Formal Integrals of Motion in Time Periodic Hamiltonian Systems Article Swipe
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· 2024
· Open Access
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· DOI: https://doi.org/10.5206/mt.v4i1.17296
· OA: W4401015318
We present an algorithm for the construction of approximate integrals of motion in time periodic Hamiltonian systems of the form H=H₀+ε H₁ where H₀=(ω₁²x²+y²)/2. We apply this algorithm in the case where H₁=−ε x³cos(ω t) with ω/ω₁ irrational and calculate approximate integrals of motion Φ=Φ₀+εΦ₁+... close to the stable periodic orbits of the system. These integrals agree approximately with the form the stroboscopic sections found by the numerical integration of the system. The agreement is better for smaller values of ε. In the resonant cases where ω/ω₁ is rational we have secular terms (terms proportional to t) in some Φᵢ. These secular terms may be avoided by using a combination of Φ and another formal integral with more complicated zero order term. We calculated explicitly such integrals in the case H₁=−ε x³ cos(ω t) the resonances with ω=2ω₁, ω=3ω₁ and in the case H₁=−ε x² cos(ω t) (in which the dynamics is governed by a Mathieu equation) the resonance ω=ω₁.
