Added section to Newtonian formalism and introduced next sections.

config/en/mkdocs.yaml

@@ -122,6 +122,10 @@ nav:
     - 'Mechanics':
       - 'Newtonian mechanics':
         - 'Newtonian formalism': physics/mechanics/newtonian-mechanics/newtonian-formalism.md
+        - 'Momentum': physics/mechanics/newtonian-mechanics/momentum.md
+        - 'Energy': physics/mechanics/newtonian-mechanics/energy.md
+        - 'Rotation': physics/mechanics/newtonian-mechanics/rotation.md
+        - 'Particle systems': physics/mechanics/newtonian-mechanics/particle-systems.md
 #      - 'Lagrangian mechanics':
 #      - 'Hamiltonian mechanics':
 #      - 'Relativistic mechanics':

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

@@ -0,0 +1,58 @@
+# Energy
+
+## Potential energy
+
+> *Definition 1*: a force field $\mathbf{F}$ is conservative if it is [irrotational](physics/mathematical-physics/vector-analysis/vector-operators/#potentials)
+>
+> $$
+>   \nabla \times \mathbf{F} = 0,
+> $$
+>
+> obtaining a scalar potential $V$ such that
+>
+> $$
+>   \mathbf{F} = - \nabla V,
+> $$
+>
+> referred to as the potential energy.
+
+## Kinetic energy
+
+> *Definition 2*: the kinetic energy $T: t \mapsto T(t)$ of a pointmass $m \in \mathbb{R}$ with position $x: t \mapsto x(t)$ subject to a force $\mathbf{F}: x \mapsto \mathbf{F}(x)$ is defined as
+>
+> $$
+>   T(t) - T(0) = \int_0^t \langle \mathbf{F}(x), dx \rangle,
+> $$
+>
+> for all $t \in \mathbb{R}$. 
+
+<br>
+
+> *Proposition 1*: the kinetic energy $T: t \mapsto T(t)$ of a pointmass $m \in \mathbb{R}$ with position $x: t \mapsto x(t)$ subject to a force $\mathbf{F}: x \mapsto \mathbf{F}(x)$ is given by
+>
+> $$
+>   T(t) - T(0) = \frac{1}{2} m \|x'(t)\|^2 - \frac{1}{2} m \|x'(0)\|^2,
+> $$
+>
+> for all $t \in \mathbb{R}$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Energy conservation
+
+> *Theorem 1*: for a pointmass $m \in \mathbb{R}$ with position $x: t \mapsto x(t)$ subject to a force $\mathbf{F}: x \mapsto \mathbf{F}(x)$ we have that
+>
+> $$
+>   T(x) + V(x) = T(0) + V(0) \overset{\mathrm{def}} = E,
+> $$
+>
+> for all x, with $T: x \mapsto T(x)$ and $V: x \mapsto V(x)$ the kinetic and potential energy of the point mass. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Obtaining conservation of energy with $E \in \mathbb{R}$ the total (constant) energy of the system.
+

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

@@ -0,0 +1 @@
+# Momentum

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

@@ -60,7 +60,7 @@ The Galilean transformations may form a Lie group.
 >
 > *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.
+> *Axiom 3:* if two particles exert forces onto 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. 
 
@@ -70,3 +70,51 @@ This implies that a inertial reference system is reference system not undergoing
 
 > *Postulate 4*: inertial reference systems exist.
 
+<br>
+
+> *Definition 3*: considering two particles $i \in \{1,2\}$ which exert forces onto each other having accelerations $\mathbf{a}_i$. Since by the 2nd and 3rd axiom we have that $\mathbf{a}_1 = - \mathbf{a}_2$ and that the ratio of their magnitudes is a constant we define the ratio of the inertial masses by
+>
+> $$
+>   \frac{m_1}{m_2} = \frac{\|\mathbf{a}_2\|}{\|\mathbf{a}_1\|}. 
+> $$
+
+A particle with a mass can be considered as a point mass, which is defined below.
+
+> *Definition 4*: a point mass is defined as a point in space and time appointed with a mass.
+
+## Forces
+
+> *Definition 5*: a force $\mathbf{F}$ is defined as
+>
+> $$
+>   \mathbf{F} = m \mathbf{a},
+> $$
+>
+> with $m \in \mathbb{R}$ the inertial mass and $\mathbf{a}$ the acceleration of the particle.
+
+For a system of particles we have the mutual forces among the selected particles referred to as internal forces, otherwise external forces. If there are no external forces, the system is called closed, otherwise open.
+
+> *Definition 6*: a central force $\mathbf{F}$ representing the interaction between two point masses at positions $\mathbf{x}_1$ and $\mathbf{x}_2$ is defined as 
+>
+> $$
+>   \mathbf{F} = F(\mathbf{x}_1,\mathbf{x}_2) \frac{\mathbf{x}_2 - \mathbf{x}_1}{\|\mathbf{x}_2 - \mathbf{x}_1\|} \overset{\mathrm{def}} = F(\mathbf{x}_1,\mathbf{x}_2) \mathbf{e}_r,
+> $$
+>
+> with $F: (\mathbf{x}_1,\mathbf{x}_2) \mapsto F(\mathbf{x}_1,\mathbf{x}_2)$ the magnitude.
+
+Which for a isotropic central force depends only on the distance between the pointmasses $\|\mathbf{x}_2 - \mathbf{x}_1\|$. 
+
+### Gravitational force of Newton
+
+> *Postulate 5*: the force $\mathbf{F}$ between two particles described by their positions $\mathbf{x}_{1,2}: t \mapsto \mathbf{x}_{1,2}(t)$ is given by
+>
+> $$
+>   \mathbf{F} = G \frac{m_1 m_2}{\|\mathbf{x}_2 - \mathbf{x}_1\|^2} \mathbf{e}_r, 
+> $$
+>
+> with $m_{1,2} \in \mathbb{R}$ the gravitational mass of both particles and $G \in \mathbb{R}$ the gravitational constant. 
+
+<br>
+
+> *Principle 2*: the inertial and gravitational mass of a particle are equal.
+

docs/en/physics/mechanics/newtonian-mechanics/particle-systems.md

@@ -0,0 +1 @@
+# Particle systems

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

@@ -0,0 +1,2 @@
+# Rotation
+