Ars Numerandi

Inertial forces

$$\vec{r}=\vec{r}'+\vec{r}_0,\quad\quad r^i\vec{e}_i={r'}^i\vec{e}'_i+r_0^i\vec{e}_i,\quad\quad\frac{d}{dt}\vec{e'}_i=\vec{\omega}\times\vec{e'}_i$$

$$\frac{dr^i}{dt}\vec{e}_i=\frac{d{r'}^i}{dt}\vec{e}'_i+\vec{\omega}\times\vec{r}'+\frac{dr_0^i}{dt}\vec{e}_i,\quad\quad\vec{v}=\vec{v}'+\vec{\omega}\times\vec{r}'+\vec{v}_0$$

$$\frac{d^2r^i}{dt^2}\vec{e}_i=\frac{d^2{r'}^i}{dt^2}\vec{e}'_i+\vec{\omega}\times\vec{v}'+\frac{d\vec{\omega}}{dt}\times\vec{r}'+\vec{\omega}\times\vec{v}'+\vec{\omega}\times\left(\vec{\omega}\times\vec{r}'\right)+\frac{d^2r_0^i}{dt^2}\vec{e}_i$$
$$\vec{a}=\vec{a}'+2\vec{\omega}\times\vec{v}'+\frac{d\vec{\omega}}{dt}\times\vec{r}'+\vec{\omega}\times\left(\vec{\omega}\times\vec{r}'\right)+\vec{a}_0$$

#physics #mechanics

Klein-Gordon equation

The field in terms of creation and annihilation operators:
$$\phi(x)=\frac{1}{(2\pi)^3}\int\frac{d^3p}{\sqrt{2{E_p}}}\left(\hat{a}_{p}e^{-i({E_p}t-\vec{p}\cdot\vec{x})}+\hat{a}_{p}^{\dagger}e^{i({E_p}t-\vec{p}\cdot\vec{x})}\right)$$
$$\pi(x)=\dot{\phi}(x)=\frac{-i}{(2\pi)^3}\int d^3p\sqrt{\frac{{E_p}}{2}}\left(\hat{a}_{p}e^{-i({E_p}t-\vec{p}\cdot\vec{x})}-\hat{a}_{p}^{\dagger}e^{i({E_p}t-\vec{p}\cdot\vec{x})}\right)$$
$$\left(\partial_{\mu}\partial^{\mu}+m^2\right)\phi(x)=0$$

#physics #fieldtheory

Lagrange's equations

Lagrangian and action
$$L(t)=\int d^3x\,\mathcal{L}(\phi_k,\partial_{\mu}\phi_k),\quad S=\int_{t_1}^{t_2}dt\,L(t)$$
\(\delta\phi_k(\vec{x},t_1)=\delta\phi_k(\vec{x},t_2)=0\)
\(\delta\left(\partial_{\mu}\phi_k\right)=\partial_{\mu}\left(\delta\phi_k\right)\)
$$\delta S=\int d^4x\,\left(\frac{\partial\mathcal{L}}{\partial\phi_k}-\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\phi_k\right)}\right)\right)\delta\phi_k+\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\phi_k\right)}\delta\phi_k\right)=0$$
Lagrange's …

#physics #fieldtheory

Banach space and Hilbert space

Banach space is a complete vector space, on which a norm \({\lVert \; \rVert }\) is defined.
- A distance between vectors:
$$d(v,w) =\lVert v-w \rVert$$
- Every Cauchy sequence of vectors always converges to a limit that is also in that space.

Hilbert space is a complete vector space on which …

#mathematics #algebra

Cauchy-Schwarz inequality

Let \(V\) be a vector space over R or C with an inner product \({\langle \;, \;\rangle} \). For any vectors \(v\) and \({w \in V}\)
$$ \lVert v \rVert \lVert w \rVert \geq \left| \langle v , w \rangle \right|$$

Proof:
For \(v= 0\) it's obvious that \(\lVert v \rVert \lVert w \rVert = 0 =\left| \langle v , w \rangle \right|\)

#mathematics #algebra

Inner product

Let V be a vector space over \(\mathrm{R}\). Inner product is an operation \({\langle \;, \;\rangle : V \times V \rightarrow \mathrm{R} }\) that satisfies the following properties:

$$\forall_{ v,w\in V}: \langle v, w \rangle =\langle w, v\rangle$$

$$\forall_{ v,w \in V; \alpha \in \mathrm{R}}: \; \langle \alpha v , w \rangle = {\alpha} \langle v , w \rangle $$

#mathematics #algebra

Vector space

A vector space over a field \((S,\cdot,+)\) is a set V with two operations, addition \({\oplus : V \times V \rightarrow V}\) and scalar multiplication \({\odot : S \times V \rightarrow V}\), that satisfy the axioms:

- associativity of addition
$$\forall_{v, w, u \in V}: (v \oplus w) \oplus u = v \oplus ( w \oplus u)$$

- commutativity …

#mathematics #algebra

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