Ivancevic_Applied-Diff-Geom

84 Applied Differential Geometry: A Modern IntroductionNow, the nature of the integrand L[w] in (6.230) depends on whether weare dealing with particles or fields. In case of particles, L[w] = L(q, ˙q) isan ordinary finite–dimensional mechanical Lagrangian (usually kineticminus potential energy), defined through mechanical (total system energy)Hamiltonian H = H(q, p) as L[w] = L(q, ˙q) = p i ˙q i − H(q, p),where q, ˙q, p are generalized coordinates, velocities and canonical momenta,respectively.In case of fields, the integrand L[w] is more involved, as fields justinfinite–dimensional particles. Thus,∫L[w] = d n q L(ϕ, ˙ϕ, ϕ q ),where the integral is taken over all n space coordinates 7 , while ϕ, ˙ϕ, ϕ qdenote field variables, their velocities and their coordinate (partial)derivatives, respectively. The subintegrand L = L(ϕ, ˙ϕ, ϕ q ) is the systemLagrangian density, defined through the system Hamiltonian densityH = H(ϕ, π, π q ) as L(ϕ, ˙ϕ, ϕ q ) = π i ˙ϕ i − H(ϕ, π, π q ), where π, π qare field (canonical) momenta, and their coordinate derivatives.(2) Variate the action A[w] using the extremal (least) action principleδA[w] = 0, (2.54)and using techniques from calculus of variations (see, e.g., [Arfken(1985); Fox (1988); Ramond (1990)]), derive classical field and motionequations, as Euler–Lagrangian equations, describing the extremalpath, or direct system path: t ↦→ w(t), from t 0 to t 1 .Again, we have two cases. The particle Euler–Lagrangian equationreads∂ t L ˙q i = L q i,and can be recast in Hamiltonian form, using the Poisson bracket (or,classical commutator) 8 , as a pair of canonical equations˙q i = [q i , H], ṗ i = [p i , H]. (2.55)7 In particular, n = 3 for the fields in Euclidean 3D space.8 Recall that for any two functions A = A(q k , p k , t) and B = B(q k , p k , t), their Poissonbracket is defined as„ ∂A ∂B[A, B] partcl =∂q k − ∂A «∂B∂p k ∂p k ∂q k .