ABOUT THE LECTURES
Mechanics is the study of the motion of bodies. Rest is a particular state of motion. By mechanics, we mean classical mechanics which can be either non-relativistic or relativistic. We will start with the study of non-relativistic mechanics and toward the end of the course, we will study the special theory of relativity. The study of non-relativistic mechanics is naturally divided into two categories namely, statics and dynamics. In statics, the goal is to determine all the forces necessary to achieve equilibrium. In dynamics, the goal is to calculate the dependence on time of the positions, velocities, and accelerations of the bodies under analysis. If for some reason the explicit dependence on time of the acceleration of each body is known, then the problem is called kinematic.
Kinematics is the easiest part of mechanics. Indeed, the physical problem is reduced to a simple mathematical one namely, the integration of the differential equations
\begin{equation}
\frac{d\vec{v}}{dt}=\vec{a}\ \ ;\ \ \frac{d\vec{r}}{dt}=\vec{v}\ ,
\end{equation}
which are called kinematic equations of motion. Let me emphasize that in kinematics the acceleration has to be an explicitly known function of time.
The physics behind kinematics resides in the cause of acceleration. But such a cause is not relevant to solve the kinematic equations. The kinematic equations are not physical laws. They are just mathematical definitions: Acceleration is the time derivative of velocity, and velocity is the time derivative of position.
There is a tendency to confuse the first kinematic equation with the second law of Newton: Just multiply by the mass on both sides, they say. The truth is that they are not equivalent. Indeed, the second law is a law of physics while the kinematic equation is a mathematical definition. The confusion arises because, in the particular case of constant force, the second law reduces to the kinematic equation. But if the force depends on the position, the velocity, or both, as happens with the gravitational force or with a mass coupled to a spring or with the magnetic force, that is no longer true.
We will study Newton’s laws of motion, their consequences, and their applications to simple systems of bodies. Important consequences of the second law are easily derived by using the properties of vector operations. The second law together with a law of force that is position-dependent lead to differential equations for the position. Thus for example, if the force is due to a spring obeying Hooke’s law, we arrive at differential equations that describe harmonic oscillations. Another example is the motion under central forces (to be defined in later lectures) that leads to differential equations whose solution is easier and more insightful to find by using polar coordinates. We first study the motion of point masses and then we extend the analysis to extended rigid bodies. This extension will require the use of integrals, simple, double, and triple. Sometimes, the integrals can be easily calculated if the use is made of either spherical or cylindrical coordinates. Clearly, we need to know the necessary mathematical tools to solve these problems. Therefore, we will start by reviewing the mathematics that we will need to solve problems in this course.
Differential calculus in one variable
In physics in general and in mechanics in particular, it is necessary to calculate what is called the instantaneous variation of a function with respect to its variable. The instantaneous variation is just what in calculus is known as the derivative of the function.
The derivative of a linear function $y=mx+b$ is just the slope $m$. This slope is the same for any value of $x$. For an arbitrary function $f$, the slope $f^{\prime}(x)$ at $x$ is defined as
\begin{equation}
f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}
\end{equation}
From the definition of derivative one can deduce the following properties: ($a$ is a constant while $f$ and $g$ are functions)
\begin{align}
(f+g)^{\prime}(x)&=f^{\prime}(x)+g^{\prime}(x)\\
(af)^{\prime}(x)&=af^{\prime}(x)\\
(fg)^{\prime}(x)&=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)\\
\left(\frac{f}{g}\right)^{\prime}(x)&=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{[g(x)]^2}\\
\left(f\circ g\right)^{\prime}(x)&=f^{\prime}(g(x))g^{\prime}(x)\ .
\end{align}
The calculation of the derivatives even of simple functions such as the trigonometric ones is tedious and sometimes awkward, but once they are calculated we can just memorize them and use them as building blocks to calculate the derivative of more complicated functions. Thus, we can build the following list of derivatives of the most common functions: ($c$ and $\alpha$ are constants)
\begin{align}
c^{\prime}&=0\\
\left(x^{\alpha}\right)^{\prime} & =\alpha x^{\alpha-1}\\
\left(e^{\alpha x}\right)^{\prime} &=\alpha e^{\alpha x}\\
\left(\alpha^{x}\right)^{\prime}&=\alpha^{x}\ln(\alpha)\ (\alpha>0)\\
\left(\ln x\right)^{\prime}&=1/x\\
\left(\sin(\alpha x)\right)^{\prime}&=\alpha\cos(\alpha x)\\
\left(\cos(\alpha x)\right)^{\prime}&=-\alpha\sin(\alpha x)\\
\left(\tan(\alpha x)\right)^{\prime}&=\alpha\sec^2(\alpha x)\ ,
\end{align}
and anything else can be calculated with help of this list and the properties of the derivative.
Taylor approximation in one variable
In physics, sometimes we know the exact analytic expression of a function, but the exact expression is often complicated and makes it difficult to understand the behavior of the system described by it. Very often the exact expression is complicated because it describes the system for a large range of values of the variable. It is common that all that is needed is to describe the system in a small range of the variable around some specific value, and then it is very convenient to use a functional form that is only an approximation but is simpler than the exact one. It turns out that polynomials are the simplest among all of the functions. This is one of the reasons to look for a polynomial approximation of a function.
\[\frac{1}{\sqrt{1+(\ln x)^2}}\approx 1-\frac{1}{2}(x-1)^2\]
We will see several examples of this situation in this curse.