Homework 3 (Extra Credit) Your job as aspiring mathematicians will be to answer similar questions for Hamiltonian vector fields. In this homework assignment you will investigate properties of Hamillonian Vector Fields. . How can we check if a vector field is Hamiltonian? Hamiltonian vector fields are the geometric interpretation of a Hamiltonian system of dif- ferential equations (remember, in class I said vector fields are the geometric interpretation . If a vector field F is Hamiltonian, how can we determine the Hamiltonian function? of ordinary differential equations). Hamiltonian systems appear frequently in physics and Is the Hamiltonian function unique? engineering. For our purposes I want to emphasize the way Hamiltonian vector fields are similar to conservative vector fields, and in what ways they are different. . Investigate integrals of Hamiltonian vector fields. When we investigated conservative vector fields we wanted to know the following things: . Is anything conserved by a Hamiltonian system? How can we check if a vector field is conservative? The answer to these questions will follow similar reasoning to the proofs we constructed . If a vector field F is conservative, how can we determine the potential function? for conservative vector fields, so this will be a good opportunity for you to review the proofs Is the potential function unique? in the textbook (on Achieve) of the relevant results. Then you can attempt to generalize . Investigate integrals of conservative vector fields. those strategies to the case of Hamiltonian vector fields. . Why do we call conservative vector fields conservative? (What is conserved by a Problem 1: In class we showed that curl F = 0