A follow on question to my previous one:

[Not sure why the Latex typesetting is not displayed throughout. Looks fine in the preview.]

Context. I would like to now extend the coordinate computation from the configuration space

${\mathbf q }= (x, y)$ to phase space $({\mathbf q, \mathbf p}) = (x, y, p_{x}, p_{y} )$

to compute the Hamiltonian vector field a.k.a. the Hamiltonian geodesic flow:

$X_k = g^{i j} p_j \frac{\partial } {\partial x^i} + g^{ kj} \Gamma^l_{j i} p_k p_l \frac{\partial } {\partial p_i}$

from the geometric Hamiltonian

$K = g^{i j}(q) p_i p_j$

with

$ \frac {dx^i} {ds} = \frac{\partial K} {\partial p_i} = g^{i j} p_j$

$ \frac {dp_i} {ds} = - \frac{\partial K} {\partial x^i} = g^{k j} \Gamma^l_{j i} p_k p_l$

where the metric is, as before,

$g_{i j} = 2[H_{o} - V(q)] \delta_v{i j}$

with a specific $V(q) = V(x, y)$ to be substituted at the end of the computation.

The code cell below is my current attempt implement $X_k$ using a separate set of indices for $q$ and $p$ so as to distinguish between the partial derivatives

$\frac{\partial } {\partial x^i}$ and $\frac{\partial } {\partial p_{\nu}}$.

However, will mixing the two sets of indices in the metric and its inverse work?

The code fragment

```
substitute(D1, h_flow);
substitute(D1, Christoffel);
evaluate(D1, metric, rhsonly=True);
```

returns $\nabla{q^{a}}$ for each line. To sum up, how would I go about implementing this?

```
# Version 2.0
# Geometric Hamiltonian dynamics
# Ref. "Geometric approach to Lyapunov analysis in Hamiltonian dynamics"
# Most recent version [V3]: https://arxiv.org/abs/nlin/0104005
# Published version: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.64.066206
{x, y}::Coordinate;
{i, j, k, l, m, n, h#}::Integer(0..1);
{i, j, k, l, m, n, h#}::Indices(values={x, y}, position=fixed);
(px, py)::Coordinate;
{\iota, \kappa, \lambda, \mu, \nu, \xi, \eta#}::Integer(0..1);
{\iota, \kappa, \lambda, \mu, \nu, \xi, \eta#}::Indices(values={px, py}, position=fixed);
\nabla::Derivative;
\partial_{#}::PartialDerivative;
g_{i j}::Metric;
g_{i j}::Depends(x, y);
g^{i j}::InverseMetric;
g^{i j}::Depends(x, y);
# HÃ©non-Heiles Hamiltonian potential: V
V::Depends(x, y);
V := (x**2 + y**2)/2 + (x**2)*y - (y**3)/3;
# Ho is a constant of the motion: conservation of energy
metric := { g_{x x} = 2*(Ho - V),
g_{y y} = 2*(Ho - V) };
complete(metric, $g^{i j}$);
Christoffel := \Gamma^{i}_{j k} = (1/2)*g^{i l}*( \partial_{k}{g_{l j}}
+ \partial_{j}{g_{l k}}
- \partial_{l}{g_{j k}} );
evaluate(Christoffel, metric, rhsonly=True);
# substitute(Christoffel, $V = @(V)$);
# map_sympy(Christoffel, 'simplify');
p_{#}::Depends(px, py);
# Hamiltonian flow template
h_flow := \nabla{v?_{h}} -> g^{i \kappa}*p_{\kappa}*\partial_{i}{v?_{h}} + g^{\kappa \lambda} * \Gamma^{\mu}_{\lambda \nu}*p_{\kappa}*p_{\mu}*\partial_{\nu}{v?_{h}};
D1 := \nabla{q^{a}};
substitute(D1, h_flow);
substitute(D1, Christoffel);
evaluate(D1, metric, rhsonly=True);
```