Word Count : 500

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For a classical one-dimensional, non-relativistic harmonic oscillator with mass m and spring constant k, Newton’s equation m d²x/dt² = −k x gives harmonic motion. The general solution is x(t) = A cos(ωt + φ), where ω = √(k/m), and the momentum is p(t) = m dx/dt = −mA ω sin(ωt + φ). The initial condition “at maximum displacement at t = 0” implies x(0) = A and p(0) = 0, so φ = 0. Therefore the explicit time dependences are x(t) = A cos(ωt) and p(t) = −mA ω sin(ωt). Energy conservation relates the amplitude to the total energy. The Hamiltonian is H = p²/(2m) + ½ k x², so at t = 0 we have E = ½ k A², giving A = √(2E/k). ___ ________ ___ _____ _______ _________ __________.
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