What is the second conserved Quantity of the Pendulum?


$Q_1$ is not a conserved quantity at all. It is just a parameter which depends on the initial conditions.


First of all, there's an error when you derived

$$\frac{1}{\sqrt{2Q_0 + \frac{g}{l} 2\cos(\theta)}} \frac{d \theta}{d t} = 1\tag{1}$$

You only took the positive square root, whereas you should have take both the possibilities of the RHS being $+1$ and $-1$. You can easily see that the equation $(1)$ never holds whenever $\theta$ is decreasing i.e. $\mathrm d \theta /\mathrm d t<0$. To correct this, we need to add a modulus around the $\mathrm d \theta/\mathrm dt$ term. Thus the corrected equation would be

$$\frac{1}{\sqrt{2Q_0 + \frac{g}{l} 2\cos(\theta)}} \left|\frac{d \theta}{d t}\right| = 1\tag{2}$$

I would advise you to re-integrate equation $(2)$ to find the correct solution which holds over the complete range of motion.

What about the other equation

The final equation which you obtained

$$\sqrt{\frac{2}{Q_0 + \frac{g}{l}}} \left(F \left[ \frac{\theta}{2} , 2 \frac{g}{l} \frac{1}{Q_0 + \frac{g}{l}} \right]\right) -t = Q_1\tag{3}$$

only holds true for the cases where $\mathrm d \theta/\mathrm dt>0$, so for now, we'll only consider cases where the pendulum is going from left to right, but the insight provided below will also help you determine the physical meaning of the new constant that you would obtaing after integrating equation $(2)$. Also, the equation $(3)$ contains an incomplete elliptic integral of the first kind. One of the important properties of this function is that


where $k$ is any real number. Thus, substituting $\theta=0$ in equation $(1)$, we get

\begin{align} \sqrt{\frac{2}{Q_0 + \frac{g}{l}}} \left(F \left[ 0 , 2 \frac{g}{l} \frac{1}{Q_0 + \frac{g}{l}} \right]\right) -t_0 &= Q_1\\ 0-t_0&=Q_1\\ Q_1+t_0&=0\tag{4} \end{align}

where $t_0$ is the time when the pendulum passes throught its equilibrium position for the first time. And since we are only considering the case where $\mathrm d\theta /\mathrm dt>0$, thus the above equation is valid only for the cases where the pendulus comes from the left and goes to the right while passing through the equilibrium position.

Physical Significance

The physical significance of the constant $Q_1$ isn't as deep and profound as you expected. $Q_1$ is just a shifting constant applied to the time. This constant will change upon changing your definition of $t=0$. Thus, it's just a parameter which adjusts/shifts the time scale of the oscillation. It adjusts according to the initial conditions and doesn't give you any more information about the dynamical parameters of the system.

Suppose that in any physical system, the solution to the equation of motion is $x(t) = f(t, x_0, v_0)$. Then $x - f = 0$, so it's conserved. In this way, you can manufacture a new conserved quantity for any physical situation. You can also add on any function $g(x_0, v_0)$ of the initial conditions, giving an infinite family of conserved quantities $x - f + g$. This is what you found.

For example, for a ball in freefall, you can easily check that $x - (x_0 + v_0 t - gt^2/2) + x_0$ is conserved, for this reason. But this isn't a new conserved quantity at all -- it's just a minor rewriting of the solution to the equation of motion, whose particular value is the initial position.

You can't use this idea to do anything. If you don't already know the general solution $f$ then you can't compute $x-f+g$, if you do know $f$ then you don't need it, and if you don't know $f$ but somehow know the numeric value of $x-f+g$, that just tells you about the initial conditions, which you already knew anyway.

The configuration space of a pendulum is 1D (in fact, a circle, $S^1$) so it's phase space is 2D (a cylinder, $S^1\times \mathbb{R}$). If there were two integrals of motion then we could label each point in the 2D phase space by those two values, and since they're meant to be conserved, the phase space dynamics would have to be trivial (i.e. position and momenta never change).

So whatever your $Q_1$ is it is either:

(a) Some function of $Q_0$ so not an independent integral of motion or

(b) A weaker kind of conserved quantity that is not just a function of phase space coordinates. For instance, the initial angle and angular velocities are strictly conserved quantities along a trajectory.

I suspect your $Q_1$ can be written in terms of the initial conditions, ie is of type (b).