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Math Test

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• Metric: $g_{\mu \nu }$
• Derivative: ${\partial }_{\mu }\varphi$
• Einstein eq.: $G_{\mu \nu }=8\pi G{T}_{\mu \nu }$
• FRW: $d{s}^2=-d{t}^2+a(t{)}^{2}d{\overrightarrow{x}}^2$

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$$g_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$$$${\mathrm{\nabla }}_{\mu }{T}^{\mu \nu }=0$$$$S=\int d^{4}x\sqrt{-g}[\frac{R}{16\pi G}-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -V(\varphi )]$$

Aligned equations

$$\begin{array}{rl}{H}^2& ={\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\rho \\ \dot{H}& =-4\pi G(\rho +p)\end{array}$$

Fractions, roots, sums

$${\int }_0^1{x}^{2}dx=\frac{1}{3},\sqrt{1+ϵ}\approx 1+\frac{ϵ}{2},\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{{\pi }^2}{6}$$

Greek + accents

$$\alpha ,\beta ,\gamma ,\mathrm{\Gamma },\mathrm{\Lambda },\mathrm{\Omega },\dot{\varphi },\ddot{\varphi },{\tilde{g}}_{\mu \nu },\overline{\rho }$$

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Definition: Phase Space

A phase space is the set of all possible states of a process or system at a specific time. For a mechanical system with $n$ degrees of freedom, the phase space is typically $2n$-dimensional, parameterized by coordinates and momenta $(q_i,p_i)$.

Example: Simple Harmonic Oscillator

For a simple 1D harmonic oscillator, the state is determined by its position $x$ and velocity $v$. The phase space is the 2D plane with coordinates $(x,v)$. The system's evolution is described by:

$$\frac{dx}{dt}=v,\frac{dv}{dt}=-{\omega }^{2}x$$

The phase curves are ellipses centered at the origin.

Theorem: Liouville's Theorem

For a Hamiltonian system, the phase-space volume is preserved along the flow of the system. If $\rho (q,p,t)$ is the probability density function in phase space, then:

$$\frac{d\rho }{dt}=\frac{\partial \rho }{\partial t}+\sum _{i=1}^n(\frac{\partial \rho }{\partial q_i}{\dot{q}}_i+\frac{\partial \rho }{\partial p_i}{\dot{p}}_i)=0$$

Important Remark: Singularities

When analyzing phase flows, special care must be taken near equilibrium points where the vector field vanishes (i.e., $v(x)=0$). Trajectories can take infinite time to reach these critical points.