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4. A Hamiltonian system in R2 is a system of the form ан ду -(x, y) == OH əx y' = -(x, y) where H R² → R is a C function called the Hamiltonian function. (Ideal Pendulum.) The system of equations 0' =v v=sin 0 describes the motion of an undamped pendulum. (a) Show that this system is a Hamiltonian system (i.e. find a Hamiltonian function for this system). (b) Determine all the critical points of the system.

4. A Hamiltonian system in R2 is a system of the form ан ду -(x, y) == OH əx y' = -(x, y) where H R² → R is a C function called the Hamiltonian function. (Ideal Pendulum.) The system of equations 0' =v v=sin 0 describes the motion of an undamped pendulum. (a) Show that this system is a Hamiltonian system (i.e. find a Hamiltonian function for this system). (b) Determine all the critical points of the system.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. A Hamiltonian system in R2 is a system of the form ан ду x' -(x, y) H y' -(x, y) where H: R² → R is a C function called the Hamiltonian function. ?x (Ideal Pendulum.) The system of equations 0' =v v=sin 0 describes the motion of an undamped pendulum. (a) Show that this system is a Hamiltonian system (i.e. find a Hamiltonian function for this system). (b) Determine all the critical points of the system. (c) Plot several trajectories of the system. Indicate the direction of the motion on each tra- jectory. (Hint: use part (a).)
Transcribed Image Text:4. A Hamiltonian system in R2 is a system of the form ан ду x' -(x, y) H y' -(x, y) where H: R² → R is a C function called the Hamiltonian function. ?x (Ideal Pendulum.) The system of equations 0' =v v=sin 0 describes the motion of an undamped pendulum. (a) Show that this system is a Hamiltonian system (i.e. find a Hamiltonian function for this system). (b) Determine all the critical points of the system. (c) Plot several trajectories of the system. Indicate the direction of the motion on each tra- jectory. (Hint: use part (a).)
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