As a simple application of the use of theangular velocity vector, we may evaluate the power being expended by the torqueacting on a rigid body. The power, of course, is the rate of change of workwith time; in three dimensions, the power turns out to be P=τ⋅ω .
关于角速度矢量，有一个简单的应用，一个扭力，对于作用于刚体之上，其功率，我们可以估计。功率，当然就是功随时间的变化率；在三维中，功率就是P=τ⋅ω 。

All the formulas that we wrote for planerotation can be generalized to three dimensions. For example, if a rigid bodyis turning about a certain axis with angular velocity ω , we might ask, “What is the velocity of a point at a certain radialposition r ?” We shall leave it as a problem for the student to show that thevelocity of a particle in a rigid body is given by v=ω×r, where ω is the angular velocity and r is the position. Also, as another example of cross products, we had aformula for Coriolis force, which can also be written using cross products: Fc=2mv×ω. That is, if a particle is moving with velocity v in a coordinate system which is, in fact, rotating with angularvelocity ω , and we want to think in terms of the rotating coordinate system,then we have to add the pseudo force Fc .
我们为平面旋转，写了若干公式，所有这些公式，都可以被普遍化到三维中。例如，如果一个刚体，以角速度ω，绕某轴旋转，我们可以问：“一点，在半径r处，其矢速为何？”我们把这个问题，留给学生，用以指出，刚体上的粒子的矢速，通过v=ω×r 来给予，这里，ω是角速度，r是位置。作为另外一个叉积的例子，我们有一个科里奥利力的公式，它也可以用叉积来写：Fc=2mv×ω。也就是说，如果一个粒子，以矢速v，在一个坐标系中运动，而坐标系，事实上是以角速度ω在旋转，我们想用旋转坐标系，来思考，因此，我们就必须加上伪力Fc。

20–3The gyroscope 20-3 陀螺仪[A1]

Fig. 20–1.Before: axis is horizontal; momentumabout vertical axis=0 . After: axis is vertical; momentum about vertical axis is still zero;man and chair spin in direction opposite to spin of the wheel. 图20-1 之前：轴是水平的；关于垂直轴的动量=0。之后：轴是垂直的；关于垂直轴的动量仍=0；人和椅子旋转的方向，与轮子旋转的方向相反。
Let us now return to the law ofconservation of angular momentum. This law may be demonstrated with a rapidlyspinning wheel, or gyroscope, as follows (see Fig. 20–1). Ifwe sit on a swivel chair and hold the spinning wheel with its axis horizontal,the wheel has an angular momentum about the horizontal axis. Angular momentumaround a vertical axis cannot change because of the (frictionless) pivotof the chair, so if we turn the axis of the wheel into the vertical, then thewheel would have angular momentum about the vertical axis, because it is nowspinning about this axis. But the system (wheel, ourself, and chair) cannothave a vertical component, so we and the chair have to turn in the directionopposite to the spin of the wheel, to balance it.
现在，让我们回到角动量守恒规律。这条规律，可以用一个快速旋转的轮子、或陀螺仪，演证如下（见图20-1）。如果我们坐在一个可来回转动的椅子上，抓着一个旋转的轮子，其轴水平，那么，这个轮子，关于水平轴，就有一个角动量。绕着一个垂直轴的角动量，并无变化，因为（无摩擦的）椅子的转轴，于是，如果我们把此轮之轴，变为垂直，那么，这个椅子，将会有关于垂直轴的角动量，因为它现在绕这个轴在转。但是，此系统（轮子、我们和椅子）并不能有一个垂直的分量，所以，轮子往一个方向转，我们和椅子，就要往相反的方向转，才能平衡。

Fig. 20–2.A gyroscope. 图20-2 一个陀螺仪
First let us analyze in more detail thething we have just described. What is surprising, and what we must understand,is the origin of the forces which turn us and the chair around as we turn theaxis of the gyroscope toward the vertical. Figure 20–2 showsthe wheel spinning rapidly about the y -axis. Therefore its angular velocity is about that axis and, it turnsout, its angular momentum is likewise in that direction. Now suppose that wewish to rotate the wheel about the x -axis at a small angular velocity Ω ; what forces are required? After a short time Δt , the axis has turned to a new position, tilted at an angle Δθwith the horizontal. Since the major part of the angular momentum isdue to the spin on the axis (very little is contributed by the slow turning),we see that the angular momentum vector has changed. What is the change inangular momentum? The angular momentum does not change in magnitude, butit does change in direction by an amount Δθ . The magnitude of the vector ΔL is thus ΔL=L0Δθ , so that the torque, which is the time rate of change of the angularmomentum, is τ=ΔL/Δt=L0Δθ/Δt=L0Ω. Taking the directions of the various quantities into account, we seethat
τ=Ω×L0. (20.15)
Thus, if Ω and L0 are both horizontal, as shown in the figure, τ is vertical. To produce such a torque, horizontal forces Fand −F must be applied at the ends of the axle. How are these forces applied?By our hands, as we try to rotate the axis of the wheel into the verticaldirection. But Newton’s Third Law demands that equal and opposite forces (andequal and opposite torques) act on us. This causes us to rotatein the opposite sense about the vertical axis z .
首先，对于上面谈到的事情，让我们更仔细地分析。有件事情，令人惊奇，也是我们必须理解的，它就是当我们把陀螺仪的轴，转到垂直方向时，有一个力，让我们和椅子旋转，此力起源为何呢？图20-2显示，轮子绕着y轴，快速转动。因此，其角速度，就是关于那个轴的，结果就是，它的角动量，同样也在那个方向。现在，假设我们希望，让轮子以小的角速度Ω，绕着x轴转动，那么，需要多大的力呢？在一个短的时间Δt之后，轴转到了一个新的位置，与水平方向夹角为Δθ。由于角动量的主要部分，要归于轴的旋转（慢速转动的贡献，非常小），我们看到，角动量矢量改变了。角动量的改变，是什么呢？角动量并没有在大小上改变，但是，它在方向上，改变了一个量Δθ。这样，矢量的大小ΔL就是ΔL=L0Δθ，于是，扭力、亦即角动量的时间变化率，就是τ=ΔL/Δt=L0Δθ/Δt=L0Ω。把不同量的方向，都考虑进来，我们就看到：
τ=Ω×L0. (20.15)
这样，如果Ω和L0都是水平的，如图所示，那么，τ就是垂直的。要产生这样的扭力，水平方向的力F和 −F，就应该被应用到，轴的末端。这些力，是如何被应用的呢？通过我们的手，由于我们尝试把轮子的轴，转到垂直方向。但是，牛顿第三规律要求，作用于我们的诸力，相等且相反（及相等且相反的扭力）。这就使得我们，在与垂直轴z相反的意义上，旋转。

Fig. 20–3.A rapidly spinning top. Note thatthe direction of the torque vector is the direction of the precession. 图20-3 一个高速旋转的陀螺。注意，扭力矢量的方向，就是旋进的方向。
This result can be generalized for arapidly spinning top. In the familiar case of a spinning top, gravity acting onits center of mass furnishes a torque about the point of contact with the floor(see Fig. 20–3).This torque is in the horizontal direction, and causes the top to precess withits axis moving in a circular cone about the vertical. If Ω is the (vertical) angular velocity of precession, we again find that
τ=dL/dt=Ω×L0.
Thus, when we apply a torque to a rapidly spinning top, the directionof the precessional motion is in the direction of the torque, or at rightangles to the forces producing the torque.
这个结果，可以推广到高速旋转的陀螺。旋转中的陀螺这种情况，我们熟悉，陀螺地板，有一个接触点（见图20-3），重力作用于其质心，提供了一个关于接触点的扭力。这个扭力，在水平方向，但却导致陀螺旋进，即让其轴，在一个关于垂直轴的圆锥中运动。如果Ω是（垂直的）旋进的角速度，我们再次发现：
τ=dL/dt=Ω×L0.
这样，当我们把扭力，应用到一个高速旋转的陀螺上时，旋进运动的方向，就是在扭力的方向，或者，与产生扭力的力，成直角。

We may now claim to understand the precessionof gyroscopes, and indeed we do, mathematically. However, this is amathematical thing which, in a sense, appears as a “miracle.” It will turn out,as we go to more and more advanced physics, that many simple things can be deducedmathematically more rapidly than they can be really understood in a fundamentalor simple sense. This is a strange characteristic, and as we get into more andmore advanced work there are circumstances in which mathematics will produceresults which no one has really been able to understand in any directfashion. An example is the Dirac equation, which appears in a very simple andbeautiful form, but whose consequences are hard to understand. In ourparticular case, the precession of a top looks like some kind of a miracle involvingright angles and circles, and twists and right-hand screws. What we should tryto do is to understand it in a more physical way.
对于陀螺的旋进，我们现在可以宣称，我们理解了，我们也确实是在数学上理解了！然而，这是一个数学上的事情，它在某种意义上，似乎表现得像个“奇迹”。随着我们在物理上，进展的越来越多，那么，将会发现，很多简单的事情，虽然从数学上，可以推出，但是，要从一个基础的、或简明的意义上，去理解它们，则要比推出，慢得多。这是一个奇怪的特点，在我们的工作，越来越深入时，将会有些情形，在其中，数学会产生一些结果，对于这些结果，没有人能够以任何直接的方式，真正地理解。迪拉克（Dirac）方程，就是一例，其形式，简明而漂亮，但其后果，很难理解。在我们的具体例子中，陀螺的旋进，看上去，像某种奇迹，牵扯到直角和圆，以及扭转和右手螺丝。我们想做的，就是尝试，以一种更物理的方式，去理解它。

How can we explain the torque in terms ofthe real forces and the accelerations? We note that when the wheel is precessing,the particles that are going around the wheel are not really moving in a planebecause the wheel is precessing (see Fig. 20–4). Aswe explained previously (Fig. 19–4), the particles which are crossing through theprecession axis are moving in curved paths, and this requiresapplication of a lateral force. This is supplied by our pushing on the axle,which then communicates the force to the rim through the spokes. “Wait,”someone says, “what about the particles that are going back on the other side?”It does not take long to decide that there must be a force in the oppositedirection on that side. The net force that we have to apply is thereforezero. The forces balance out, but one of them must be applied at oneside of the wheel, and the other must be applied at the other side of thewheel. We could apply these forces directly, but because the wheel is solid weare allowed to do it by pushing on the axle, since forces can be carried upthrough the spokes.
我们如何用力和加速度，来解释扭力呢？我们注意到，当轮子在旋进时，跟着轮子走的粒子，并没有真正在一个平面上移动，因为轮子在旋进（见图20-4）。正如我们前面（图19-4）所解释，那些穿越过旋进轴的粒子，是在曲线上运动，这就要求，有一个横向力的应用。这个过程，如此完成：我们推轴，轴经轮辐，把力传给边缘。或曰：“等等，粒子回到另一边，又是怎么回事？”。在另一边，必有一力，方向相反，决定这点，无需花费很长时间。我们要应用的净力，因此就是零。诸力平衡抵消了，但是，其中一个，必须被应用于轮子的一边，而另一个，则必须在轮子的另一边。我们可以直接地应用这些力，但是，因为轮子是固体，所以允许我们，通过推轴，来做此事，由于，力可经过轮辐，被传到边缘。

What we have so far proved is that if thewheel is precessing, it can balance the torque due to gravity or some otherapplied torque. But all we have shown is that this is a solution of an equation.That is, if the torque is given, and if we get the spinning started right,then the wheel will precess smoothly and uniformly. But we have not proved (andit is not true) that a uniform precession is the most general motion a spinningbody can undergo as the result of a given torque. The general motion involvesalso a “wobbling” about the mean precession. This “wobbling” is called nutation.
迄今为止，我们所证明的就是，如果轮子是在旋进，那么，它就可以平衡：由重力或其他被应用的力所带来的扭力。但是，我们业已指出的就是：这是一个方程的解答。也就是说，如果扭力，被给予了，如果我让旋转正确地启动了，那么，轮子将会平稳地、均匀地旋进。但是，我们尚未证明（且这也不是真的），对于一个物体，如果给定扭力，那么，该物体所要经历的最普遍的运动，就是一个均匀的旋进。普遍运动，也会牵扯到关于平均旋进的“摇晃”。这种摇晃，被称为章动。

Some people like to say that when one exertsa torque on a gyroscope, it turns and it precesses, and that the torque producesthe precession. It is very strange that when one suddenly lets go of agyroscope, it does not fall under the action of gravity, but movessidewise instead! Why is it that the downward force of the gravity,which we know and feel, makes it go sidewise? All theformulas in the world like (20.15)are not going to tell us, because (20.15)is a special equation, valid only after the gyroscope is precessing nicely.What really happens, in detail, is the following. If we were to hold the axisabsolutely fixed, so that it cannot precess in any manner (but the top isspinning) then there is no torque acting, not even a torque from gravity,because it is balanced by our fingers. But if we suddenly let go, then therewill instantaneously be a torque from gravity. Anyone in his right mind wouldthink that the top would fall, and that is what it starts to do, as can be seenif the top is not spinning too fast.
有些人会说，当某人在一个陀螺仪上，施加一个扭力时，那么，它就会转动、旋进，扭力产生了旋进。但有件事，很奇怪，就是当某人，突然放手一个陀螺仪，则它在重力的作用下，并不倒下，而是会向边上运动！重力是向下的，这我们都知道、能感觉到，但为什么重力会让它向边上走呢？所有类似（20.15）的公式，都不能告诉我们，因为，（20.15）是一个特殊的方程，它只是在陀螺仪正常旋进时，才有效。真正发生的事情如下。如果我们把轴抓牢，绝对地固定住，让它不能以任何方式旋进（但陀螺是在旋转的），因此，就没有扭力在起作用，甚至都没有来自重力的扭力，因为它被我们的指头，给抵消了。但是，如果我们突然放手，那么一瞬间，就会有一个来自重力的扭力。任何思维正常的人，都会想到，陀螺将会倒下，它也是这么做的，如果陀螺仪，不是旋转的很快的话，这一幕，就会看到。

The gyro actually does fall, as we wouldexpect. But as soon as it falls, it is then turning, and if this turning were tocontinue, a torque would be required. In the absence of a torque in thisdirection, the gyro begins to “fall” in the direction opposite that of themissing force. This gives the gyro a component of motion around the verticalaxis, as it would have in steady precession. But the actual motion “overshoots”the steady precessional velocity, and the axis actually rises again to thelevel from which it started. The path followed by the end of the axle is acycloid (the path followed by a pebble that is stuck in the tread of anautomobile tire). Ordinarily, this motion is too quick for the eye to follow,and it damps out quickly because of the friction in the gimbal bearings,leaving only the steady precessional drift (Fig. 20–5). Theslower the wheel spins, the more obvious the nutation is.
陀螺仪实际上是倒下了，正如我们所期待的那样。但是，刚倒下时，它仍在转动，如果这个转动持续，那么，就要求，有一个扭力。在这个方向上的扭力的缺失，导致陀螺仪开始“倒下”，与那个失踪的力所导致的扭力，方向相反{？}。这给了陀螺仪一个绕着垂直轴的运动分量，正如它将有一个稳定旋进一样。但是，实际的运动，“超过了”稳定的旋进矢速，使得轴，又开始往起抬，达到它开始的那个水平。轴的尾端所经过的路径，是一条转轮线（在汽车轮胎上，塞一个鹅卵石，该石所走过的路径）。通常，这个运动，对于眼睛来说，太快，因为万向支架轴承的摩擦力，它很快就会阻尼掉，只留下，稳定的旋进漂移（图20-5）。轮子旋转地越慢，章动就越明显。

Fig. 20–5.Actual motion of tip of axis ofgyroscope under gravity just after releasing axis previously held fixed. 图20-5 把陀螺仪的轴，抓牢，然后释放，在重力的影响下，陀螺仪的轴的尖端的实际运动。

When the motion settles down, the axis ofthe gyro is a little bit lower than it was at the start. Why? (These are themore complicated details, but we bring them in because we do not want thereader to get the idea that the gyroscope is an absolute miracle. It isa wonderful thing, but it is not a miracle.) If we were holding the axis absolutelyhorizontally, and suddenly let go, then the simple precession equation wouldtell us that it precesses, that it goes around in a horizontal plane. But thatis impossible! Although we neglected it before, it is true that the wheel has somemoment of inertia about the precession axis, and if it is moving about thataxis, even slowly, it has a weak angular momentum about the axis. Where did itcome from? If the pivots are perfect, there is no torque about the vertical axis.How then does it get to precess if there is no change in the angularmomentum? The answer is that the cycloidal motion of the end of the axis dampsdown to the average, steady motion of the center of the equivalent rolling circle.That is, it settles down a little bit low. Because it is low, the spin angularmomentum now has a small vertical component, which is exactly what is neededfor the precession. So you see it has to go down a little, in order to goaround. It has to yield a little bit to the gravity; by turning its axis down alittle bit, it maintains the rotation about the vertical axis. That, then, is theway a gyroscope works.
当陀螺仪最终停止不动时，其轴要比开始时，略低一点。为什么？（这些是更复杂的细节，之所以说起，因为我们不希望读者，有这样的想法，即陀螺仪是一个绝对的神迹。它确实很奇妙，但并不是一个神迹。）如果我们绝对水平地拿着轴，然后，突然放手，那么，简单的旋进方程，将会告诉我们，它在旋进。它绕着一个水平平面在转。但这是不可能的！因为有一点，我们以前忽略了，但它是真的，即，轮子关于旋进轴，有一些惯性力矩，如果它绕此轴在运动，即便很慢，那么，它也有一个关于此轴的很弱的角动量。此角动量，从何而来？如果支点轴承，是完美的，那么，关于垂直轴，就没有扭力。因此，如果角动量没有变化，那么，它是如何旋进的呢？答案就是，轴的端点的旋轮线运动，减慢到平均水平，即同等滚动圆的圆心的稳定运动。也就是说，它平静下来，要低一点。因为，它是低了点，旋转角动量现在就有了一个小的垂直分量，这正是旋进所需要的。于是，你看，为了转着走，它必须要低一点。在重力面前，它必须屈服一点；通过把它的轴降低一点，它保持了关于垂直轴的旋转。这就是陀螺仪的工作方式。

Fig. 20–6.The angular momentum of arotating body is not necessarily parallel to the angular velocity. 图20-6 一个旋转物体的角动量，并不必然地平行于角速度。
Before we leave the subject of rotations inthree dimensions, we shall discuss, at least qualitatively, a few effects that occurin three-dimensional rotations that are not self-evident. The main effect is that,in general, the angular momentum of a rigid body is not necessarily inthe same direction as the angular velocity. Consider a wheel that is fastenedonto a shaft in a lopsided fashion, but with the axis through the center ofgravity, to be sure (Fig. 20–6).When we spin the wheel around the axis, anybody knows that there will be shakingat the bearings because of the lopsided way we have it mounted. Qualitatively,we know that in the rotating system there is centrifugal force acting on thewheel, trying to throw its mass as far as possible from the axis. This tends toline up the plane of the wheel so that it is perpendicular to the axis. Toresist this tendency, a torque is exerted by the bearings. If there is a torqueexerted by the bearings, there must be a rate of change of angular momentum.How can there be a rate of change of angular momentum when we are simply turningthe wheel about the axis? Suppose we break the angular velocity ωinto components ω1 and ω2 perpendicular and parallel to the plane of the wheel. What is theangular momentum? The moments of inertia about these two axes are different,so the angular momentum components, which (in these particular, special axes only)are equal to the moments of inertia times the corresponding angular velocitycomponents, are in a different ratio than are the angular velocitycomponents. Therefore the angular momentum vector is in a direction in space notalong the axis. When we turn the object, we have to turn the angular momentumvector in space, so we must exert torques on the shaft.
三维中的旋转，是一个主题，在我们离开它之前，我们将讨论一些效果，至少是定性地；这些效果，出现在三维旋转中，但并非不证自明。一般来说，主要的效果就是，一个刚体的角动量，并非必然地与其角速度，在同一个方向。考虑一个轮子，以一种一侧高一侧低的方式，捆了一个杆子，但保证轴，过其重心（图20-6）。当我们让轮子，绕轴旋转时，任何人都会知道，因为我们绑它的方式，所以在支撑处，会有摇晃。在旋转系统中，我们可以定性地知道，有离心力，作用于轮子，尝试把其质量，从轴处，往外抛。这趋向于把轮子的平面，排起来，使它垂直于轴。要抵抗这个趋势，支撑点就要施加一个扭力。如果，通过支撑点，施加了一个扭力，那么，就必然会有角动量的变化率。当我们只是让轮子绕轴转时，怎么就会有角动量的变化率呢？假设我们把角速度ω，分成分量ω1 和 ω2，分别垂直于、和平行于轮子的平面。那么，角动量是什么呢？关于这两个轴的惯性力矩，是不同的，于是，角动量分量（在这个具体的、特殊的轴中）就等于惯性力距，乘以相应的角速度分量，且角动量分量的比率，与角速度分量的比率，不同。因此，角动量矢量，在空间的方向，并非沿着轴的。当我们转动对象时，我们必然会转动空间中的角动量矢量，所以，我们对于杆子，必然施加了扭力。

Although it is much too complicated toprove here, there is a very important and interesting property of the moment ofinertia which is easy to describe and to use, and which is the basis of ourabove analysis. This property is the following: Any rigid body, even anirregular one like a potato, possesses three mutually perpendicular axesthrough the CM, such that the moment of inertia about one of these axes has thegreatest possible value for any axis through the CM, the moment of inertiaabout another of the axes has the minimum possible value, and the momentof inertia about the third axis is intermediate between these two (or equal to oneof them). These axes are called the principal axes of the body, and theyhave the important property that if the body is rotating about one of them, itsangular momentum is in the same direction as the angular velocity. For a bodyhaving axes of symmetry, the principal axes are along the symmetry axes.
惯性力矩，有一个属性，非常重要和有趣，它很容易描述和使用，且它是我们上面分析的基础；虽然，在这里证明它，太过困难。这个属性如下：任何刚体，甚至一个不规则的刚体，比如土豆，都拥有三个相互垂直的轴，通过质心，这样，对于任何通过质心的轴来说，关于这三个轴中的一个的惯性力距，就有着最大可能的值，关于这三个轴中的另外一个的惯性力矩，则有着最小可能的值，关于其中第三个轴的惯性力矩，在前两个之间（或等于其中一个）。这些轴，被称为物体的主轴，它们有着重要的属性，即如果物体绕着其中之一旋转，其角动量就与角速度，在同一个方向。如果一个物体有对称轴，那么，主轴就是沿着对称轴的。

Fig. 20–7.The angular velocity and angularmomentum of a rigid body (A>B>C) . 图20-7 一个刚体的角速度和角动量(A>B>C)。

If we take the x -, y -, and z -axes along the principal axes, and call the corresponding principalmoments of inertia A , B , and C , we may easily evaluate the angular momentum and the kinetic energyof rotation of the body for any angular velocity ω . If we resolve ω into components ωx , ωy , and ωz along the x -, y -, z -axes, and use unit vectors i , j , k , also along x , y , z , we may write the angular momentum as
L=Aωxi+Bωyj+Cωzk. (20.16)
The kinetic energy of rotation is
(20.17)
如果我们，沿着主轴，取x、y、z轴，并称相应的主惯性力距为A , B , 和C，那么，对于一个物体，角速度为ω，我们很容易估计其角动量、和旋转的动能。如果我们把ω，分解成沿着x、y、z轴的分量ωx , ωy , 和 ωz，且使用也是沿着x、y、z轴的单位矢量i , j , k，那么，我们可以把角动量写为：
L=Aωxi+Bωyj+Cωzk. (20.16)
把旋转的动能写为：
(20.17)
1、That this is true can be derived by compounding the displacements ofthe particles of the body during an infinitesimal time Δt . It is not self-evident, and is left to those who are interested totry to figure it out.
脚注1、在极短的时间内，物体的粒子，各有移动过的距离，通过混合这些距离，可以推出，它是真的。它不是不证自明的，留给感兴趣的人去思考。

Chapter21.The Harmonic[A1] Oscillator第21章 谐波振荡
21–1Linear differential equations 21-1 线性微分方程
In the study of physics, usually the courseis divided into a series of subjects, such as mechanics, electricity, optics,etc., and one studies one subject after the other. For example, this course hasso far dealt mostly with mechanics. But a strange thing occurs again and again:the equations which appear in different fields of physics, and even in othersciences, are often almost exactly the same, so that many phenomena have analogsin these different fields. To take the simplest example, the propagation of soundwaves is in many ways analogous to the propagation of light waves. If we study acousticsin great detail we discover that much of the work is the same as it would be ifwe were studying optics in great detail. So the study of a phenomenon in one fieldmay permit an extension of our knowledge in another field. It is best torealize from the first that such extensions are possible, for otherwise onemight not understand the reason for spending a great deal of time and energy onwhat appears to be only a small part of mechanics.
研究物理，通常会把课程，分成若干主题，例如力学、电学、光学等，然后，一个一个研究。例如，本课程，目前处理的，都是力学。但是，有个奇怪的事情，会不断地发生：在物理学的不同领域中、甚至在其他科学中，有些方程，不断出现，它们几乎一样，于是，在这些领域中，有很多现象，可以类比。举一个最简单的例子，声波的传播，在很多方面，可与光波的传播，相类比。如果我们仔细研究声学，我们将会发现，其中很多工作，与光学中的一样。所以，一个领域中，有若干现象，研究它们，允许我们，把我们的知识，扩展到另外一个领域。在开始的时候，就意识到这种扩展是可能的，当然最好，否则的话，人们可能就不会理解，为什么要把大量的时间和精力，花在似乎只是一个小的力学的问题上。