 # Number Theory Question

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## Transcribed Text

(a) Consider the Lagrangian n n L = [ - 4j + (g) t2 1 [ ajkqjik, j=1 where Ej. ajk, j,k = 1, n are constants and ajk = -akj. We assume that q;>0,j=1, = n. Find the Euler-Lagrange equations and write them in the form 4j = Fi(q,q). (b) Compute the canonically conjugate variables Pj and obtain the Hamiltonian of the system. Check explicitely that Hamilton's equations are equivalent to the Euler- Lagrange equations of part (a). (c) Suppose now Ej = 0, j = 1, n. Consider the n functions n = Pj + k=1 where bjk are constants. Determine the constants bjk such that each I¿ is a first integral {Ij, H} = 0, j = 1, , n. Deduce the Poisson bracket {1j,Jk} in terms of ajk

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