Prisoner of Azkaban is commonly referred to as the best of the Potter movies. My personal favourite, which I genuinely think is the best movie, is Goblet of Fire. Although I'm glad that Cuarón took a less literal approach to adapting the books than Columbus, producing a more streamlined, movie-like plot with much better pacing, I think the movie really suffers from leaving out any explanation of who Mooney, Wormtail, Padfoot and Prongs are and the significance of Harrry's patronus being a deer. It would hardly have taken very long to pop in a quick explanation during Harry and Lupin's final conversation in Lupin's office and surely this must be an obvious gap in the plot to those who haven't read the books. The movie is also a little bit too arty for my taste sometimes. There are lots of extra little touches that I love, including the Night Bus squeezing and jerking its way around, the man who can't be bothered to twirl his coffee spoon with his fingers but is using magic instead, the repeated motif of the clocks and timepieces that foreshadows the final act and the long-suffering cleaning lady at the Leaky Cauldron. But sometimes the camera does an especially swirly move, or there's a very obviously carefully staged moment, like the slow pull back from Uncle Vernon waving at Aunt Marge while Dudley appears to be watching Strictly Come Dancing on television (clever, since it hadn't started yet), that's just so obviously staged and designed to be terribly arty, it takes me out of the story. And why has Dean Thomas been virtually replaced by a completely new student, and CrabbeOrGoyle by a ganglier student? Every time I see one of these random new people it sets my teeth on edge (besides, I like Dean Thomas, he does normal things like supporting a football team).
Cuarón is also responsible for casting Micheal Gambon, a decision I am Not Happy With. Gambon is a fantastic actor (brilliant in Gosford Park, for example) but he just doesn't get Dumbledore. Richard Harris sometimes lacked a bit of sparkle, but you could always feel the repressed power of Dumbledore when he played him. Gambon has a bit more twinkle - though not much, really - but provides no feeling of hidden power or depth at all (it doesn't help that he's made it clear in interviews that he doesn't really care about the material. If you do an acting job for money, fine, but if you take on a role that means so much to hundreds of kids, you should at least make some effort to understand the character). And his accent is awful, veering all over the place and most often sounding like some bizarre combination of Irish and Canadian.
Still, Prisoner of Azkaban is a very, very good Potter film and features one of my favourite Potter-y things, the choir singing 'Double, double, toil and trouble' (with toads!) to John Williams' excellent soundtrack - Williams seems to have been much more inspired on this one, providing a wonderfully rich variety of styles in the soundtrack while maintaining an overall unity (I especially like the Tudor-like bits for the castle). David Thewlis is not how I pictured Lupin - I had quite the thing for Lupin in the books and pictured him as, well, no offence to Thewlis but a bit sexier. Still, his performance is excellent. Some of Cuarón's changes to the overall setting seem a bit random - the bridge thing has proved popular in subsequent movies but doesn't actually have much of a purpose beyond being pretty, and why he changed Professor Flitwick's look is anyone's guess - but it's wonderful to see Hogwarts looking like it's really in Scotland, complete with Scottish weather and beautiful Highland landscapes, and lovely steep slopes for the kids to run around on.
Anyway, on to Classics! Yet another element redesigned by Cuarón is the Fat Lady, who plays a bigger role in this story than any of the others. In Philosopher's Stone, she looked just the way I pictured her from the book, a large grey-haired lady in a pink dress, imagined as vaguely nineteenth century. Here, she has been re-cast, presumably because she needed a comic scene to re-establish her importance before she is attacked, so comic actress Dawn French was cast instead of Elizabeth Spriggs. However, since Cuarón can't leave anything alone, she's also been re-imagined and is now a neo-Classical figure. The style of painting fits the eighteenth or nineteenth century, a self-consciously Classicizing image which represents how eighteenth and nineteenth century artists viewed Greece and Rome (gorgeous landscape with vaguely Classical architecture in the background, figure in the foreground in white tunic and with her head covered in fruit - exactly who the Fat Lady is supposed to be, mortal or goddess, is not clear and presumably, since she's known as the Fat Lady, no one knows).
Intriguingly, the Fat Lady appears to consider herself actually to be Roman - judging by her criticism of the kids as 'plebs' when they fail to appreciate her singing (though, to be fair, a well educated person from the eighteenth or nineteenth century would probably use the same insult, so it's not a definitive conclusion!). I'm always a little confused about how far the various portraits in Harry Potter are supposed to think and feel as their subjects, or how connected they are with their status as paintings or photographs (I think it generally depends on the painting or photograph). The Fat Lady certainly seems to have chosen to identify with her Classical setting, even though it represents a later fantasy of the Classical world rather than any Classical reality.
Much more important to the plot, of course, is Buckbeak the Hippogriff. Hippogriffs are not really known from Classical Mythology, though name is half-Greek (hippo, from horse) and there is a line in Virgil's Eclogues that refers to mating a Gryphon and a mare. The Potterverse Hippogriff, however, is basically the same creature as the Classical Griffin, described by Aeschylus, Herodotus and Pausanias, among others. Werewolves (the name correctly identified by Snape as being from Anglo-Saxon, in which wer means man) are also talked about in ancient texts, most notably in the werewolf-story in Petronius' Satyricon. Potterverse werewolves are similar to Buffyverse werewolves in that they are completely normal outside the period of the full moon but uncontrollable monsters when wolves, though in the Potterverse, lycanthropy (werewolfism) is treated much more like a serious chronic illness than anything else (and there's a distinct metaphor about misunderstood and socially unacceptable illnesses going on there). Ancient texts tend not to present werewolves in such a friendly light. (I've also discussed Remus Lupin's name before).
Other bits and pieces of Classical Stuff get the odd mention. Hermione points out that the Egyptians used to worship cats. The current password for the Gryffindor common room, appropriately enough, is fortuna major (literally 'big luck', lots of luck, great luck or Great Luck, the personified goddess of luck - though Google suggests it may have more to do with a divinatory practice I'm not familiar with). Sybil Trelawny is, of course, named after the Sibyl, but the forms of divination she teaches are distinctly newish - crystal balls and tea leaves are slightly more modern obsessions, at least in the West (some say Druids used crystals, but somehow I doubt they were using lovely smooth spherical crystal balls). One of the many classes Hermione is taking is Ancient Runes - you'd think, considering the nature of most of their spells, that Latin would be more useful but, of course, Latin is a subject some children might learn in real schools. For her ancient language to sound sufficiently weird and magical, it had to be runes.
I can see why so many people list this as their favourite Harry Potter film. It's adopting one of the best books, which helps (I was genuinely shocked to discover that Ron had literally been sleeping with a bad guy for years). It is also truly magical and there are so many lovely touches - the giant giraffe wandering through the portraits is awesome and Hogsmeade in the snow is beautiful. Snape as Grandma is hilarious too (as is my favourite exchange 'Spiders! They want me to tap-dance! I don't wanna tap-dance!' 'You tell those spiders, Ron'). I love the way Snape throws himself in front of the kids he hates to protect them from the werewolf as well. Unfortunately, for me, there are just a few too many deeply irritating ticks (Gambon's accent, random extra students, unexplained plot points) for it to be my favourite, but it's a very, very good adaptation and certainly stands up best as a film by itself, outside of the series as a whole.