# Energy, Momentum, and Conservation Laws

Saturday, January 22, 2022

## New Definitions of Energy and Momentum

Dynamical quantities such as energy and momentum rely on length and time, and so have issues working with special relativity.

### Momentum

Momentum is classically defined as $\boldsymbol{\overrightarrow{p}}=m\boldsymbol{\overrightarrow{v}}$. An observer moving with respect to a collision would believe that momentum was not conserved due to relativistic effects.

The equation for momentum must suffice two criteria:

- The principle of relativity (it should be impossible to differentiate between inertial reference frames)
- At low speeds, the equation should reduce to the classical version

Linear momentum is therefore defined as:

$\boldsymbol{\overrightarrow{p}}=\frac{m\boldsymbol{\overrightarrow{v}}}{\sqrt{1-\frac{v^2}{c^2}}}$

Or, component-wise:

$\boldsymbol{p_x}=\frac{m\boldsymbol{v_x}}{\sqrt{1-\frac{v^2}{c^2}}} \newline \boldsymbol{p_y}=\frac{m\boldsymbol{v_y}}{\sqrt{1-\frac{v^2}{c^2}}}$

Note: the velocity, $v$, in the denominator is the velocity of the object with respect to a certain inertial reference frame, not the velocity of the reference frame itself.

### $K$

Kinetic Energy,In classical mechanics, work is defined as such:

$W=\int Fdx=\int\frac{dp}{dt}~dx=\int\frac{dx}{dt}~dp=\int v~dp=pv-\int p~dv$

Using the new definition for momentum, we get:

$K=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}v-\int_0^v \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}~dv=\frac{mv^2}{\sqrt{1-\frac{v^2}{c^2}}}+mc^2\sqrt{1-\frac{v^2}{c^2}}-mc^2 \newline K=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}-mc^2$

It may not look like it, but this simplifies to $K=\frac{1}{2}mv^2$ for small values of $v$.

### $E$

Relativistic Total Energy,Relativistic kinetic energy, $K$, can also be defined as such:

$K=E-E_0$

Where $E$ is the total relativistic energy and $E_0$ is the *rest energy*.

Total relativistic energy and rest energy are defined as:

$E=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} \newline E_0=mc^2$

For high-energy collisions, kinetic energy is not always conserved (since rest energies can change), but total relativistic energy *is* always conserved.

Using the equation for momentum, we get another fundamental relationship:

$E^2=(mc^2)^2+(pc)^2$

For particles approaching the speed of light, energy can be approximated as:

$E\cong pc$

For massless particles (like light), energy is exactly defined as

$E=pc$

## Conservation Laws

In an isolated system of particles, the total linear momentum remains constant.

Classical collisions:

- For elastic collisions, the kinetic energy of the particles is the only form of energy
- For inelastic collisions, the extra energy is stored in the particles

Relativistic collisions:

- Internal stored energy contributes to rest energy
- Total energy consists of kinetic energy and rest energy
- The total amount of energy does not change

In an isolated system of particles, the total relativistic energy remains constant.