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@ -2946,7 +2946,7 @@ The differential work is now the true derivative of the potential function.
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Conservative forces enjoy a number of remarkable properties. Initially, conservative forces are defined as forces that perform the same work along any path joining the same initial and final points, as expressed by eq. (3.13). Simple calculus reasoning is then used to prove that a force is conservative if and only if the work it performs vanishes over any arbitrary closed path, see eq. (3.14). Finally, conservative forces are shown to be derivable from a potential, as expressed by eq. (3.16). Consequently, the work done by a conservative force along any path joining two points can be evaluated as the difference between the potential function evaluated at these two points, see eq. (3.18).
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保守力具有一些显著的特性。首先，保守力被定义为连接相同始点和终点的任意路径上所做的功相同，如公式(3.13)所示。随后，利用简单的微积分推理证明，一个力是保守力，当且仅当它在任意闭合路径上所做的功为零，参见公式(3.14)。最后，证明保守力可以从势函数导出，如公式(3.16)所示。因此，保守力沿连接两个点的任意路径所做的功，可以评估为在这些点处势函数之差，参见公式(3.18)。
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# Examples of conservative forces
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### Examples of conservative forces
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To illustrate these concepts, consider the gravity force acting on a particle of mass $m$ located at the surface of the earth. It can easily be shown that this force is conservative. Therefore, the scalar potential, $V$ , of the gravity forces is $V=m g\,\underline{{{r}}}{\cdot}\bar{\imath}_{3}=m g x_{3}$ , where ${\underline{{r}}}=x_{1}{\bar{\imath}}_{1}+x_{2}{\bar{\imath}}_{2}+x_{3}{\bar{\imath}}_{3}$ is the position vector of the particle. The gravity force, $\underline{{F}}_{g}$ , acting on the particle can be obtained from this potential using eq. (3.17) to find $\underline{{\tilde{F_{g}}}}=-\nabla V=-\partial V/\partial x_{3}\,\bar{\iota}_{3}=-m g\bar{\iota}_{3}$ , and the gravity forces is said to be “derived from a potential.”
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@ -2968,7 +2968,7 @@ At first glance, the potential of a gravity force and the strain energy of an el
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乍一看，重力势能和弹性弹簧的应变能似乎是截然不同、无关的概念。然而，这两个量都共享一个共同的属性：力可以从这些标量势能中推导出来。考虑一个质量为 $m$ 的粒子，该粒子连接到刚度常数为 $k$ 的弹性弹簧，并且受到一个在弹簧方向上的重力作用。粒子的向下位移 $u$ 测量了弹簧的伸长量和粒子的高度。施加的外部重力可以从势能 $V\,=\,m g u$ 中推导出来，得到 $F_{g}\,=\,-\partial V/\partial u\,=\,-m g$；弹簧中的回复力可以从应变能 $V=1/2\,k u^{2}$ 中推导出来，后者也可以被视为内部力的势能，得到 $F_{s}=-\partial V/\partial u=-k u.$ 。因此，作用在粒子上的两个力可以从势能中推导出来。
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# 3.2.1 Principle of conservation of energy
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### 3.2.1 Principle of conservation of energy
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The forces applied to a particle can be divided into two categories: the conservative forces, which can be derived from a potential, and the non-conservative forces, for which no potential function exists. The principle of work and energy, eq. (3.11), now becomes
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作用于颗粒的力可以分为两类：保守力，它可以从势函数导出；以及非保守力，对于非保守力不存在势函数。做功与能量原理，公式 (3.11)，现在变为
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$$
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If the particle is acted upon by conservative forces only, the principle of work and energy reduces to
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如果该叶片仅受保守力作用，工作与能量原理简化为：
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$$
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E_{f}=E_{i}.
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$$
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This statement is known as the principle of conservation of energy.
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这个声明被称为能量守恒定律。
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**Principle 2 (Principle of conservation of energy for a particle)** If a particle is subjected to conservative forces only, the total mechanical energy is preserved.
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Clearly, the term “conservative forces” stems from the fact that in the sole presence of such forces, the total mechanical energy of the particle is conserved.
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In view of the principle of work and energy, work, kinetic energy, potential energy, and total mechanical energy all share the same units, force times distance, $\mathbf{N}{\cdot}\mathbf{m}$ . A Joule is defined as $1\;\mathrm{J}=1\;\mathrm{N}{\cdot}\mathrm{m}$ . Although the moment of a force has the same units, $\mathbf{N}{\cdot}\mathbf{m}$ , Joules are used only when dealing with energy; in other words, a $10\,\mathrm{{N\cdotm}}$ moment should not be referred to as a $10\,\mathrm{J}$ moment.
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In view of the principle of work and energy, work, kinetic energy, potential energy, and total mechanical energy all share the same units, force times distance, $\mathbf{N}{\cdot}\mathbf{m}$ . A Joule is defined as $1\;\mathrm{J}=1\;\mathrm{N}{\cdot}\mathrm{m}$ . Although the moment of a force has the same units, $\mathbf{N}{\cdot}\mathbf{m}$ , Joules are used only when dealing with energy; in other words, a $10\mathbf{N}{\cdot}\mathbf{m}$ moment should not be referred to as a $10\,\mathrm{J}$ moment.
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The work done by force over a period of time from $t_{i}$ to $t_{f}$ , see eq. (3.12), can be written as
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**原理 2 (单个粒子的能量守恒原理)** 如果一个粒子仅受到保守力作用，则其总机械能守恒。
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显然，“保守力”一词源于这样一个事实：仅在这些力单独存在的情况下，粒子的总机械能才得以守恒。
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鉴于功与能量的原理，功、动能、势能和总机械能都具有相同的单位，即力乘以距离，$\mathbf{N}{\cdot}\mathbf{m}$。 一焦耳被定义为 $1\;\mathrm{J}=1\;\mathrm{N}{\cdot}\mathrm{m}$。 尽管力矩也具有相同的单位，$\mathbf{N}{\cdot}\mathbf{m}$，但焦耳仅用于处理能量；换句话说，一个 $10\,\mathrm{{N\cdotm}}$ 的力矩不应被称为 $10\,\mathrm{J}$ 的力矩。
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鉴于功与能量的原理，功、动能、势能和总机械能都具有相同的单位，即力乘以距离，$\mathbf{N}{\cdot}\mathbf{m}$。 一焦耳被定义为 $1\;\mathrm{J}=1\;\mathrm{N}{\cdot}\mathrm{m}$。 尽管力矩也具有相同的单位，$\mathbf{N}{\cdot}\mathbf{m}$，但焦耳仅用于处理能量；换句话说，一个 $10\mathbf{N}{\cdot}\mathbf{m}$ 的力矩不应被称为 $10\,\mathrm{J}$ 的力矩。
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在时间 $t_{i}$ 到 $t_{f}$ 期间，力所做的功，见公式 (3.12)，可以写为</text>
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在时间 $t_{i}$ 到 $t_{f}$ 期间，力所做的功，见公式 (3.12)，可以写为
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$$
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W_{t_{i}\rightarrow t_{f}}=\int_{t_{i}}^{t_{f}}\underline{{F}}^{T}\mathrm{d}\underline{{r}}=\int_{t_{i}}^{t_{f}}\underline{{F}}^{T}\frac{\mathrm{d}\underline{{r}}}{\mathrm{d}t}\,\mathrm{d}t=\int_{t_{i}}^{t_{f}}\underline{{F}}^{T}\underline{{v}}\,\mathrm{d}t.
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$$
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The last integrand, $\underline{{F}}^{T}\underline{{v}}.$ , is the power of the externally applied forces; it is a measure of the work done by the forces per unit time. Power has units of work divided by time, J/s. A Watt is defined as $1\ \mathrm{W}=1\ \mathrm{J}/\mathrm{s}=1\ \mathrm{N}{\cdot}\mathrm{m}/\mathrm{s}$ .
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# 3.2.2 Potential of common conservative forces
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最后一个积分项，$\underline{{F}}^{T}\underline{{v}}.$，代表**外力所做的功的功率；它是衡量单位时间内所做功的指标**。功率的单位是功除以时间，即 J/s。瓦特被定义为 **$1\ \mathrm{W}=1\ \mathrm{J}/\mathrm{s}=1\ \mathrm{N}{\cdot}\mathrm{m}/\mathrm{s}$** 。
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### 3.2.2 Potential of common conservative forces
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In the previous section, it was shown that conservative forces are associated with special functions called potential functions, from which they can be derived. A few commonly used potential functions will be derived in this section.
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在上一节中，已经论证了保守力与称为势函数（potential functions）的特殊函数相关联，并且可以从这些函数中推导出来。本节将推导几个常用的势函数。
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#### Work done by a central force
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# Work done by a central force
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Fig. 3.5. A central force.
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Because the differential work can be expressed as an exact differential, the central force is a conservative force, and its potential is the integral of the magnitude of the central force. The potential is defined within a constant: adding a constant to the potential does not alter the magnitude of the central force.
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# The potential of gravity forces
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首先，我们将评估由中心力所做的功。中心力是指其作用线通过空间固定惯性点的力，且其大小仅取决于粒子与该固定点之间的距离 $r$。图 3.5 所示的是一个质量为 $m$ 的粒子，它受到一个作用线通过点 O（惯性系原点）的中心力 $\underline{{F}}$ 的作用。由于粒子到原点之间的距离是中心力定义的固有属性，因此使用第 2.7.2 节中定义的球坐标系来表达粒子的位置似乎是自然的选择。
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在球坐标系中表达的粒子速度，见公式 (2.95b)，为 $\underline{{v}}=\dot{r}\;\bar{e}_{1}+\dot{r}\dot{\phi}\;\bar{e}_{2}+\dot{r}\dot{\theta}\sin\phi\;\bar{e}_{3}$ 。将此关系乘以 $\mathrm{d}t$ 揭示了粒子位置的增量 $\mathrm{d}\underline{{r}}$ 与坐标增量 $\mathrm{d}r,\mathrm{d}\theta$ 和 $\mathrm{d}\phi$ 之间的关系，即 $\mathrm{d}\underline{{r}}=\mathrm{d}r\;\bar{e}_{1}+r\mathrm{d}\phi\;\bar{e}_{2}+r\mathrm{d}\theta\sin\phi\;\bar{e}_{3}$ 。另一方面，中心力可以表示为 $\underline{{F}}=-f(r)\bar{e}_{1}$ ，其中 $f(r)$ 是仅取决于 $r$ 的大小，而 $\bar{e}_{1}$ 是其作用线，始终通过点 $\mathbf{o}$。
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现在，由中心力所做的微分功变为 $\mathrm{d}W=\underline{{F}}^{T}\mathrm{d}\underline{{r}} = -f(r){\bar{e}}_{1}^{T}(\mathrm{d}r\;{\bar{e}}_{1}+r\mathrm{d}\phi\;{\bar{e}}_{2}+r\mathrm{d}\theta\sin\phi\;{\bar{e}}_{3})=-f(r)\mathrm{d}r$ 。中心力的势能 $V$ 定义为
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$$
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f(r)={\frac{\mathrm{d}V}{\mathrm{d}r}}.
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$$
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根据这个定义，由中心力所做的微分功变为
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$$
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\mathrm{d}W=-\frac{\mathrm{d}V}{\mathrm{d}r}\mathrm{d}r=-\mathrm{d}(V).
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$$
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由于微分功可以表示为全微分，因此中心力是一个保守力，其势能是中心力大小的积分。势能定义在常数范围内：给势能添加一个常数不会改变中心力的数值大小。
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#### The potential of gravity forces
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An important example of central forces are gravitational forces, as described by Newton’s gravitation law. The magnitude of the gravitational force is given by eq. (3.5) as $f(r)=G M m/r^{2}$ . The gravitational force acts on an particle of mass $m$ due to the presence of another particle of mass $M$ assumed to be fixed with respect to an inertial frame; fig. 3.1 shows that such force is a central force. The potential function for the gravity forces then follows from eq. (3.27) as
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@ -3078,10 +3099,25 @@ $$
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This potential function is the potential of gravity forces for particles located near the surface of the earth. The height, $h$ , of the particle is measured from a reference elevation, called the datum, which is selected in an arbitrary manner. Indeed, changing the datum is equivalent to adding a constant to the potential function, leaving the gravitation forces unchanged.
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# The strain energy function of an elastic spring
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一个重要的中心力例子是万有引力，正如牛顿万有引力定律所描述的。万有引力的大小由公式 (3.5) 给出，为 $f(r)=G M m/r^{2}$ 。由于存在固定在惯性参考系中的质量为 $M$ 的另一个粒子，质量为 $m$ 的粒子会受到万有引力作用；图 3.1 显示这种力是一种中心力。万有引力势函数则可以从公式 (3.27) 导出，为
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$$
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V(r)=-G{\frac{M m}{r}}.
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$$
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这个势函数被称为万有引力势。
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现在考虑一个位于地球表面上方高度 $h$ 的粒子；这意味着 $\boldsymbol{r}\,=\,\boldsymbol{r}_{e}\,+\,\boldsymbol{h}$ ，其中 $r_{e}$ 是地球的半径。如果粒子靠近地球表面，则 $h\ll r_{e}$ 且 $1/r\,=\,1/[r_{e}\,\left(1+h/r_{e}\right)]\approx\,(1-h/r_{e})/r_{e}$ 。势函数现在变为：$V(r)=-G M m/r_{e}+G M m h/r_{e}^{2}$ 。由于这个表达式的第一项是一个常数，它可以被省略，从而得到势函数为
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$$
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V(r)=G\frac{M m h}{r_{e}^{2}}=m g h.
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$$
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这个势函数是位于地球表面附近的粒子的万有引力势。粒子的高度，$h$ ，是从一个参考高程，称为基准面，测量得到的，基准面的选择是任意的。事实上，改变基准面等同于在势函数中添加一个常数，而不改变万有引力。
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#### The strain energy function of an elastic spring
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Consider now a particle of mass $m$ connected to a rectilinear spring; the other end of the spring is attached to inertial point O, as depicted in fig. 3.6. The spring can stretch elastically, but is massless; it practice, this means that the mass of the spring is negligible with respect to that of the particle.
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现在考虑一个质量为 $m$ 的粒子，它与一个直线弹簧相连；弹簧的另一端固定在惯性点 O，如图 3.6 所示。该弹簧可以弹性伸展，但质量可以忽略不计；实际上，这意味着弹簧的质量相对于粒子的质量可以忽略不计。
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Fig. 3.6. Particle connected to an elastic spring.
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@ -3105,7 +3141,7 @@ Next, the set of external forces that maintained the steady deformation $\varDel
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Both kinetic and strain energy functions are positive-definite functions, i.e., $K=$ $1/2\:m v^{2}>0$ for any arbitrary speed of the particle $v\neq0$ and $V=1/2\;k\varDelta^{2}>0$ for any stretch of the elastic spring $\varDelta\neq0$ . Consider a strain energy function of the form $\dot{V}\,=\,1/2\,\,k_{0}\varDelta^{2}+1/3\,\,\dot{k}_{1}\varDelta^{3}$ ; this strain energy function vanishes for $\varDelta_{\mathrm{cr}}\,=$ $-3/2\,k_{0}/k_{1}$ . For stretches $\varDelta<\varDelta_{\mathrm{cr}}$ , the strain energy becomes negative, hence this strain energy function is invalid because it is not positive-definite. For $\varDelta<\varDelta_{\mathrm{cr}}$ , the spring will add energy to the system; energy is being created, a physical impossibility for a passive device.
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# The strain energy function of a torsional spring
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#### The strain energy function of a torsional spring
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Consider the planar problem depicted in fig. 3.7: a particle of mass $m$ is connected to a rigid rod of length $\ell$ . The rod pivots about inertial point O, where a torsional spring of stiffness constant $k$ is located. The torsional spring applies a moment to the rigid rod about point O, which is then transmitted to the particle in the form of a force $\underline{{F}}_{:}$ , acting in the direction normal to the rod; this force is clearly not a central force. The position of the particle will be represented by polar coordinates, $r$ and $\theta$ , see section 2.7.1. The velocity of the particle is $\underline{{v}}\;=\;\dot{r}~\bar{e}_{1}\:+\:r\dot{\theta}~\bar{e}_{2}$ , see eq. (2.91b). Because the rod is rigid, $\dot{r}=0$ , and multiplying the velocity relationship by $\mathrm{d}t$ implies $\underline{{\mathrm{d}}}\underline{{r}}=\ell\mathrm{d}\theta\;\bar{e}_{2}$ . The force vector, $\underline{{F}}$ , has a line of action along $\bar{e}_{2}$ and its magnitude is a function of the sole angle $\theta:\underline{{F}}=-f(\theta)\bar{e}_{2}$ . The differential work done by this force now becomes
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