Updated environments and added first part of Newtonian mechanics.

config/en/mkdocs.yaml

@@ -119,8 +119,9 @@ nav:
 
   - 'Physics':
     - physics/index.md
-#    - 'Mechanics':
+    - 'Mechanics':
-#      - 'Newtonian mechanics':
+      - 'Newtonian mechanics':
+        - 'Newtonian formalism': physics/mechanics/newtonian-mechanics/newtonian-formalism.md
 #      - 'Lagrangian mechanics':
 #      - 'Hamiltonian mechanics':
 #      - 'Relativistic mechanics':
@@ -129,6 +130,7 @@ nav:
 #      - 'Electrostatics':
 #      - 'Magnetostatics': 
       - 'Maxwell-equations': physics/electromagnetism/maxwell-equations.md
+#      - 'Electrodynamics': 
       - 'Optics':
         - 'Waves': physics/electromagnetism/optics/waves.md
         - 'Electromagnetic waves': physics/electromagnetism/optics/electromagnetic-waves.md

docs/en/mathematics/index.md

@@ -2,14 +2,15 @@
 
 Welcome to the mathematics page. Some special mathematical environments that will be used in this section are listed and explained below.
 
-* *Principles*: a fundamental rule or concept in mathematics serving as a basis for reasoning.
+* *Axioms*: fundamental assumptions or self-evident truths that serve as the basis for mathematical reasoning within a particular system. Axioms are not proved within the system but are taken as starting points from which other mathematical statements are deduced.
+* *Postulates*: a statement that is accepted without proof, typically serving as starting assumptions in a specific mathematical theory or system. Postulates are similar to axioms but are often specific to a particular branch in mathematics.
+* *Principles*: a fundamental rule or concept that govern mathematical reasoning. Principles may be derived from axioms, postulates, or empirical observations and are used to guide mathematical analysis or argumentation.
 * *Definitions* : a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.
 * *Theorems* : a mathematical statement that is proved to be true using rigorous mathematical reasoning. In a mathematical text, the term theorem is often reserved for the most important results.
 * *Propositions* : an often interesting result, but generally less important than a theorem.
 * *Lemmas* : a minor result whose purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem.
 * *Corollaries* : a result in which the proof relies heavily on a given theorem.
-* *Proofs* : a convincing argument that a certain mathematical statement is necessarily true. A proof
+* *Proofs* : a convincing argument that a certain mathematical statement is necessarily true. A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language.
-generally uses deductive reasoning and logic but also contains some amount of ordinary language.
 
 The mathematics sections of this wiki are based on various books and lectures. A comprehensive list of references can be found below.
 

docs/en/mathematics/set-theory/additional-axioms.md

@@ -2,7 +2,7 @@
 
 ## Axiom of choice
 
-> *Principle*: let $C$ be a collection of nonempty sets. Then there exists a map 
+> *Axiom*: let $C$ be a collection of nonempty sets. Then there exists a map 
 >
 >$$
 >    f: C \to \bigcap_{A \in C} A
@@ -22,6 +22,6 @@ The following statements are equivalent to the axiom of choice.
 
 ## Axiom of regularity
 
-> *Principle*: let $X$ be a nonempty set of sets. Then $X$ contains an element $Y$ with $X \cap Y = \varnothing$. 
+> *Axiom*: let $X$ be a nonempty set of sets. Then $X$ contains an element $Y$ with $X \cap Y = \varnothing$. 
 
 As a result of this axiom any set $S$ cannot contain itself.

docs/en/physics/index.md

@@ -2,15 +2,15 @@
 
 Welcome to the physics page. Some special physical environments that will be used in this section are listed and explained below.
 
-* *Principles*: a fundamental rule or concept in physics serving as a basis for reasoning. 
+* *Axioms*: fundamental assumptions or self-evident truths that serve as the basis for physical reasoning within a particular system. Axioms are not proved within the system but are taken as starting points from which other physical statements are deduced.
-* *Assumptions*: a less fundamental rule or concept in physics that is taken to be true such that certain phenoma can be simplified. 
+* *Postulates*: a statement that is accepted without proof, typically serving as starting assumptions in a specific physical theory or system. Postulates are similar to axioms but are often specific to a particular branch in physics.
-* *Definitions*: a precise and unambiguous description of the meaning of a physical term. It characterizes the meaning of a word by giving all the properties an only those properties that must be true.
+* *Principles*: a fundamental rule or concept that govern physical reasoning. Principles may be derived from axioms, postulates, or empirical observations and are used to guide mathematical analysis or argumentation.
-* *Theorems*: a mathematical statement that is proved to be true using rigorous mathematical reasoning. In a mathematical text, the term theorem is often reserved for the most important results.
+* *Definitions* : a precise and unambiguous description of the meaning of a physical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.
-* *Laws*: a physical statement that is proved to be true, under the made assumptions and posed principles. In a physical text, the term law is often reserved for the most important results.
+* *Theorems* : a mathematical statement that is proved to be true using rigorous mathematical reasoning. In a mathematical text, the term theorem is often reserved for the most important results.
-* *Propositions*: an often interesting result, but generally less important than a theorem or law.
+* *Propositions* : an often interesting result, but generally less important than a theorem.
-* *Lemmas*: a minor result whose purpose is to help in proving a theorem or law. It is a stepping stone on the path to proving a theorem or law.
+* *Lemmas* : a minor result whose purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem.
-* *Corollaries*: a result in which the proof relies heavily on a given theorem or law.
+* *Corollaries* : a result in which the proof relies heavily on a given theorem.
-* *Proofs*: a convincing argument that a certain physical statement is necessarily true. A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language.
+* *Proofs* : a convincing argument that a certain mathematical statement is necessarily true. A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language.
 
 The physics sections of this wiki are based on various books and lectures. A comprehensive list of references can be found below.
 

docs/en/physics/mechanics/newtonian-mechanics/newtonian-formalism.md

@@ -0,0 +1,72 @@
+# Newtonian formalism
+
+## Fundamental assumptions
+
+> *Postulate 1*: there exists an absolute space in which the axioms of Euclidean geometry hold.
+
+The properties of space are constant, immutable and entirely independent of the presence of objects and of all dynamical processes that occur within it.
+
+> *Postulate 2*: there exists an absolute time, entirely independent.
+
+From postulate 1 and 2 we obtain the notion that simultaneity is absolute. In the sense that incidents that occur simultaneously in one reference system, occur simultaneously in all reference systems, independent of their mutual dynamic states or relations.
+
+The definition of a reference system will follow in the next section.
+
+> *Principle of relativity*: all physical axioms are of identical form in all **inertial** reference systems.
+
+It follows from the principle of relativity that the notion of absolute velocity does not exist.
+
+> *Postulate 3*: space and time are continuous, homogeneous and isotropic.
+
+Implying that there is no fundamental limit to the precision of measurements of spatial positions, velocities and time intervals. There are no special locations or instances in time all positions and times are equivalent. The properties of space and time are invariant under translations. And there are no special directions, all directions are equivalent. The properties of space and time are invariant under rotations and reflections.
+
+## Galilean transformations
+
+> *Definition 1*: a **reference system** is an abstract coordinate system whose origin, orientation, and scale are specified by a set of geometric points whose position is identified both mathematically and physically.
+
+From the definition of a reference system and postulates 1, 2 and 3 the Galilean transformations may be posed, which may be used to transform between the coordinates of two reference systems. 
+
+> *Principle 1*: let $(\mathbf{x},t) \in \mathbb{R}^4$ be a general point in spacetime. 
+> 
+> A uniform motion with velocity $\mathbf{v}$ is given by
+>
+> $$
+>   (\mathbf{x},t) \mapsto (\mathbf{x} + \mathbf{v}t,t),
+> $$
+>
+> for all $\mathbf{v}\in \mathbb{R}^3$.
+>
+> A translation by $(\mathbf{a},t)$ is given by
+>
+> $$
+>   (\mathbf{x},t) \mapsto (\mathbf{x} + \mathbf{a},t + s),
+> $$
+>
+> for all $(\mathbf{a},t) \in \mathbb{R}^4$. 
+>
+> A rotation by $R$ is given by
+>
+> $$
+>   (\mathbf{x},t) \mapsto (R \mathbf{x},t),
+> $$
+>
+> for all orthogonal transformations $R: \mathbb{R}^3 \to \mathbb{R}^3$. 
+
+The Galilean transformations may form a Lie group. 
+
+## Axioms of Newton
+
+> *Axiom 1*: in the absence of external forces, a particle moves with a constant speed along a straight line. 
+>
+> *Axiom 2:* the net force on a particle is equal to the rate at which the particle's momentum changes with time.
+>
+> *Axiom 3:* if two particles exert forces on each other, then the mutual forces have equal magnitudes but opposite directions.
+
+From axiom 1 and the principle of relativity the definition of a inertial reference system may be posed. 
+
+> *Definition 2*: an **inertial reference system** is a reference system in which the first axiom of Newton holds.
+
+This implies that a inertial reference system is reference system not undergoing any acceleration. Therefore we may postulate the following. 
+
+> *Postulate 4*: inertial reference systems exist.
+