Consider the pendulum equation (w = 1 case),0 = v, v = – sin (0). There are "two" ways of implementing the forward Euler method: An+1 vn+1 = An+1 vn +1 = = = θη + Δt.vn vn - At sin (en) or if we use the most up-to-date value of the angle for the second equation, 1 θη + Δt. vn vn_ At sin (n+¹). Code each method and plot (t) and H (0 (t), v (t)) =v² - co t = [0, 10], and for an appropriate At. 1 What are your conclusions? 2 Try to show that the second approach guarantees that the discrete Hamil- tonian, Hn is bounded Vtn = n. At. cos (6), for 3 Using the second approach, write a code that "draws numerically" the phase portrait for the pendulum equation (e.g. figure 6.7.3 in the book).
Consider the pendulum equation (w = 1 case),0 = v, v = – sin (0). There are "two" ways of implementing the forward Euler method: An+1 vn+1 = An+1 vn +1 = = = θη + Δt.vn vn - At sin (en) or if we use the most up-to-date value of the angle for the second equation, 1 θη + Δt. vn vn_ At sin (n+¹). Code each method and plot (t) and H (0 (t), v (t)) =v² - co t = [0, 10], and for an appropriate At. 1 What are your conclusions? 2 Try to show that the second approach guarantees that the discrete Hamil- tonian, Hn is bounded Vtn = n. At. cos (6), for 3 Using the second approach, write a code that "draws numerically" the phase portrait for the pendulum equation (e.g. figure 6.7.3 in the book).
Computer Networking: A Top-Down Approach (7th Edition)
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![Consider the pendulum equation (w = 1 case),Ỏ = v, v = − sin (0). There are
"two" ways of implementing the forward Euler method:
An+1
vn +1
An+1
vn +1
=
=
or if we use the most up-to-date value of the angle for the second equation,
=
θη + Δt · υπ
vn At sin (n)
1
= θη + Δt.·υη
vn
At sin (0+¹).
Code each method and plot 0 (t) and H (0 (t), v (t)) = ¼½v² -
t = [0, 10], and for an appropriate At.
1 What are your conclusions?
2 Try to show that the second approach guarantees that the discrete Hamil-
tonian, Hn is bounded Vtn = n. At.
cos (0), for
3 Using the second approach, write a code that "draws numerically" the
phase portrait for the pendulum equation (e.g. figure 6.7.3 in the book).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F680f25e8-67cc-4689-b50c-53b5c12fc146%2Fd53845aa-2fe6-41d6-a4a0-0e2863031a76%2Fymbboi_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the pendulum equation (w = 1 case),Ỏ = v, v = − sin (0). There are
"two" ways of implementing the forward Euler method:
An+1
vn +1
An+1
vn +1
=
=
or if we use the most up-to-date value of the angle for the second equation,
=
θη + Δt · υπ
vn At sin (n)
1
= θη + Δt.·υη
vn
At sin (0+¹).
Code each method and plot 0 (t) and H (0 (t), v (t)) = ¼½v² -
t = [0, 10], and for an appropriate At.
1 What are your conclusions?
2 Try to show that the second approach guarantees that the discrete Hamil-
tonian, Hn is bounded Vtn = n. At.
cos (0), for
3 Using the second approach, write a code that "draws numerically" the
phase portrait for the pendulum equation (e.g. figure 6.7.3 in the book).
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