: find better explanation or don't include this! BEYOUND THE SCOPE OF A FIRST MECHANICS CLASS!

Physics has been traditionally a male-dominated field. It is typically men that sit down with the hairy equations and slave away calculating integrals. Some of the biggest ideas in physics are due to women, and in this section we reach one of the most beautiful insights that you can possibly imagine.

Amelie Emmy Noether came up with a theorem which deduces the existence all conserved quantities from a simple observation about symmetry. The theorem states that for each symmetry in the equations of physics we use, there exists a corresponding conserved quantity in the physics equations.

Consider the equation of the total energy of an object: \[ E_T = K + U(\vec{x}) = \frac{1}{2}mv^2 + U(\vec{x}), \] where in the right-hand side we have shown how explicitly how the energy depends on the velocity $v\equiv |\vec{v}|$, and the position $\vec{x}$ of the particle.

In general both $v$ and $x$ may vary as a function of time in a complicated way. In all of the physical situations we have seen in this book, however, the total energy $E_T$ stays constant in time. This is called the time invariance symmetry of the energy equation.

Mathematically we have \[ E_T(t) \equiv E_T. \] The total energy is constant in time. Or we can say energy is a conserved quantity.

It just happens that when first introducing the concept of energy it is easy to state this observation as a law: the law of conservation of energy.

Consider a physical system with no external forces, and therefore no potential \[ E_T = \frac{1}{2}mv^2 + \underbrace{ U(\vec{x}) }_{=0} = \frac{1}{2}m v^2. \]

So basically the only kind of energy there is kinetic energy. This is the situation we previously called uniform velocity motion or UVM.

A physics situation with UVM in it, has a translational symmetry of the space coordinate $\vec{x}$. What this means that I can always add the physical description of the system in coordinates $\vec{x}_1 = \vec{x}$ and in coordinates $\vec{x}_2 = \vec{x} + \vec{s}$ where $\vec{s}$ is any constant translation will not change, because the absolute value of $\vec{x}$ never appears in the energy equation.

\[ \vec{v}_2 = \frac{ d \vec{x}_2(t) }{ dt } = \frac{ d (\vec{x}_1(t) + \vec{c} }{ dt } = \frac{ d (\vec{x}_1(t) }{ dt } = \vec{v}_1, \] since $\frac{d\vec{c}}{dt}=0$.

In this case Noether's theorem tells us that there is a conserved quantity that corresponds to the translational invariance symmetry of the energy. The conserved quantity is the momentum $\vec{p} = m\vec{v}$.

Rotational symmetry. When an object is turning, you don't really care what its initial $\theta_0$ was, you just care about its angular velocity $\omega$.

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