\Delta t{\bigg ]}_{t_{1}}^{t_{2}}\\\end{aligned}}$

Hence, Hamilton's action principle can be extended to Weiss action principle as the dynamical trajectory in configuration space is that which only provides end-point contributions to $\Delta \Phi$.

References

Sudarshan, E C George; Mukunda, N (2010). Classical Dynamics: A Modern Perspective. Wiley. ISBN 9780471835400.