The Electromagnetic Field of a Known Charge Distribution

Field theory may be regarded as a generalization of the mechanics of point particles, in which the dynamical variables q i (t) are replaced with fields Ф(x, t), such as E(x, t) and B(x, t). The discrete index i goes over to the continuos variable x, and, accordingly, the sum ∑ i is replaced with an integral ∫d 3 x. A direct transcription of the formalism of I, §3, leads to infinite-dimentional manifolds, which we would prefer to avoid. Instead, we merely generaliza the stationary-action principle (1:2.3.20) in order to find the analogues of the constant arising from the invariance properties. It is clear that in field theory the action % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWaa8qaaeaaiqGacaWFKbGaa8hDaiaa-bcacaWFmbac % eaGaa4hkaiaa-fhacaGFSaGab8xCayaacaGaa4xkaaWcbeqab0Gaey % 4kIipaaaa!4412! $$ \int {dt L(q,\dot q)} $$ involves an integral over a four- dimentional submanifold N 4, and thus requires a 4-form, which allows the construction of a chart-independent integral.