In every set of five, the first three Lagrange points are slightly unstable.

在点中，前三拉格朗日点略微不稳定。

Every Lagrange point in our solar system has its quirks.

太阳系中的拉格朗日点都有其特点。

Gunthorp Farms in Lagrange, Indiana is selling chicken directly to customers.

位于印第安纳州拉格朗日的冈索普农场直接向顾客出售鸡肉。

I'm looking for Trojan asteroids at Earth's LFive Lagrange point.

在地球的L5拉格朗日点寻找特洛伊小行星。

Consider the five Lagrange points generated by the Sun and the Earth.

以太阳和地球产生的拉格朗日点为例。

However, Lagrange points are places where all these forces balance out.

但是，拉格朗日点是所有这些力平衡的地方。

Euclid is orbiting the sun about 1.5 million kilometers away from Earth at " Lagrange Point Two, " or L2.

欧几里得在距离地球约150万公里的“拉格朗日L2点”处绕太阳运行。

The James Webb space telescope has arrived at its final destination, a region of space called the second Lagrange point.

詹姆斯·韦伯太空望远镜已经抵达它的最终目的地，一被称为第二拉格朗日点的空间区域。

Exactly like Newtonian physics or its higher versions of the Lagrange or Hamiltonians.

就像牛顿或其更高版本的拉格朗日或哈密顿力一样。

And then later, Euler and Lagrange reformulated Newtonian mechanics in a totally different way.

后来，欧拉和拉格朗日以完全不同的方式重新表述了牛顿力。

Pairs of massive bodies in our solar system generate sets of five Lagrange points.

太阳系中对大型天体会产生拉格朗日点。

So I guess Hamilton is building on Lagrange in that way.

所以我猜想汉密尔顿是以这种方式借鉴拉格朗日的。

Space telescopes are often placed near Lagrange points for their orbits around Earth and the Sun.

太空望远镜常被安置在拉格朗日点附近，以便其围绕地球和太阳的轨道运行。

Euclid launched on the 1st July 2023, and has  since arrived at its designated L2 Lagrange point.

Euclid 于 2023 年 7 月 1 日发射，现已到达指定的 L2 拉格朗日点。

We send it to a place called Lagrange Point 2, about a million miles farther out from the Sun than we are.

我们把它送到一叫第二拉格朗日点的地方，一比我们距离太阳还要远大约一百万英里的地方。

Sure, so I mentioned the classical mechanics and the Euler-Lagrange formulation.

当然，我提到了经典力和欧拉-拉格朗日方程。

And around five years later, just a year after Maupertuis' death, Lagrange succeeded in providing a general proof.

大约年后，也就是莫佩尔蒂去世一年后，拉格朗日成功地给出了普遍的证明。

First we'll explain the general approach Euler and Lagrange came up with.

首先，我们将解释 Euler 和 Lagrange 提出的一般方法。

Named after the 18th century mathematician who deduced their positions, Lagrange points are rare places of equilibrium in our constantly shifting universe.

拉格朗日点是以推导出其位置的 18 世纪数家拉格朗日的名字命名的，拉格朗日点是我们不断变化的宇宙中罕见的平衡点。

[Derek] Now I wanna explain Lagrange's proof, but I don't want to do it the way he did.

- [德里克] 现在我想解释拉格朗日的证明，但我不想按照他的方式去做。