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Subject: Numerical Methods for Ordinary Differential Equations: Initial Value Problems.

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4. Consider the Kepler system that has the following Hamiltonian: H (P1, P2,91,92) = 3 (Pi + p) - Apply the Verlet scheme to this problem for a variety of initial conditions. Produce a single plot of these various solutions in the 9192 plane. Briefly summarize and explain your results. Ok) Verlet Scheme for General Hamiltonians: [Implicit] $*** = p ba (25+3,') q*** =q' +* { Soy (183,9") + 2 * (**, g'*')} [implicit] p1+1 = p'+: - (ont, qu+1) [Explicit] O(K2) Verlet Scheme for Separable Hamiltonians: H(p, q) = {(p) + V(g) [Explicit] [Explicit] g** = g' +km (2"+) p=+* = p++? (2+") [Explicit] 4. Consider the Kepler system that has the following Hamiltonian: H (P1, P2,91,92) = 3 (Pi + p) - Apply the Verlet scheme to this problem for a variety of initial conditions. Produce a single plot of these various solutions in the 9192 plane. Briefly summarize and explain your results. Ok) Verlet Scheme for General Hamiltonians: [Implicit] $*** = p ba (25+3,') q*** =q' +* { Soy (183,9") + 2 * (**, g'*')} [implicit] p1+1 = p'+: - (ont, qu+1) [Explicit] O(K2) Verlet Scheme for Separable Hamiltonians: H(p, q) = {(p) + V(g) [Explicit] [Explicit] g** = g' +km (2"+) p=+* = p++? (2+") [Explicit]