The phase portraits - 在E型代數結構下之N相黎曼空間的單擺運動之確切理論與數值運算

The ordinary diﬀerential equation we had discussed is the mathematical model of ideal pendulum. Now we try to plot the relation between U (t) and U_t and the graph is called phase portrait. Before drawing the phase portrait, we see back to the equation (5.2.8) as

2u²_t − cos u = E , where E is a constant.

It shows that 1

2u²_t− cos u is a constant. It can be regarded as a conservation law in the viewpoint of mathematics since− cos u is not always larger than 0.

This constant and the former part 1

2u²_t can be regarded as kinetic energy and the latter part − cos u can be regarded as potential energy.

We will discuss the potential energy and phase portrait with diﬀerent cases.

Case I. −1 < E < 1

We set E = 0 to analyze this case. By the equation (5.2.8), we have the equation u_t =±√

2 cos u .

The following graphs are potential energy and phase portrait respec-tively. This means that they are the relation between u and cos u and the relation between u and u_t.

Figure 5.4.1. The potential energy and phase portrait for E = 0

Remark 5.5.

(1) 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 counterclockwise and the negative velocity is deﬁned by rotating clockwise.

(2) By the graph of potential energy, we can ﬁnd out that the maxi-mum of amplitude, u(t) , for the pendulum is π

2 and it oscillates forth and back.

Case II. E = 1

Now we focus on the case with E = 1 . By the equation (5.2.8), we have u_t =±√

2(1 + cos u) . We see the potential energy and phase portrait as following.

Figure 5.4.2. The potential energy and phase portrait for E = 1

By the graph of potential energy, we can ﬁnd out that the maximum of amplitude, u(t) , for the pendulum is π. If we release the pendulum at position π , the particle will approach to the position −π after inﬁnite time.

Case III. E > 1

Last, we see the case E > 1 with E = 3

2. By the equation (5.2.8), we have u_t =±

√ 2(3

2+ cos u) . We see the potential energy and phase portrait as following.

Figure 5.4.3. The potential energy and phase portrait for E = 3 2

Remark 5.6.

(1) From the graph of the phase portrait, we know that the pendu-lum of this case will never stop since the phase portrait has no intersection with the u-axis.

(2) By the graph of potential energy, we observe that the kinetic en-ergy is never equal 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 ﬁn-ishing the section, we combine the three phase portraits and the vector ﬁeld together.

Figure 5.4.4. Global phase portrait

5.5 Related knowledge

We can connect with a partial diﬀerential equation

u_tt− uxx+ sin u = 0 , (5.5.28) which is called sine-Gordon equation.

Firstly, we can simplify the equation (5.5.28). Assume that θ = kx− ωt with ω²− k² = 1 .

And by chain rule, we have

u_t = ∂u

∂θ

∂t = (−ω)uθ

u_x = ∂u

∂θ

∂x = k u_θ Using the same way, we get

u_tt = ω²u_θθ uxx = k²uθθ

Then (5.5.28) can be transferred to be an ordinary diﬀerential equation

u_θθ+ sin(u) = 0 . (5.5.29)

That is the pendulum motion we discussed.

Chapter 6 Conclusion

In this paper, we study the ideal pendulum equation u^′′+ sin u = 0 , which can be translated into integral form

∫ 1

√2(E + cos u) du =±

∫ dt

where E is the integration constant. The integrator involve √

2(E + cos u) where E + cos u is a transcendental function and it has inﬁnitely many zeros, so u resides on Riemann surface of genus∞.

Hence, we study its nonlinear approximation, namely u^′′+ P_{2N +1}(u) = 0 ,

where P_{2N +1}(u) is the {2N+1}-th Taylor expansion of sin u . Then this O.D.E. has the integral form

∫ 1

√2(E− P2N +2(u)) du ,

where u now resides on Riemann surface of genus N , then we can analyze and compute it.

We then study the classical elliptic function and apply to analyze the exact theory of pendulum motions with a table given as follows:

Energy E −1 < E < 1 E = 1 E > 1

Solution U (t) 2 sin⁻¹(κsn(t, κ)) 2 sin⁻¹(sn(t, 1)) 2 sin⁻¹(sn(κ⁻¹t, κ)) Modulus κ

√E + 1

2 1

√ 2

E + 1

Periods T 4K ∞ No periodicity

For further study, we may consider the more complicated sine-Gorden equa-tion

utt− uxx+ sin u = 0 , where u^′′+ sin u = 0 is just a special case.

Appendix A

The process of computation

A.0.1

A.0.3

=−0.768165 + 0.242221 i

A.0.4

= 0.170019− 0.268168 i