T. S. Eliot's poem “Burnt Norton,” the first of his Four Quartets, is, to put it mildly, not an easy poem to understand, but it contains plenty of material for analysis. Let's look at some possibilities you might include in your detailed analysis of this poem.
First, you should examine the poem's structure. Eliot divides this long piece into five sections of varying lengths and complexity. For much of the poem, he uses free verse with no meter or rhyme scheme, and his line lengths vary widely. Yet in the second section, he includes a fifteen-line portion that does show rhythm and rhyme, although both of these are rather strange. The rhythm is not consistent, and the rhyme scheme scans as abaccbdcbbdeeec. The d rhymes are not full rhymes but merely suggestive, and the pattern as a whole is unusual.
You should also talk about the poem's themes in your analysis. Time and its mysteries stand at the center of the poem, but Eliot also delves into reflections about the seeming purposeless of human life, especially when it deviates from the “still point of the turning world.” This “still point” is God, and Eliot struggles to capture His essence (knowing all the while that a human being never fully can). He also ponders words and music and their seeming but not actual disappearance into silence. You should discuss how Eliot weaves these themes throughout each section of the poem.
Your analysis can also focus in on some of the details of each of the poem's sections. You might, for instance, discuss the imagery of the Chinese jar and the “shaft of sunlight” in the fifth section or the long list of vivid words Eliot uses to describe words and voices. You might point out the parallelism in the final stanza of the third section and reflect on its meaning; you might note the use of the German word Erhebung in the second section; or you might examine the imagery of the birds and the pool in the first section. The possibilities are endless.