Does it make sense to iterate phase space?

Question

I am studying Hamiltonian mechanics, where the phase space of configuration space $M$ is abstractly represented by the cotangent bundle $ T^*(M) $. Hamiltonian vector fields naturally arise in terms of certain sections $T^*(M)\to T(T^*(M))$ of the tangent bundle on phase space.

My question is as follows; we can treat the phase space itself as the configuration space and iterate the cotangent bundle to obtain $T^*(T^*(M))$, and of course we can construct vector fields as sections of the tangent bundle $T^*(T^*(M))\to T(T^*(T^*(M)))$ . Perhaps this is a naïve question, but does this "iterated phase space" have any physical interpretation, and moreover, do Hamiltonian flows on $T^*(T^*(M))$ correspond to anything useful in physics?

Answer 1

Yes, double tangent bundles, double cotangent bundles, and so forth, do appear in the literature.

Examples:

• Ref. 1 states on p.33: In 1977, W.M. Tulczyjew [38] gave a geometric interpretation of the Legendre transform as a canonical symplectomorphism between the cotangent bundles $T^{\ast}(TM)$ and $T^{\ast}(T^{\ast}M)$.

• In the Hamiltonian version of the Batalin-Vilkovisky (BV) formalism, one introduces momenta to both the fields and antifields, i.e. one considers $T^{\ast}(\Pi T^{\ast}M)$. Here $\Pi T^{\ast}M$ denotes the Grassmann-parity-reversed cotangent bundle [since the antifields have opposite Grassmann-parity] with a Grassmann-odd Poisson structure [=the antibracket].

References:

1. D. Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, Ph.D. thesis, arXiv:math/9910078; p.33.