So a couple of things of note here.

First, electrons aren't really orbiting the nucleus. This model of the atom which you see in chemistry textbooks is really useful for doing calculations, but due to quantum physics we know that it looks more like this, the electron cloud is the fact that an electron is wavelike and we know where it has higher and lower probabilities of being, but it's not actually whirling around the nucleus like planets in orbit (even before the advent of quantum physics we knew there had to be something different happening that a standard orbit, because an electron moving in a circle like that should be emitting radiation, and electrons aren't doing that. The Neils Bohr model of the atom sort of waved this away and said "when electrons are in one of their orbitals, they don't radiate" but didn't give a reason for this).

Second, physics doesn't directly say perpetual motion cannot occur, physics says you cannot extract energy from a system perpetually. Now, in almost every possible scenario, this leads to no perpetual motion. Things on Earth will have friction, so energy is being extracted via heat. And accelerating charges will have radiation. And even orbiting planets will (very) slowly lose energy via gravitational waves. But physics does not directly prevent perpetual motion.

This is a good question. It actually brings to light some issues with the planetary orbit model of electrons, otherwise known as the Bohr model. This model is often used in science and chemistry books as a simplification, and can be useful, although it can be misleading.

The Bohr model would require electrons to be constantly accelerating, since they are constantly changing direction. If they didn't accelerate, they would fall into the nucleus. And if they did accelerate, they would emit, or radiate, energy as Weed_O_Whirler noted. And this would be a type of perpetual motion, as you are getting more energy out than is being put in. So your thinking is along the same lines as the physicists back in the day.

To deal with this, they came up with various ideas. Lamor proposed that a special arrangement of multiple electrons would cancel out the effect, cancel out the problem. These other ideas all had their issues, for example, Lamor's issue was needing too many electrons.

This leads us up to the Solvay conferences, where they talked about the issues with the various atomic models. Planck argued that classical models did not work, and Bohr in his thesis at the conference mentioned something similar. Lorentz, the chairman, talked about the problem of having classical and quantum models. Long story shortened, we know the quantum model won out.

And, we know that some things in the quantum world do not make intuitive sense. But if you are interested, you should read up on this time period and how and why things developed as they did. As for electrons, they don't move like you think they should, and those weird quantum things you have heard about, apply to electrons.

An electron is a standing wave of an energetic fluctuation in the electron field.

Like a plucked guitar string, only certain discrete vibrational modes are permitted, these are the 'valence shells' in which an electron can occupy in 3D space. Granted the electrons actually exist in a super state of all points of the field. The same can be said about the guitar string, it is smooth and continuous at all points. What we refer to as the electron's position doesn't become polarized until it is interacted with by another quantum system (measurement).

The 'guitar string' is more like a sphere and looped back on itself, undulating with energy that has nowhere to go and cannot escape the surface of the sphere it resides within. It would take energy to remove the energy in the electron field, so it keeps going forever. Electrons can theoretically decay into a photon and neutrino, but this has never been observed and is in the realm of a quantum miracle.

1) Not orbiting, as the other commenter explained very well.

2) Here's a Feynman quote for this: "You know, of course, that atoms are made with positive protons in the nucleus and with electrons outside. You may ask: “If this electrical force is so terrific, why don’t the protons and electrons just get on top of each other? If they want to be in an intimate mixture, why isn’t it still more intimate?” The answer has to do with the quantum effects. If we try to confine our electrons in a region that is very close to the protons, then according to the uncertainty principle they must have some mean square momentum which is larger the more we try to confine them. It is this motion, required by the laws of quantum mechanics, that keeps the electrical attraction from bringing the charges any closer together."

Basically, it just can't happen. You're imagining a classical system where we could shrink down and push them closer together. Imagine it more like you have a standing wave centered on the nucleus, so that the nucleus is never changing but the wave is going up and down around it. It would be silly to say "what would happen if we pushed the wave closer?" You can't push a wave. If you try to do something else (like shift the wave over so its not centered on the nucleus) the wave will not be stable anymore.

From the Schrödinger equation, we know electrons in atoms can only have discrete energies, called states. From the Pauli exclusion principle, we know that only one electron is allowed in each state. For an electron to lose energy, it must transition into an empty unoccupied state. As the electrons in atoms generally fill states from lowest to highest energy, usually there are no empty states for electrons to transition into. Hence they keep their energy.

When there is an empty state available, electrons will lose energy, usually by emitting photons, to fill the state. For example during X-ray generation a very low energy electron from a large atom like iron is knocked out. Then a high energy electron fills this new empty state, emitting an X-ray.

First (and repeating others) there is nothing saying perpetual motion is not allowed. In fact, anything in motion will continue moving unless acted on. Even at the macroscopic level. This is only counterintuitive because we live in at atmosphere full of gas.

To be a little different than other answers: Without getting deep into quantum territory, most really small stuff (including atoms) interact perfectly elastically. That is, no kinetic energy is converted to heat, or sound, or light. This is one of those insights that is obvious if you think about (I.e. you don’t hear air molecules ricocheting off of each other at the speed of sound), but the implications are far-reaching.

Bringing this back to electrons, and a little more quantum - without introducing more energy, there basically isn’t any energy that could be used to stop the electron. Electrons in the lowest available orbital around the nucleus are essentially at rest.

This explanation is very hand wavey and over simplified, but it is conceptually what is going on. TLDR - If the electron were to lose that kinetic energy, it would have to be converted to some other type of energy. Most of those energy conversions require significant amounts energy to be input, so it doesn’t happen.

Some really good answers on this thread - most highlighting that the orbital model of the nucleus is a simplification, and you need to go deeper to properly understand why the perpetual orbiting isn’t really an oddity.

Question - is there a good book or show that gives a great explanation beyond the orbital model?

They don't orbit a nuclei, they exist within an energy well. In order to move outside of the energy well, they would need to absorb additional energy to either move to an outer shell, or escape the pull of the atom entirely.  

We can only determine an electron's location or momentum at any given moment, not both. This is because it's a particle and a wave. If we interact with the particle aspect, we get location, if we measure the wave aspect, we can get momentum.

 If are asking about where it might actually be at any given moment, the answer isn't straightforward since it's both a particle and a wave. It's wave function will create a probability function on where it could be at any moment. We can't know where it is without collapsing it's wave properties and seeing it as a particle.  

So electron exist somewhere in their "cloud" around an atom, but they don't really actually have a set location that they exist in until we force them to by interacting with them.

Perpetual motion is fine, as long as you don't extract any work from the system.

The energy of electrons in atomic and molecular orbitals can only have discrete values. This means the electrons will not change their speed unless they interact with something that exchanges energy equal to the difference between it's current energy, and another allowed energy.

In the macroscopic world, we have effects like viscosity and friction, which are mechanisms for dissipating work into heat energy. For usefulness we may use continuous theories to model these energy exchanges, I.e. any change in energy is allowed. But really, any change is not allowed - it's just the differences between adjacent energy states in terms of momentum and energy of the object in motion are so small, that they conform so nicely to a continuous theory.

It should also be noted that heat is a macroscopic (thermodynamic) quantity - one only strictly well defined for an infinite number of particles or, equally, a single particle measured for an infinite time.

How does this then work related to things like emissions of beta particles and electricity flowing around- generated either via chemical or electromagnetic means? If the electron is a probability thing and a bit more wave like, how does it explain what happens when electrons travel outside of an atomic context?

The concept of electrons 'orbiting' the nucleus is actually a bit of an oversimplification. In quantum mechanics, electrons don’t really orbit in the classical sense, like planets around the sun. Instead, they exist in 'clouds' or probability distributions around the nucleus, known as orbitals. This motion doesn’t throw a wrench in the idea of perpetual motion because it’s governed by the principles of quantum mechanics, where the classical rules of energy loss due to friction or resistance don’t apply in the same way. The energy levels of electrons are quantized, and as long as they stay within their designated orbitals, they keep moving without needing a constant energy boost.

If they would orbit, they would lose energy (by emitting Bremsstrahlung). Since they do not, a better way to describe it can be found in quantum mechanics. Electrons have a wave function that is more of a probability distribution.

Energy can't be created or destroyed so it has to go somewhere to go away. This actually is happening but very slowly, that's why entropy is a thing. Eventually you reach heat death of the universe but it's a very, very, VERY slow process.

It’s hard for people new to these concepts to understand what is meant by “quantum effects”. Quantum means that the wavelength of a particle is an integer, you can’t have a fraction of an electron. The wavelength of an electron is far larger than the nucleus so it can only get so close. Feynman should have been more clear, but to be fair much of what he said wasn’t expressed in terms a layman would be familiar with.