Galileo’s theory of relativity states there is no absolute or preferred:

• Origin of time

• Orientation in space

• Position in space

• Inertial frame of reference

What we can and cannot distinguish are

❌ Time

❌ Position

❌ Rotation

❌ constant velocity

✅ Relative Time

✅ Relative Position

✅ Rotating (fictional forces)

✅ Acceleration (cant be discerned with gravity)

💼 Case: consider an isolated system of $N$ particles with vector coordinates $\vec r_j$ and momenta $\vec p_j$ with $j=1,\ldots ,N$ labelling the particles

we have $3N$ DoF and $6N$ Hamilton’s equations

Each statement of the theory of relativity has a direct impact on the Hamiltonian of the system

1. Equations of motion are unchanged by a displacement in time

   $$ t\to t+\delta t $$

   This requires $\partial H/\partial t=0$ (no explicit time dependence). This implies that $H$ is a conserved quantity, and for an isolate system, that the total energy $E$ is conserved

2. Equations of motion are unchanged by a displacement of the entire system in space

   $$ \vec r_j\to \vec r_j+\delta \vec r \quad \forall j $$

   This can only be the case if the Hamiltonian depends only on relative displacements

   $$ \vec r_{jk}\equiv \vec r_j-\vec r_k $$

   since the $\vec r_{jk}$ ‘s are unchanged by translations of the entire system

3. Since $\vec r_{jk}$ has a direction, if the Hamiltonian depended directly on it, then the equations of motion would have a preferred direction. Thus the Hamiltonian can depend only on scalar quantities such as

   $$ r^2_{jk}=|\vec r_{jk}|^2=\vec r_{jk}\cdot \vec r_{jk} \quad \text{or} \quad \vec r_{jk}\cdot \vec r_{lm} $$

Time dependence

💼 Case: consider a system of $N$ point particles moving in three dimensions. We consider calculating some function $F$ which is a function of the positions and momenta of all the particles, and possibly time: $F(\{q_i\},\{p_i\},t)$. We call $F$ an observable

• The general expression for the rate of change of $F$ can be written as

  $$ \begin{aligned} \frac{\text d F}{\text dt}&=\sum_i \left ( \frac{\partial F}{\partial q_i}\dot q_i +\frac{\partial F}{\partial p_i}\dot p_i \right )+\frac{\partial F}{\partial t} \\ &=\sum_i\left ( \frac{\partial F}{\partial q_i}\frac{\partial H}{\partial p_i} -\frac{\partial F}{\partial p_i}\frac{\partial H}{\partial q_i} \right )+\frac{\partial F}{\partial t}

  \end{aligned} $$

  where we have used Hamilton’s equations

• For any two observables $F$ and $G$ we define the following quantity as their Poisson bracket $[F,G]$

  $$ [F,G]=\sum_i \left ( \frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_1} \right ) $$

  We therefore have

  $$ \frac{\text dF}{\text dt}=[F,H]+\frac{\partial F}{\partial t} $$

  Since the Poisson bracket of $H$ with any observable $F$ generates the implicit time dependence of $F$, we say that “$H$ generates a displacement in time”.

🗒️ Note: if $\frac{\partial F}{\partial t}=0$ and $[F,H]=0$ then $F$ is a conserved quantity

In particular since $[H,H]=0$, if $\frac{\partial H}{\partial t}=0$ then $H$ is a conserved quantity and we already said that the principle of relativity guarantees $\frac{\partial H}{\partial t}=0$

Spatial Dependence

Our system of $N$ particles has $3N$ degrees of freedom $q_i$ where $i=1,\ldots , 3N$

• We can define an alternative labelling that distinguishes the three coordinates of each of the $N$ particles

  $$ \begin{aligned} q_{sj} \quad\text{where}&\quad s=x,y,z \quad r,\theta,\phi \quad \rho,\theta,z \quad \ldots \\ \text{and}&\quad j=1,\ldots ,N

  \end{aligned} $$

  and likewise for $p_{sj}$. Therefore we can use interchangeably

  $$ \sum_i \leftrightarrow \sum_s \sum_j $$

💼 Case: Consider a displacement of all $N$ particles by a distance $\delta x$ in the $x$ direction