In (classical) mechanics we usually deal with the concept of a particle. A particle is a body in which we neglect its size. A planet can be considered as a particle if we are dealing with its motion around the Sun. But not if we are studying its rotation around its axis.
The position of a particle can be written as the radius vector |r>, whose components are the Cartesian coordinates (x,y,z). Since the Unicode arrow for vectors is not very good, x⃗, X⃗, we choose |x> as the notation for vectors. It is a ket notation taken from Dirac's way of writing quantum mechanics. We can also write <x| for a vector (in fact it would be a form, but it does not matter here). In classical mechanics you can use |x> and <x| without distinction, but surely you will prefer to write a scalar product as <x|y>. Another way of seeing this is considering <x| as a row vector and |x> as a column vector, although we usually deal with |x> and write it in a row! But if we were to be very formal, all |x> should be column vectors and <x| row vectors. This way, <x|x> is the scalar product giving the squared magnitude of the vector, x², while |x><x| would be a matrix, or more appropriately, a tensor.
The total time derivative of |r> is d|r>/dt = |v>, which is the velocity. Velocity is always a vector, and its magnitude, v, is called speed, although many times we are sloppy with these terms and it is not very important.
The second total time derivative of the position vector is d²|r>/dt² = |a>, which is the acceleration of the particle.
For time derivatives we can use Newton's dot notation. It consists in painting little dots *above* the letters. Leibniz's notation, d/dt, is sometimes very unpleasant. For dx/dt we write ẋ and for d²x/dt² we write ẍ. We can also do the third derivative d³x/dt³ = x⃛ and the fourth derivative d⁴/dt⁴ = x⃜. From third derivative above, there is no need to write all the dots in a single row. You can write ᪴, ⁖, ⁘, ⁙, or what you fancy. Very rarely we use third or fourth derivatives. Fortunately, for first and second derivatives, the Unicode dot notation works well even for capital letters (̇Ȧ,Ä). Third and fourth derivatives don't work well with capital (A⃛, A⃜). An alternative notation can be to write [x·] = ẋ, [x:] = ẍ and so on. Newton invented this notation and he didn't call them derivatives, but *fluxions*. For example, ẋ is the fluxion of x. Conversely, x is the *fluent* of ẋ. Since x, y and z are the most used letters for fluents, you can now understand the name of this site, fluents.xyz, clearly inspired in Newton's nomenclature.
With vectors, we can write |v> = |ẋ> and |a> = |v̇> = |ẍ>.
For N particles, we need 3N coordinates to specify the positions of all of them.
The total number of *independent* quantities needed to specify the *positions* of all the particles in a system is called the number of *degrees of freedom* (dof). There quantities are not necessarily the Cartesian coordinates of the particles. Any set of independent quantities q₁, q₂, ..., qₛ which completely determine the position of a system with s dof is called a set of *generalised coordinates*. Their fluxions, q̇ᵢ, are called its generalised velocities.
When all the values of qᵢ are given, we still have not determined the "mechanical state" of the system. For any given set of qᵢ, the particles can have any set of velocities.
If all qᵢ and all q̇ᵢ are specified, then the mechanical system is completely determined. Then, we can predict all future motion of the system, and also all past motion. Mathematically, this means that once all q,q̇ (we already omit subscripts to refer to the whole set) are given, then all q̈ are uniquely specified as well.
The relation between q's, q̇'s and q̈'s are called *equations of motion*. They are *second-order* differential equations for q(t).By integrating them, we can obtain q(t), q̇(t) and q̈(t).