New canonical perturbation method for complete set of integrals of motion

Abstract

The solution of the equations of motion defined by the Hamiltonian function $\text{H = H}{}_{\mathrm{o}}\text{(q}{}_{\mathrm{k}}\text{,p}{}_{\mathrm{k}}\text{) +}\text{∑}\text{n=1}\text{λ}{}^{\mathrm{n}}\text{H}{}_{\mathrm{n}}\text{(q}{}_{\mathrm{k}}\text{,p}{}_{\mathrm{k}}\text{) = E, k = 1,…,N}$ is found through the use of power series expansions in the perturbation parameter λ. The solution is in the form of 2N independent integrals of motion $\text{j}{}_{\mathrm{k}}\text{= P}{}_{\mathrm{k}}\text{+}\text{∑}\text{n=1}\text{λ}{}^{\mathrm{n}}\text{J}{}_{\mathrm{kn}}\text{(Q}{}_{\mathrm{l}}\text{,P}{}_{\mathrm{l}}\text{) =}\text{const}$ $\text{W}{}_{\mathrm{k}}\text{= Q}{}_{\mathrm{k}}\text{+}\text{∑}\text{n=1}\text{λ}{}^{\mathrm{n}}\text{W}{}_{\mathrm{kn}}\text{(Q}{}_{\mathrm{l}}\text{,P}{}_{\mathrm{l}}\text{) = δ}{}_{\mathrm{kN}}\text{·t +}\text{const}$ where the perturbation terms of these integrals are given by recursion formulas $\text{J}{}_{\mathrm{kn}}\text{=}\text{∫}\text{C}\text{Q}{}_{\mathrm{N}}\text{δH}{}_{\mathrm{n}}\text{δQ}{}_{\mathrm{k}}\text{−}\text{∑}\text{s=1}\text{n−1}\text{{J}{}_{\mathrm{ks}}\text{, H}{}_{\mathrm{n-s}}\text{}}\text{dQ′}{}_{\mathrm{N}}\text{for}\text{k≠N, J}{}_{\mathrm{N\; n}}\text{= H}{}_{\mathrm{n}}\text{(Q}{}_{\mathrm{l}}\text{,P}{}_{\mathrm{l}}\text{)}$ $\text{W}{}_{\mathrm{kn}}\text{=}\text{∫}\text{C}\text{Q}{}_{\mathrm{N}}\text{δH}{}_{\mathrm{n}}\text{δP}{}_{\mathrm{k}}\text{+}\text{∑}\text{s=1}\text{n−1}\text{{W}{}_{\mathrm{ks}}\text{, H}{}_{\mathrm{n-s}}\text{}}\text{dQ′}{}_{\mathrm{N}}$ and P_k(q_l, p_l), Q_k(q_l, p_l) is the canonical transformation for which P_N = H₀(q_l, p_l), (δ_kN = Kronecker delta, {} = Poisson bracket). Quantities J_k, w_k are the canonical variables giving the perturbed Hamiltonian the form H = J_N. There are 2N − 1 time-independent integrals: J₁ ,…, J_N, W₁ ,…, W_N−1 describing the trajectory in 2N-dimensional phase space. The last integral W_N − t = const, determines the position on the trajectory at any given time.