Exact Theory of the Simple Pendulum Motion
2. By conservation of energy:
5.4 Periods and phase portraits with different total energy
Since the solutions of u′′+ sin u = 0 had been found in terms of Jacobian elliptic function with different E, we want to further know the period of the so-lution if it is periodic.
Moreover, we try to plot the relation between U and U′, i.e., the phase portrait. Before drawing the phase portrait, we see back to the equation (5.4) first. It shows that 1
2(u′)2− cos u is a constant. It can be regarded as a conserva-tion law in the view point of mathematics since− cos u is not always larger than 0. (But this case can be transferred to the conservation law in the view point of physics by plus a constant 1 for equation (5.4).) This means that its total energy is a constant and the former part 1
2(u′)2 can be regarded as kinetic energy and the latter part − cos u can be regarded as potential energy.
1. −1 < E < 1 :
The solution with−1 < E < 1 is given by
U (t) = 2 arcsin (k sn(t, k)) where k =
√2E + 2
2 . Therefore, by subsection 4.4.4, the period is T = 4
By the assumption we defined before, there is
−1 < E1 < E2 < 1 ⇒ k1 < k2, and, hence,
√ 1
(1− z2)(1− k12z2) < 1
√(1− z2)(1− k22z2), i.e. K1 < K2.
In short, if there are two different E1 and E2, where −1 < E1 < E2 < 1, the comparison with two periods is
T1 < T2.
Figure 5.4: Solution curves with E = −1
2, E = 0, and E = 1 2
In the sense of pendulum motion, the greater total energy means the higher initial position, and it is naturally that the time pendulum returns to the initial position is longer if the initial position is higher. Thus we have the result as above, E1 < E2 implies T1 < T2.
We set E = 0 to analyze the phase portrait. By the equation (5.4), we have u′ = ±√
2 cos u. The following graphs are potential energy and phase portrait, respectively. Those graphs show the relation between u and cos u and the relation between u and u′.
Figure 5.5: Potential energy and phase portrait with E = 0
From the graph of the phase portrait, the red curve means that the velocity at those position are positive and the blue curve means that the velocity at those position are negative. The positive velocity is defined by rotating counterclock-wise and the negative velocity is defined by rotating clockcounterclock-wise.
2. E = 1 :
The solution with E = 1 is given by
U (t) = 2 arcsin (sn(t, 1)).
T = 4
∫ 1
0
√ 1
(1− z2)(1− z2)dz
= 4
∫ 1 0
1 1− z2dz
=∞.
Figure 5.6: Solution curve with E = 1
In the sense of pendulum motion, since the total energy E + 1 = 2, the potential energy must be 2 and the kinetic energy must be 0 somewhere. By the language we used before, that is the greatest potential energy means the highest place in the pendulum motion, surely the top of the circular path. Therefore it implies that if we release the pendulum at the top of the circular path, it will return to the initial position after travelling the time infinity.
Now we focus on the phase portrait with E = 1. By the equation (5.4), we have u′ =±√
2(1 + cos u), and phase portrait as following.
3. E > 1 :
By (5.5), there is u′ > 0 if E > 1. This means that for any time t, the velocity of pendulum is always greater than 0. That is, the pendulum will never stop. So the motion is no periodicity.
Figure 5.8: Solution curves with E = 3
2, E = 2, and E = 5 2 Last, we see the phase portrait with E = 3
2. By the equation (5.4), we have u′ =±
√ 2(3
2+ cos u), and phase portrait as following.
Figure 5.9: Potential energy and phase portrait with E = 3 2
From the graph of the phase portrait, we know that the pendulum of this case will never stop since the phase portrait has no intersection with the u-axis.
And by the graph of potential energy, we observe that the kinetic energy is never equal to 0. This implies that the case has no periodic solution and the result is corresponded to the property which we had discussed.
By our discussion, there are three kinds of the phase portraits. Before finishing the section, we combine the three phase portraits and the vector field together.
Figure 5.10: Global phase portrait
There are three different kinds of phase portraits with different energy E.
The outer curve corresponds to larger energy E. They are separated by the phase portrait with E = 1 and the phase curve is called the separatrix with periods∞.
The phase curves outer the separatrix are called the wave train and they has no period. The phase curves inside the separatrix are periodic and their period T
5.5 Summary
The mathematical model of the simple pendulum motion is a nonlinear second order differential equation. There are k corresponding to the given E, and thus the solutions of the simple pendulum motion is expressed by Jacobian elliptic function sn(t, k) within different cases of E.
Together the consequences in all cases we considered, there is the following table.
- −1 < E < 1 E = 1 E > 1
Modulus k
√2E + 2
2 1 2
√2E + 2 Solution U (t) 2 arcsin (k sn(t, k)) 2 arcsin (sn(t, 1)) 2 arcsin (sn(t
k, k))
Period T 4K ∞ No periodicity
Table 5.1: Summary about the simple pendulum motion within different E
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