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Newtonian mechanics is the standard theory that allows us to describe every day object like an apple that falls to the ground.
We do this by solving, so called equation of motion. The solutions of these equations are particle paths which describe how a particle will move as time passes on.
For example, the path of a point-like object could look as follows:
The basis of Newtonian Mechanics is summarized by three laws, commonly called "Newton's laws of motion":
The equations and the theory of Newtonian mechanics were deduced historically from experiments. This worked pretty good but is highly unsatisfactory from a theoretical point of view. Newton proposed his second law
\begin{equation} \label{newtonssecond} \tag{1} F = m \frac{d²}{dt²}q =m \ddot q, \end{equation}
where $m$ is the mass, $\ddot q$ the acceleration and $F$ the force that acts on the object in question.
To describe some object we simply have to deduce equations for the forces $F$ that act on the object from experiments and put them on the left-hand side of the equation. This yields a differential equation, which we must solve for $q=q(t)$.
The solution is called the trajectory of the object and describes the position of the object for every moment in time. This is one framework for classical mechanics and it‘s useful for many, many things.
Every student of physics has to solve Newton's second law for many different situations.
First law:
If the forces acting on an object are balanced, i.e. the total force is zero $\vec F=0$, the velocity of the object will remain constant: $\vec v=\text{const}$. So when the velocity is zero, it will remain zero. If the velocity has some other value it will keep it.
Forces are balanced $ \vec F=0$ | ||||||||||||||||||||||||
$ \vec a=0$ | ||||||||||||||||||||||||
Object at rest: $\vec v=0$ | Object in motion $ \vec v\neq 0$ | |||||||||||||||||||||||
Object stays at rest: $\vec v=0$ | Object remains in motion $ \vec v \neq 0$; same $ \vec v$. | |||||||||||||||||||||||
Second law:
$$ \vec F = m \vec a$$
Forces are unbalanced $ \vec F\neq 0$ | ||||||||||||||||||||||||
$ \vec a \neq 0$ | ||||||||||||||||||||||||
acceleration $\vec a$ depends directly on the net-force $\vec F$ that acts on the object | acceleration $\vec a$ depends inversly on the mass $m$ of the object | |||||||||||||||||||||||
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Newton's second law is \begin{equation} F=ma \end{equation} which describes a particle moving in $\mathbb{R}^n$.
Its position, which we call $q$, depends on the time $t\in\mathbb{R}$.
Therefore, the position defines a function, \[ q \colon \mathbb{R}\longrightarrow\mathbb{R}^n. \] Using this function $q$ we can define the corresponding velocity, \[ v=\dot{q} \colon \mathbb{R}\longrightarrow\mathbb{R}^n \] where $\dot{q}=\frac{dq}{dt}$. Analogously, we can define the acceleration \[ a=\ddot{q} \colon \mathbb{R}\longrightarrow\mathbb{R}^n. \] We call $m>0$ be the mass of the particle. The last puzzle piece in the equation $F$ is a vector field on $\mathbb{R}^n$ which we call the force. Newton second law is 2nd-order differential equation for $q\colon\mathbb{R}\rightarrow\mathbb{R}^n$. It has a unique solution given some $q(t_0)$ and $\dot{q}(t_0)$, provided the vector field $F$ is smooth and bounded (i.e., $|F(x)|<B$ for some $B>0$, for all $x\in\mathbb{R}^n$).
Classical Mechanics is the framework that we use to describe the behaviour of objects we meet in everyday life: from the famous apple that falls from a tree, to the motion of planets in our solar system.
In addition to these "school physics" application, there is a high relevance to almost all topics in modern physics. Namely, it turns out that there is not one unique way to describe classical mechanics but several. Instead, we can describe classical mechanics using:
Each approach has unique strengths and it usually depends on the system that we want to describe which approach is best suited. It is important to note that these three approaches are completely equivalent in what they do. This means, the physics they describe is exactly the same. However, dependending on the system in question, sometimes calculations and the interpretation is easier in Lagrangian Mechanics and sometimes in Hamiltonian Mechanics.
It makes sense to study these various approaches instead of just one because:
Moreover, classical mechanics is an ideal playground to get familiar with many important mathematical tools and concepts.
Nature and Nature’s laws lay hid in night:
God said, "Let Newton be!" and all was light. Alexander Pope
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