Variation of Action with time coordinate variations

This is the Variational Problem with free end-time and one proceeds like this:

$$\delta S= \int_{t_i}^{t_f+\delta t_f} L \left(q+\delta q,\dot{q}+\delta \dot{q},t\right) dt - \int_{t_i}^{t_f} L \left(q, \dot{q}, t\right) dt$$

After several transformations and integration by parts one finally gets the usual Euler-Lagrange diff eq plus a boundary condition involving $\delta t_f$:

$$0 = L(q, \dot{q}, t) \delta t_f + \frac{\partial L}{\partial \dot{q}}(\delta q_f - \dot{q} \delta t_f)$$

Derivation steps:

a. Expand the first integral in a Taylor series and keep terms of 1st order and splitting the limits of integration (and doing any cancelations):

$$ \delta S= \int_{t_f}^{t_f+\delta{t_f}} \left[ L + \frac{\partial{L}}{\partial{q}}\delta{q} + \frac{\partial{L}}{\partial{\dot{q}}}\delta{\dot{q}} \right] dt + \int_{t_i}^{t_f} \left[ \frac{\partial{L}}{\partial{q}}\delta{q} + \frac{\partial{L}}{\partial{\dot{q}}}\delta{\dot{q}} \right] dt $$

b. The total variation consists of 2 variations; $\delta{q}$ and $\delta{t_f}$. Integrating over a small interval i.e $[t_f, t_f + \delta{t_f}]$ is effectively equivalent to multiplication by $\delta{t_f}$:

$$ \delta S= \delta{t_f} \left[ L + \frac{\partial{L}}{\partial{q}}\delta{q} + \frac{\partial{L}}{\partial{\dot{q}}}\delta{\dot{q}} \right] + \int_{t_i}^{t_f} \left[ \frac{\partial{L}}{\partial{q}}\delta{q} + \frac{\partial{L}}{\partial{\dot{q}}}\delta{\dot{q}} \right] dt $$

c. Terms like $\delta{t_f}\delta{q}$ or $\delta{t_f}\delta{\dot{q}}$ are 2nd order variations and can be dropped:

$$ \delta S= \delta{t_f} L + \int_{t_i}^{t_f} \left[ \frac{\partial{L}}{\partial{q}}\delta{q} + \frac{\partial{L}}{\partial{\dot{q}}}\delta{\dot{q}} \right] dt $$

d. Integration by parts yields a usual Euler-Lagrange diff eq. plus the boundary condition.

e. On the boundary condition at time $t_f$ one has:

Total variation of $q$ at $t_f$ is

$$\delta{q_f} = \delta{q(t_f)} + (\dot{q} + \delta{\dot{q}})\delta{t_f} = \delta{q(t_f)} + \dot{q}\delta{t_f}$$

or

$$\delta{q(t_f)} = \delta{q_f} - \dot{q}\delta{t_f}$$

Hint for @Y2H:

The total variation at the boundary $\delta{q_f}$ is simply the sum of the path and time variations at the boundary (since these can be considered as independent variations), ie $(\delta{q(t_f)}) + (q(t_f+\delta{t_f})+\delta{q(t_f+\delta{t_f})}-q(t_f)-\delta{q(t_f)})$ and the latter can be decomposed (up to 1st order variations) as $(\dot{q}(t_f) + \delta{\dot{q}(t_f)})\delta{t_f}$

PS: There has been a long time since posted this answer and I dont have my notes handy, but hope the above give you a hint.

PS2: Here are some synoptic lecture notes on generalised calculus of variations with free end points

I) Hint: Decompose the full infinitesimal variation

$$ \tag{A} \delta q~=~\delta_0 q + \dot{q} \delta t $$

in a vertical infinitesimal variation $\delta_0 q$ and a horizontal infinitesimal variation $\delta t$. Similarly the full infinitesimal variation becomes

$$ \tag{B} \delta I~=~\delta_0 I + \left[ L ~\delta t \right]_{t_1}^{t_2}, $$

where the vertical piece follows the standard Euler-Lagrange argument

$$ \tag{C} \delta_0 I~=~ \int_{t_1}^{t_2}\! dt~\left[\frac{\partial L}{\partial q}-\dot{p} \right] \delta_0 q + \left[ p ~\delta_0q \right]_{t_1}^{t_2}, $$

and we have for convenience defined the Lagrangian momenta

$$ \tag{D} p~:=~\frac{\partial L}{\partial \dot{q}}. $$

Now combine eqs. (A-D) to derive eq. (65) in Ref. 1:

$$ \tag{65} \delta I~=~ \int_{t_1}^{t_2}\! dt~\left[\frac{\partial L}{\partial q}-\dot{p} \right] \delta_0 q + \left[ p ~\delta q - (p\dot{q}-L)\delta t\right]_{t_1}^{t_2}, $$

II) Ideologically, we should stress that Ref. 1 is not interested in proposing a variational principle for non-vertical variations (such as, e.g., Maupertuis' principle, or a variant of Pontryagin's maximum principle, etc). Ref. 1 is merely calculating non-vertical variations within a theory that is still governed by the principle of stationary action (for vertical variations).

III) Ref. 1 mainly uses eq. (65) to deduce properties of the on-shell Dirichlet action $^1$

$$ \tag{E} S(q_2,t_2;q_1,t_1)~:=~I[q_{\rm cl};t_1,t_2],$$

cf. e.g. this Phys.SE post.

References:

1. L.B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article, Living Rev. Relativity 7 (2004) 4.

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$^1$ Ref. 1 calls $S(q_2,t_2;q_1,t_1)$ the Hamilton-Jacobi principal function. Although related, the Hamilton-Jacobi principal function $S(q,P,t)$ is strictly speaking another function, cf. e.g. this Phys.SE post.