The action is not a well defined function

The action of moving from q_a to q_b in time t is not a well defined function

for example, consider the harmonic oscillator

if q_a= q_b, then only for t being the multiple of the harmonic oscillator period will the particle return to its initial position

Therefore, for such kind of q_a and q_b, the action is ill-defined for an arbitrary t

but people are always taking the derivatives of the action to the time!!!!!!

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vanhees71
Gold Member
2019 Award
Who does this? Of course the action is not a function but a functional, i.e., it maps from an appropriate space of functions into the real numbers. It also depends on whether you work in the Lagrangian formulation (then the action is a functional over the configuration-space trajectories) or the Hamiltonian formulation (then it's a functional over the phase-space trajectories).

stevendaryl
Staff Emeritus
The action of moving from q_a to q_b in time t is not a well defined function

for example, consider the harmonic oscillator

if q_a= q_b, then only for t being the multiple of the harmonic oscillator period will the particle return to its initial position
That's not true. Suppose the particle is undergoing simple harmonic motion

$x = A cos(\omega t + \phi)$

If $t_1$ and $t_2$ are two different times, then $x(t_1) = x(t_2)$ in the following cases:

$t_1 = t_2 + \dfrac{2 n \pi}{\omega}$
$t_1 = - t_2 + \dfrac{2 (n \pi - \phi)}{\omega}$

In the latter case, $t_2 - t_1$ will not be a multiple of the period.

That's not true. Suppose the particle is undergoing simple harmonic motion

$x = A cos(\omega t + \phi)$

If $t_1$ and $t_2$ are two different times, then $x(t_1) = x(t_2)$ in the following cases:

$t_1 = t_2 + \dfrac{2 n \pi}{\omega}$
$t_1 = - t_2 + \dfrac{2 (n \pi - \phi)}{\omega}$

In the latter case, $t_2 - t_1$ will not be a multiple of the period.
you are right.

however, t2-t1 still can only take discrete values.

stevendaryl
Staff Emeritus
you are right.

however, t2-t1 still can only take discrete values.
No, the constant $\phi$ is arbitrary. By adjusting $\phi$, you can get $t_2 - t_1$ to be anything you like.

I shouldn't say it's arbitrary; it's determined by the boundary conditions. The harmonic oscillator $m \ddot{x} + m \omega^2 x = 0$ has solutions of the form:

$x(t) = A cos(\omega t + \phi)$

That has two constants, $A$ and $\phi$. If you pick two times, $t_1$ and $t_2$, and specify that $x(t_1) = x_1$ and $x(t_2) = x_2$, then that uniquely determines $A$ and $\phi$.

The action of moving from q_a to q_b in time t is not a well defined function
Examine this expression for the action for a system with 1 generalized coordinate.

$S = \int_{t_0}^{t_1} \mathcal{L}(q, \dot{q}, t) dt$

The action is well enough defined for different functions q(t), even ones that aren't valid solutions to the problem I'm trying to solve! To use your example, the action can have a value even for space endpoints q_a and q_b, even when there is no valid solution to the problem which connects q_a and q_b in time t1 - t0.

The valid solutions are the ones for which the action is stationary.

If we find the conditions for the action being stationary, we still haven't specified a particular path q(t) with starting and ending points!

These would be related the the initial conditions.

We may get some differential equations by solving the Euler Lagrange equations, which give us the conditions for which the action is stationary. But differential equations don't specify a single exact solution; there can be more than one function which is a solution to a differential equation. But once you fully specify the initial conditions, the values after time t are going to be set.

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stevendaryl
Staff Emeritus
Examine this expression for the action for a system with 1 generalized coordinate.

$S = \int_{t_0}^{t_1} \mathcal{L}(q, \dot{q}, t) dt$

The action is well enough defined for different functions q(t), even ones that aren't valid solutions to the problem I'm trying to solve! To use your example, the action can have a value even for space endpoints q_a and q_b, even when there is no valid solution to the problem which connects q_a and q_b in time t1 - t0.
Hmm. If $t_1 > t_0$ and the potential is not infinite, isn't there always a solution connecting $q_a$ at time $t_0$ to $q_b$ at time $t_1$? I think there is, in the one-dimensional case, anyway.

stevendaryl
Staff Emeritus
Hmm. If $t_1 > t_0$ and the potential is not infinite, isn't there always a solution connecting $q_a$ at time $t_0$ to $q_b$ at time $t_1$? I think there is, in the one-dimensional case, anyway.