The movie is based around the Beatles' adventures with Young Fred as they try to save his beautiful homeland, Pepperland, from eternal silence without music following a takeover by the evil Blue Meanies. On the surface the film has an unthinkable number of errors in its physics. However, there is some evidence that the filmmakers have an idea of what physics is.
Consequently, I have divided the physics of Yellow Submarine into three categories: "normal" errors, flashes of insight into the real world, and errors which are due to the fact that the film was animated before they knew what animation was. We'll overlook that last category because they simply couldn't help it.
The first serious problem with the film is the location of Pepperland. "80,000 leagues beneath the sea it lay," says the voiceover. 80,000 leagues is very approximately equal to 400,000 km (one marine league being 5.56 km). The diameter of the Earth is only 12,740 km, so if Pepperland were 400,000 km below the surface of the ocean, it would be beyond the moon (which is 384,000 km away).
Even so, Pepperland is "an earthly paradise", and we are taken on a tour of the happy country. Everyone seems to be playing musical instruments, but these instruments certainly do not behave like earthly ones. We see two people playing a duet on a piano. Their hands continually hit the keyboard in exactly the same places, yet the chords they are playing change. How this happens we do not know, but can only assume that in Pepperland the pianos are controlled by the pedals and that the players were merely exercising their hands in a rhythmical manner.
This strange instrument behaviour continues when we come upon Sergeant Pepper's Lonely Hearts' Club Band, the stars of Pepperland. This is a brass band with four members, playing slide trombones and trumpets. At first glance they look fine, and the crowd certainly seem to be enjoying their music, but from where is this music coming? Not from the trombones or the trumpets, that's for sure. While the players constantly wave their instruments around with great flourishes through the air, the slides and keys are not moving at all.
We have hardly witnessed this cheerful scene, however, before disaster strikes. Declaring, "that country is a tickle of happiness upon the blueberry of the universe. It must be scratched," the Chief Blue Meanie launches a merciless attack upon Pepperland, aiming to make it blue and empty of music forever. The attack begins in the heart of Pepperland - the auditorium at which Sergeant Pepper's Lonely Hearts' Club Band is at that moment playing. A large, blue "anti-music missile" lands on the stage, instantly silencing the music. The audience scream with horror, their eyes reflecting the blue colour of the missile. As they turn to face the camera it becomes obvious that in Pepperland they have colourless retina as well as foot-powered pianos, for every eye is blue from edge to edge. In reality, even if the colour was strong enough for it to be reflected in the whites of the eyes, it would not show at all in the dark-coloured retina. The darker colour would simply absorb most of the light, so that any visible reflection from the centre of the eye would be negligible.
Young Fred alone manages to escape from the site of the missile landing. Pursued by Apple-Bonkers and shot at by Blue Meanies, he makes straight for the secluded glen where the Lord Mayor of Pepperland plays with his string quartet. However, he seems almost certain not to get there, for the Blue Meanies hound him every inch of the way. The Chief Blue Meanie even sends the Terrible Flying Glove after him. Pointing and punching, it destroys every building or structure he tries to take refuge in, and escape seems almost impossible for Young Fred as he finds himself trapped by the ruins.
"Not polite to point!" he stammers as the Glove catches him standing on the lower end of a fallen pillar, lying crosswise across another. The two pillars together form a seesaw-like lever, the effectiveness of which is demonstrated when the Glove punches down on the higher end of the pillar on which Fred is standing, sending him flying up into the air. The force of the punch is such that the whole seesaw rolls sideways a little. A moment later Fred comes down again, landing squarely on the higher end of the lever and sending the Glove flying in turn. However, due to the sideways roll of the seesaw and the angle at which he was thrown up, it is more likely that Young Fred would have missed the seesaw altogether and come down on the ground, to face the mercy of the waiting Glove.
Young Fred is obviously a man of miracles, for he escapes the Glove and makes it to the glen where his Lord Mayor plays, the Meanies in hot pursuit. As he explains the situation to the disbelieving old man, the Meanies pick off his fellow musicians one by one. Young Fred catches each dropped instrument as the musicians turn silently to stone, and it isn't long before he is playing three violins with one bow. Even assuming that his arms are long enough to hold the three violins, there is no way that a bow intended for use on one violin would be long enough to play three at once. Despite this, Young Fred produces quite a good sound from his difficult instruments. So they have magically extending bows as well?
Obviously the Pepperlanders need some expert help, both to drive off the Blue Meanies and to rescue their warped physics. Escaping in the Yellow Submarine, Young Fred soon finds assistance ("H for hurry, E for urgent, L for love me and P for please help!") in the form of the Beatles themselves. As he whisks them away from Liverpool Fred's Newtonian physics soon appears to be in perfect working order, but quantum physics and relativity begin to play a part in making the journey an increasingly strange one.
At this point the film undergoes an unexpected reversion to the real world, beginning with a clever observation from John: "We've become involved in Einstein's time-space continuum theory... relatively speaking." Time slows down, stops completely and begins to go backwards as the Submarine gets lost on its way back to Pepperland. Certainly, the distortions of space necessary to get a submarine to a place 400,000 km away, which is somehow on (or inside) the Earth, in a reasonable time would cause the fabric of space-time to warp so severely that time could quite possibly stop. It is now known that a particle travelling forward in time is identical to an antiparticle travelling backward in time, and so time travel is happening at every second of every day. This is an example of good physics, as is the sudden speeding-up of time which follows when the Submarine finds its course again.
The strange things which the Fab Four and Fred see in the ensuing journey to Pepperland are often explained by George as being "all in the mind". (This may or may not have something to do with the test-tube of chemicals that John swallowed when he was Frankenstein earlier in the film.) When researchers were attacking the most difficult problems of quantum physics in the 1930s, they found that fundamental particles such as electrons appear to alter, not only their behaviour, but also their very nature according to whether they are being observed. The most understandable explanation of what happens to them when we aren't looking is that they are not actually there, and that they materialise at a point decided by quantum probabilities as soon as we look for them.
With this in mind, the idea that reality as well as perception is "all in the mind" is not merely a philosophical one. When testing to find out whether light is made of waves or particles, the researchers found that it behaved like waves when they were testing for waves, and like particles when they were testing for particles. In fact, it seemed that the nature of light really was "all in the mind" and based on the observers' expectations. Not long afterwards it was found that photons (light-particles) are fundamental particles just like electrons, and therefore that the entire universe (consisting of these fundamental particles) can be said to be "all in the mind".
After arriving in Pepperland, the Beatles dress up as Sergeant Pepper's Lonely Hearts' Club Band and make their way to the auditorium, where the real Band still stand frozen inside the Blue Meanies' Anti-Music Missile. Wondering how to break the Missile open to let out the Band, they tap on it and search fruitlessly through their pockets for tools. Then Ringo comes up with an idea: "I've got a hole in my pocket," he says, as he draws out the hole he salvaged earlier from the Sea of Holes and puts it on the side of the dome. The gas inside immediately begins to pour out, the Missile collapses and the band are soon free.
The casual audience may think that this is fairly far-fetched, but in actual fact it has its roots deeply embedded in established quantum theory. The antimatter antiparticle corresponding to an electron is called a positron, or a "hole", as it is merely an absence of an electron that behaves like a positive charge. To all intents and purposes the positron would be merely a very light proton, were it not for one important feature. When a negatively charged particle meets a particle with an equivalent positive charge under normal circumstances, they bond and go around as a pair with overall neutrality - but the charge on each particle is not destroyed, merely cancelled out by the charge on the other. Not so with electrons and positrons. Because positrons are really absences of electrons, or "holes", when an electron meets a positron it "falls" into the "hole" and both disintegrate in a burst of energy.
When Ringo puts his hole onto the side of the dome, a very similar thing happens. The dome takes some time to "fall" through the "hole" as it is so large, but it does do it. The resulting red smoke is the visible manifestation of the energy created in such an interaction.
Obviously, the problems that Pepperland had with its physics could not have been solved without the help of the Beatles. It doesn't take the Fab Four long to vanquish the Blue Meanies, turning the Chief Blue Meanie into a mobile, flowering, singing rose bush, and everyone lives happily every after in a perfectly musical Pepperland (that may have a few insignificant problems with its biology).
It may seem that the problems they have at the start of the film with Newtonian physics do not allow my rating of Good Physics. However, the filmmakers demonstrate such a good grasp of the more abstract concepts of quantum physics towards the end that I feel sure the errors at the beginning were deliberately included to show how essential the Beatles are to the sanity of the universe.
Perhaps it's just as well we aren't so dependent in the "real" world.