As a Cadabra 2 neophyte, with experience in symbolic computation using Sympy, I'm attempting my first component computation after reading the reference manuals. The application is to geometric-classical Hamiltonian dynamics. I've tried to follow and adapt the simple 2-sphere example provided. However, I've run into an issue and it's not clear to me what aspect(s) of Cadabra 2 I'm misunderstanding.

Running the Cadabra 2 kernel in Jupyter Notebook using Firefox 75.0. Ran the 2-sphere coordinate computation example and it worked as expected.

The code, so far, is as follows:

```
# Component computations
# 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);
\partial{#}::PartialDerivative;
g_{i j}::Metric.
g^{i j}::InverseMetric.
# Hénon-Heiles potential
V := (x**2 + y**2)/2 + (x**2)*y - (y**3)/3;
# Ho is a constant: 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);
```

After calling "evaluate", all the components of the Christoffel symbol are returned as being identically zero, whereas a calculation by hand yields, for example,

Christoffel^{0}_{0 0} = -[1 / (Ho - V)] * (x + 2*x*y)

Once V is substituted into the metric, is it being treated a constant in the above code?

Any insight into what I have misunderstood with regards to Cadabra Coordinates and Indices would be appreciated.

System configuration:

OS. Ubuntu 20.04 LTS WSL
Python. Anaconda 3.7.7
cadabra2-jupyter-kernel 2.2.9

jupyter 1.0.0

jupyter_client 6.1.2

jupyter_console 6.1.0

jupyter_core 4.6.3
jupyterlab 1.2.6
jupyterlab_server 1.1.0