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# Energy
## Potential energy
> *Definition 1*: a force field $\mathbf{F}$ is conservative if it is [irrotational](../../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.

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# Momentum
> *Definition 1*: the **momentum** $\mathbf{p}$ of a particle is defined as the product of the mass and velocity of the particle
>
> $$
> \mathbf{p} = m \mathbf{v},
> $$
>
> with $m$ the mass of the particle and $\mathbf{v}$ the velocity of the particle.
For the case that $\mathbf{v}: t \to \mathbf{v}(t) \implies \mathbf{v}'(t) = \mathbf{a}(t)$ we have the following theorem.
> *Theorem 1*: let $\mathbf{v}$, $\mathbf{a}$ be the velocity and acceleration of a particle respectively, if we have
>
> $$
> \mathbf{v}: t \to \mathbf{v}(t) \implies \forall t \in \mathbb{R}: \mathbf{v}'(t) = \mathbf{a}(t),
> $$
>
> then
>
> $$
> \mathbf{p}'(t) = \mathbf{F}(t),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.

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# Newtonian formalism of mechanics
## 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 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.
> *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.
<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.
Definition 5 also implies the equation of motion, for a constant force a second order ordinary differential equation of the position.
> *Proposition 1*: in the case that a force only depends on position, the equation of motion is invariant to time inversion and time translation.
??? note "*Proof*:"
Will be added later.
This implies that for a moving a particle in a force field it can not be deduced at what point in time it occured and whether it is moving forward or backward in time.
> *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.
According to the observation of Galilei; all object fall with equal speed (in the absence of air friction), which implies that the ratio of inertial and gravitational mass is a constant for any kind of matter.
> *Principle 2*: the inertial and gravitational mass of a particle are equal.

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# Particle systems
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 1*: the internal interaction forces $\mathbf{F}_i$ in a system of $n \in \mathbb{N}$ particles with position $\mathbf{x}_i$ may be approximated by pairwise interaction forces given by
>
> $$
> \mathbf{F}_i (\mathbf{x}_i) = \sum_{j=1}^n \mathbf{F}_{ij}(\mathbf{x}_i, \mathbf{x}_j) \epsilon_{ij},
> $$
>
> for all $\mathbf{x}_i$ with $\mathbf{F}_{ij}$ the pairwise interaction force between particle $i$ and $j$.
For high density systems this approximation diverges.
## Systems with conservative internal forces
Considering a system of $n \in \mathbb{N}$ particles with position $\mathbf{x}_i$ and mass $m_i \in \mathbb{R}$ with conservative external forces $\mathbf{F}_i$. For each particle an equation of motion can be formulated using the pairwise interaction approximation (definition 1), obtaining
$$
m_i \mathbf{x}_i''(t) = \mathbf{F}_i(\mathbf{x}_i(t)) + \sum_{j=1}^n \mathbf{F}_{ij}(\mathbf{x}_i, \mathbf{x}_j) \epsilon_{ij},
$$
for all $t \in \mathbb{R}$ with $\mathbf{F}_{ij}$ the pairwise interaction force.
> *Definition 2*: the total mass $M$ of the system is defined as
>
> $$
> M = \sum_{i=1}^n m_i.
> $$
<br>
> *Definition 3*: the center of mass $\mathbf{R}: t \mapsto \mathbf{R}(t)$ of the system is defined as
>
> $$
> \mathbf{R}(t) = \frac{1}{M} \sum_{i=1}^n m_i \mathbf{x}_i(t),
> $$
>
> for all $t \in \mathbb{R}$.
<br>
> *Definition 4*: the total momentum $\mathbf{P}$ and angular momentum $\mathbf{J}$ of the system are defined as
>
> $$
> \begin{align*}
> \mathbf{P} &= \sum_{i=1}^n \mathbf{p}_i, \\
> \mathbf{J} &= \sum_{i=1}^n \mathbf{x}_i \times \mathbf{p}_i,
> \end{align*}
> $$
>
> with $\mathbf{p}_i$ the momentum of each particle.
We have for $\mathbf{P}: t \mapsto \mathbf{P}(t)$ the total momentum equivalently given by
$$
\mathbf{P}(t) = M \mathbf{R}'(t),
$$
for all $t \in \mathbb{R}$ with $\mathbf{R}: t \mapsto \mathbf{R}(t)$ the center of mass.
> *Definition 5*: the total external force $\mathbf{F}$ and torque $\mathbf{\Gamma}$ of the system are defined as
>
> $$
> \begin{align*}
> \mathbf{F} &= \sum_{i=1}^n \mathbf{F}_i, \\
> \mathbf{\Gamma} &= \sum_{i=1}^n \mathbf{x}_i \times \mathbf{F}_i,
> \end{align*}
> $$
>
> with $\mathbf{F}_i$ the conservative external force.
<br>
> *Proposition 1*: the total momentum $\mathbf{P}: t \mapsto \mathbf{P}(t)$ is related to the total external force $\mathbf{F}: t \mapsto \mathbf{F}(t)$ by
>
> $$
> \mathbf{P}'(t) = \mathbf{F}(t),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be adder later.
> *Proposition 2*: the total angular momentum $\mathbf{J}: t \mapsto \mathbf{J}(t)$ is related to the total external torque $\mathbf{\Gamma}: t \mapsto \mathbf{\Gamma}(t)$ by
>
> $$
> \mathbf{J}'(t) = \mathbf{\Gamma}(t),
> $$
>
> for all $t \in \mathbb{R}$ if the internal forces are central forces.
??? note "*Proof*:"
Will be adder later.
### Orbital and spin angular momentum
Considering internal position vectors $\mathbf{r}_i$ relative to the center of mass $\mathbf{r}_i = \mathbf{x}_i - \mathbf{R}$. I propose that the total angular momentum $\mathbf{J}$ can be expressed as a superposition of the orbital $\mathbf{L}$ and spin $\mathbf{S}$ angular momentum components given by
$$
\mathbf{J} = \mathbf{L} + \mathbf{S}.
$$
??? note "*Proof*:"
Will be added later.
> *Definition 6*: the orbital angular momentum $\mathbf{L}$ of the system is defined as
>
> $$
> \mathbf{L} = \mathbf{R} \times \mathbf{P},
> $$
>
> with $\mathbf{R}$ the center of mass and $\mathbf{P}$ the total momentum of the system.
<br>
> *Definition 7*: the spin angular momentum $\mathbf{S}: t \mapsto \mathbf{S}(t)$ of the system is defined as
>
> $$
> \mathbf{S}(t) = \sum_{i=1}^n \mathbf{r}_i(t) \times m_i \mathbf{r}'_i(t)
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbb{r}_i$ the internal position.
Analoguosly the orbital and spin torque may be defined.
> *Definition 8*: the orbital and spin torque $\mathbf{\Gamma}_{o,s}$ of the system are defined as
>
> $$
> \begin{align*}
> \mathbf{\Gamma}_o &= \mathbf{R} \times \mathbf{F}, \\
> \mathbf{\Gamma}_s &= \sum_{i=1}^n \mathbf{r}_i \times \mathbf{F}_i,
> \end{align*}
> $$
>
> with $\mathbf{R}$ the center of mass, $\mathbf{r}_i$ the internal position and $\mathbf{F}_i$ the conservative external force.
Similarly, the total torque $\mathbf{\Gamma}$ of the system is the superposition of the orbital and spin torque $\mathbf{\Gamma}_{o,s}$ given by
$$
\mathbf{\Gamma} = \mathbf{\Gamma}_o + \mathbf{\Gamma}_s.
$$
??? note "*Proof*:"
Will be added later.
> *Proposition 3*: let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the orbital angular momentum and let $\mathbf{S}: t \mapsto \mathbf{S}(t)$ be the spin angular momentum. Then we have
>
> $$
> \begin{align*}
> \mathbf{L}'(t) &= \mathbf{\Gamma}_o(t), \\
> \mathbf{S}'(t) &= \mathbf{\Gamma}_s(t),
> \end{align*}
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{\Gamma}_o: t \mapsto \mathbf{\Gamma}_o(t)$ and $\mathbf{\Gamma}_s: t \mapsto \mathbf{\Gamma}_s(t)$ the orbital and spin torque.
### Energy
> *Definition 9*: the total kinetic energy $T$ of the system is defined as
>
> $$
> T = \sum_{i=1}^n \frac{1}{2} m_i \|\mathbf{x}_i'\|^2,
> $$
>
> with $\mathbf{x}_i$ the position of each particle.
<br>
> *Definition 10*: the orbital and internal kinetic energy $T_{o,r}$ of the system are defined as
>
> $$
> \begin{align*}
> T_o = \frac{1}{2} M \|\mathbf{R}\|^2, \\
> T_r = \sum_{i=1}^n \frac{1}{2} m_i \|\mathbf{r}_i'\|^2,
> \end{align*}
> $$
>
> with $M$ the total mass, $\mathbf{R}$ the center of mass and $\mathbf{r}$ the internal position of each particle.
<br>
> *Proposition 4*: the total kinetic energy $T$ of the system is a superposition of the orbital and internal kinetic energy given by
>
> $$
> T = T_o + T_r.
> $$
??? note "*Proof*:"
Will be added later.
> *Proposition 5*: the dynamics of the orbital and kinetic energy $T_o: t \mapsto T_o(t)$ is decoupled
>
> $$
> T_o'(t) = \langle \mathbf{F}, \mathbf{R}'(t) \rangle,
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{F}$ the total external force and $\mathbf{R}$ the center of mass.
>
> The dynamics of the internal kinetic energy $T_r: t \mapsto T_r(t)$ is not decoupled
>
> $$
> T_r'(t) = \sum_{i=1}^n \langle \mathbf{f}_i, \mathbf{r}_i'(t) \rangle,
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{f}_i$ the sum of both external and internal forces for each particle.
??? note "*Proof*:"
Will be added later.

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# Rotation
Rotation is always viewed with respect to the axis of rotation, therefore in the following definitions the origin of the position is always implies to be the axis of rotation.
## Angular momentum
> *Definition 1*: the angular momentum $L$ of a point mass with position $\mathbf{r}$ and a momentum $\mathbf{p}$ is defined as
>
> $$
> \mathbf{L} = \mathbf{r} \times \mathbf{p},
> $$
>
> for all $\mathbf{r}$ and $\mathbf{p}$.
## Torque
> *Definition 2*: the torque $\mathbf{\Gamma}$ acting on a point mass with position $\mathbf{r}$ for a force $\mathbf{F}$ os defined as
>
> $$
> \mathbf{\Gamma} = \mathbf{r} \times \mathbf{F},
> $$
>
> for all $\mathbf{r}$ and $\mathbf{F}$.
The torque is related to the angular momentum by the following proposition.
> *Proposition 1*: let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the angular momentum of a point mass, then it holds that
>
> $$
> \mathbf{L}'(t) = \mathbf{\Gamma}(t),
> $$
>
> for a constant $\mathbf{r}$ and all $t \in \mathbb{R}$ with $\mathbf{\Gamma}: t \mapsto \mathbf{\Gamma}(t)$ the torque acting on the point mass.
??? note "*Proof*:"
Let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the angular momentum of a point mass and suppose $\mathbf{r}$ is constant, then
$$
\mathbf{L}'(t) \overset{\mathrm{def}} = d_t (\mathbf{r} \times \mathbf{p}(t)) = \mathbf{r} \times \mathbf{p}'(t),
$$
by [proposition](momentum.md) we have $\mathbf{p}'(t) = \mathbf{F}(t)$, therefore
$$
\mathbf{L}'(t) = \mathbf{r} \times \mathbf{F}(t) \overset{\mathrm{def}} = \mathbf{\Gamma}(t).
$$