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 Hamiltonian 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- erential 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 # F = Vf. In this problem you will conservative system?) show that Hamiltonian vector fields salisly an analogous relationship but with divergence Definition. A vector field F(r,y) is called Hamiltonian if there exits a function H (called instead of the curl. the Hamiltonian function) such that First assume F is Hamiltonian and show that the divergence is zero and the flux is path independent. F(x,y) = (Hy(z, y), -H=(x, ")) - (i) Show that if F is a Hamiltonian vector field, then div(F) = 0. Suggested strategy: Definition. A vector field F(r, y) is called conservative if there exists a function f (called direct calculation; compute div (F). a potential function) such that (i) Show that if ?'(t) = (r(1), y(1)) is a parametrization of the curve C from 7*(0) = P to 7(1) = Q, then the flux of F through C is path independent. Suggested strategy: try F(z, y) = (1=(x, y), f,(z, #)) = VS. to repeat the steps in the proof (in the book or my class notes) of the fundamental theorem for line integrals, but replace the line integrals of F with flux integrals, and Compare the two definitions. Both define the field F in terms of the partial derivatives of a (obviously) use that F is Hamiltonian instead of conservative. scalar function; Hamiltonian functions are to Hamiltonian vector fields as potential functions Now show the reverse implication: assume div( F) = 0 and show that F is Hamiltonian. are to conservative vector fields. In fact we could write the Hamiltonian vector field using the same "del" operator (ii) Show that if div( F') = 0 then F is a Hamiltonian vector field. Suggested strategy: F(I, y) = J(VH) first show that if div( F) = 0 then the flux integral of F is path independent (use the where / is the rotation matrix Divergence Theorem). Then use path independence (of the flux) to reconstruct the Hamiltonian function H (see the proof that path independence of the line integral implies F is conservative in the textbook). This / matrix is the same rotation operation that we used to define the normal vector in 2d, Problem 2: In class we learned that conservative fields are called conservative because J((z,y)) = (1, -x) and mechanical energy E is conserved when an object moves subject to a conservative force: N(t) = JP(1). we used chain rule to find # E, and then used 7* = F (Newton's Second law) to simplify this expression yielding zero. (i) Use a similar strategy to show that # is conserved along solutions to the differential equation ?"(t) = F, where F is Hamiltonian. (i) Give a geometric interpretation of this fact