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Action is defined as,

$$S ~=~ \int L(q, q', t) dt,$$

but my question is what variables does $S$ depend on?

Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$?

In Wikipedia I've read that $S = S(q(t))$ and I think that suppose, $q$ and $t$ are considered as independent coordinates. Then $S$ should depend on $q'$ also because, for the typical Lagrangian

$$L ~=~ \frac{q'^2}{2} - V(q).$$

Qmechanic
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user5198
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1 Answers1

13

  1. Firstly, the Lagrangian $L(q(t),v(t),t)$ at some time $t$ is a function of:

    1. the instantaneous position $q(t)$ at the time $t$;
    2. the instantaneous velocity $v(t)$ at the time $t$; and
    3. the time $t$ (also known as explicit time-dependence).

  2. Secondly, the (off-shell) action $$ S[q]~:=~ \left. \int_{t_i}^{t_f}\! \mathrm{d}t \ L(q(t),v(t),t)\right|_{v(t)=\dot{q}(t)} \tag{1}$$ is a functional of the full position curve/path $q:[t_i,t_f] \to \mathbb{R}$ for all times $t$ in the interval $[t_i,t_f]$.

  3. Thirdly, if one imposes boundary conditions (B.C.), e.g. Dirichlet B.C., $$ q(t_i)~=~q_i \qquad \text{and}\qquad q(t_f)~=~q_f,\tag{2}\label{eq:2} $$ then there is also a notion of a (Dirichlet) on-shell action $^1$ $$ S(q_f,t_f;q_i,t_i)~:=~S[q_{\rm cl}]\tag{3}$$ where $q_{\rm cl}:[t_i,t_f] \to \mathbb{R}$ is the classical path, which satisfies Euler-Lagrange equations with the Dirichlet B.C. $\eqref{eq:2}$. The on-shell action $S(q_f,t_f;q_i,t_i)$ is a function of

    1. the initial time $t_i$;
    2. the initial position $q_i$;
    3. the final time $t_f$; and
    4. the final position $q_f$.

$^1$ See also e.g. MTW Section 21.1. For the on-shell action $S(q_f,t_f;q_i,t_i)$ to be well-defined, there should exist a unique classical path with the B.C. $\eqref{eq:2}$. (Here the words on-shell and off-shell refer to whether the Euler-Lagrange equations are satisfied or not.)

Qmechanic
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