Non-Euclidean Phase Space? In classical mechanics, the canonical equations of motion can be rendered in terms of Poisson Brackets:
$$\begin{align}
\left\{q_i, F(\mathbf{q},\mathbf{p})\right\} &= \frac{\partial F}{\partial p_i}, \\
\left\{p_i, F(\mathbf{q},\mathbf{p})\right\} &= -\frac{\partial F}{\partial q_i},\ \mathrm{and} \\
\left\{H, F(\mathbf{q},\mathbf{p})\right\} &= -\frac{\operatorname{d} F}{\operatorname{d} t}.
\end{align}$$ 
This is taken to mean that the $q_i$ generates translations in the $-p_i$ direction, $p_i$ in the $q_i$ direction, and $H$ (the Hamiltonian) through time. Is there anything that can be gained by adding a Christoffel symbol like connection to the canonical equations (ie translating the phase space gradient into a covariant derivative)?
Concretely, say $V_j$ is in a vector space tangent to the phase space manifold (in some combination of $\mathbf{q}$ and $\mathbf{p}$ directions, or in an entirely unrelated vector space). Is it possible to construct a meaningful phase space by defining the Poisson brackets as:
$$\begin{align}
\left\{q_i, V_j(\mathbf{q},\mathbf{p})\right\} &= \frac{\partial V_j}{\partial p_i} + \left[\Gamma_p\right]_{i\hphantom{k}j}^{\hphantom{i}k} V_k, \\
\left\{p_i, V_j(\mathbf{q},\mathbf{p})\right\} &= -\frac{\partial V_j}{\partial q_i} - \left[\Gamma_q\right]_{i\hphantom{k}j}^{\hphantom{i}k} V_k,\ \mathrm{and} \\
\left\{H, V_j(\mathbf{q},\mathbf{p})\right\} &= -\frac{\operatorname{d} V_j}{\operatorname{d} t}- \left[\Gamma_t\right]_{i\hphantom{k}j}^{\hphantom{i}k} V_k,
\end{align}$$
or some analogous construction?
Is the resulting curved phase space always expressible, through some transformation of coordinates and Hamiltonian, using ordinary canonical equations of motion?
 A: *

*Given a symplectic manifold $(M,\omega)$, it is natural to ponder what tangent bundle connection $$\nabla: \Gamma(TM)\times\Gamma(TM)\to \Gamma(TM) \tag{1}$$ to chose?


*Generically, it is natural to choose $\nabla$ to be torsionfree
$$T~=~0,\tag{2}$$
and compatible
$$\nabla \omega~=~0\tag{3}$$
with the symplectic $2$-form $\omega$.


*One may show (via partition of unity) that a torsionfree & compatible connection $\nabla$ exists on a paracompact manifold, cf. e.g. this Phys.SE post. Be aware that a such a connection $\nabla$ is far from being unique.


*The triple $(M,\omega,\nabla)$ is called a Fedosov manifold, and it is the geometric input for the Fedosov star product $\star$ in deformation quantization.


*Fedosov quantization can be used to define covariant derivatives and time evolution for tensor fields, cf. Refs. 1-2. The classical construction can be extracted in the $\hbar\to 0$ limit.


*In special cases the symplectic manifold $(M,\omega)$ is endowed with a compatible metric $g$, cf. Kähler manifold. In such situations, the metric $g$ uniquely singles out the Levi-Civita connection.
See also this related Phys.SE post.


*Finally, let us mention that if the symplectic manifold $M=T^{\ast}Q$ is a cotangent bundle equipped with the tautological symplectic structure (cf. e.g. this Phys.SE post), and the base manifold $Q$ is endowed with a connection, this also leads to interesting possibilities, e.g. a super-Poisson bracket, cf. Ref. 3.
References:

*

*B.V. Fedosov, A simple geometrical construction of deformation quantization, J.  Diff. Geom. 40 (1994) 213.


*B.V. Fedosov, Deformation quantization and index theory, Mathematical Topics, Vol. 9, Akademie Verlag, Berlin, 1996.


*B. DeWitt, Supermanifolds, Cambridge Univ. Press, 1992; Section 6.7.
