非线性影响单摆

非线性影响单摆
Nonlinearity Affects a Pendulum

原始链接: https://www.johndcook.com/blog/2026/04/24/nonlinear-pendulum/

简单摆的运动方程在 introductory physics 中通常通过用 θ 代替 sin θ 来简化，假设角度很小。这引发了关于*为什么*做出这种近似以及它的*准确性*如何的问题。核心原因是原始方程由于正弦函数而*非线性*，无法用标准的 introductory calculus 技术求解。近似 sin θ ≈ θ 对于小角度是有效的，但随着角度增大，其准确性会降低。精确（非线性）和近似（线性）解之间的关键区别在于振荡周期。非线性摆的周期*更长*，并随着初始角度的增大而增加。虽然一个具有调整后的更长周期的线性方程可以近似非线性解，但仍然存在差异——尽管通常很小。例如，在 60° 时，非线性周期比线性近似预测的周期长约 7.32%。本质上，这种简化允许得到一个可解的方程，但理解其局限性以及对周期的影响对于准确的分析至关重要。

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原文

The equation of motion for a pendulum is the differential equation

$\theta'' + \frac{g}{\ell}\sin \theta = 0$

where g is the acceleration due to gravity and ℓ is the length of the pendulum. When this is presented in an introductory physics class, the instructor will immediately say something like “we’re only interested in the case where θ is small, so we can rewrite the equation as

$\theta'' + \frac{g}{\ell} \theta = 0$

Questions

This raises a lot of questions, or at least it should.

Why not leave sin θ alone?
What justifies replacing sin θ with just θ?
How small does θ have to be for this to be OK?
How do the solutions to the exact and approximate equations differ?

First, sine is a nonlinear function, making the differential equation nonlinear. The nonlinear pendulum equation cannot be solved using mathematics that students in an introductory physics class have seen. There is a closed-form solution, but only if you extend “closed-form” to mean more than the elementary functions a student would see in a calculus class.

Second, the approximation is justified because sin θ ≈ θ when θ is small. That’s true, but it’s kinda subtle. Here’s a post unpacking that.

The third question doesn’t have a simple answer, though simple answers are often given. An instructor could make up an answer on the spot and say “less than 10 degrees” or something like that. A more thorough answer requires answering the fourth question.

I address how nonlinear affects the solutions in a post a couple years ago. This post will expand a bit on that post.

Longer period

The primary difference between the nonlinear and linear pendulum equations is that the solutions to the nonlinear equation have longer periods. The solution to the linear equation is a cosine. Solving the equation determines the frequency, amplitude, and phase shift of the cosine, but qualitatively it’s just a cosine. The solution to the corresponding nonlinear equation, with sin θ rather than θ, is not exactly a cosine, but it looks a lot like a cosine, only the period is a little longer [1].

OK, the nonlinear pendulum has a longer period, but how much longer? The period is increased by a factor f(θ₀) where θ₀ is the initial displacement.

You can find the exact answer in my earlier post. The exact answer depends on a special function called the “complete elliptic integral of the first kind,” but to a good approximation

$f(\theta) \approx \frac{1}{\sqrt{\cos(\theta/2)}}$

The earlier post compares this approximation to the exact function.

Linear solution with adjusted period

Since the nonlinear pendulum equation is roughly the same as the linear equation with a longer period, you can approximate the solution to the nonlinear equation by solving the linear equation but increasing the period. How good is that approximation?

Let’s do an example with θ₀ = 60° = π/3 radians. Then sin θ₀ = 0.866 but θ₀, in radians, is 1.047, so we definitely can’t say sin θ₀ is approximately θ₀. To make things simple, let’s set ℓ = g. Also, assume the pendulum starts from rest, i.e. θ'(0) = 0.

Here’s a plot of the solutions to the nonlinear and linear equations.

Obviously the solution to the nonlinear equation has a longer period. In fact it’s 7.32% longer. (The approximation above would have estimated 7.46%.)

Here’s a plot comparing the solution of the nonlinear equation and the solution to the linear equations with period stretched by 7.32%.

The solutions differ by less than the width of the plotting line, so it’s too small to see. But we can see there’s a difference when we subtract the two solutions.

Here’s a plot of the solutions to the nonlinear and linear equations.

Update: The plot above is misleading. Part of what it shows is numerical error from solving the pendulum equation. When we redo the plot using the exact solution the error is about half as large. And the error is periodic, as we’d expect. See this post for more on the exact solution using Jacobi functions and the differential equation solver that was used to make the original plot.

[1] The period of a pendulum depends on its length ℓ, and so we can think of the nonlinear term effectively replacing ℓ by a longer effective length ℓ_eff.