(1) Let M be a fixed Riemannian manifold, and let X : R → M be a map from the t-line R (endoowed with the metric (dt)^{2}) to M.

Let

L = \frac{1}{2} ∫ dtg_{IJ} \frac{dX^{I}}{dt} \frac{dX^{j}}{dt} .  (1)

(a) Describe the space W of critical points of L. (You should find that it is closely related to the space of geodesics on M.)

(b) Show that given any choice of a point t_{0} ∈ R, W has a natural identification with T^{∗}M. (Use the metric on M to identify T M and T^{∗}M.)

(c) Compute the symplectic structure ω on W. Show that for any choice of t_{0}, W becomes identified (under the identification in (b)) with T^{∗}M with its natural symplectic structure.

(c) The vector field d/dt on R induces a vector field v on W. The “Hamiltonian” is a function H on W such that

dH = i_{v}ω.  (2)

This is usually expressed by saying that “H generates time translations via Poisson brackets.” Compute H.

(2) Consider R^{1,1} with even and odd coordinates t and θ and an odd distribution A generated by the vector field

D = \frac{∂}{∂θ} − θ \frac{∂}{∂t}.  (3)

Thus, D^{2} = −∂/∂t.

As in the last exercise, let M be a fixed Riemannian manifold. Let X be a map from R^{1,1} to M. Consider the Lagrangian

L = \frac{1}{2} ∫_{R^{1,1}} g_{IJ} \frac{∂X^{I}}{∂t} DX^{j}.  (4)

(a) Make sense of the definition of L by interpreting the integrand as a section of the Berezinian of the tangent bundle of R^{1,1}.

(b) Setting X^{I} = x^{I} + θψ^{I}, write L explicitly as an ordinary integral, over the t-line R^{1,0}, of a function of x^{I} and ψ^{I}.

(c) Describe the space Y of critical points of L. You should get a description of the following kind. Describe the reduced space Y_{red} as an ordinary manifold, and the normal bundle to Y_{red} in Y as ΠV, where V is a vector bundle over Y_{red}. What are the “initial data” at a given point t = t_{0} in R^{1,1} needed to determine a critical point? (This is the analog of problem 1(b) above.)

(d) Compute the symplectic structure on Y.

(e) The vector fields

\frac{∂}{∂θ} + θ \frac{∂}{∂t} and \frac{∂}{∂t}  (5)

induce vector fields on Y. Find the functions Q and H that generate these vector fields by Poisson brackets. Explain why {Q, Q} = 2H.