Fix all examples for PR #29

[Tejas7007]

Got all 44 examples in PR #29 running end-to-end (was 3/44).

Verified locally: Fig 3 (gc_vs_fo), Fig 5 (poincare_plots), and the autodiff figure (gradients) all reproduce correctly.

Source-level changes (all additive / backward-compatible):

• essos/coils.py: override setters for Curves gamma / gamma_dash / gamma_dashdash (unblocks coil perturbation with the immutable Curves model)

• essos/surfaces.py: SurfaceRZFourier.init accepts filename string or Vmec object as first arg (existing explicit-arg calls unchanged)

• essos/dynamics.py: optional solver= kwarg in Tracing (defaults preserve Dopri8); energy() now handles FullOrbit_Boris

• essos/objective_functions.py: restored loss_coil_curvature_new, loss_coil_length_new, loss_particle_r_cross_max_constraint as flat-vector wrappers; loss_BdotN accepts surface= alternative to vmec=; fixed shape mismatch in loss_optimize_coils_for_particle_confinement

• essos/augmented_lagrangian.py: 16 jax.jax.tree_util -> jax.tree_util typo fixes

• essos/coil_perturbation.py: jnp.clip(eigvals, a_min=0) -> min=0 (NumPy 2.x API)

Example fixes:

• Path bugs (10+ files): name -> file typos, "../examples/input_files" -> "../input_files", missing "../" when running from subdirs

• API renames: Coils_from_json -> Coils.from_json (7 files), Coils_from_simsopt -> Coils.from_simsopt (2 files)

• Tracing API alignment in poincare_plots.py (timesteps=count -> timestep=dt, tol_step_size -> atol/rtol)

• gradients.py: hardcoded 8 processors -> 1 (was failing on shape mismatch)

• Rewrote paper/fo_integrators.py and paper/gc_integrators.py for current keyword API + new solver= kwarg (Fig 4 unblocked)

• .energy attribute -> .energy() method form

• _new aliasing in adam / augmented-lagrangian examples

Optional deps the examples need: simsopt, h5py, plotly, booz_xform (latter needs libnetcdf-dev + gfortran on the system). Happy to add these as extras to pyproject.toml in a follow-up if useful.

Caveats: every example now starts and runs without crashing. End-to-end completion verified for the 3 paper figures above. The longer-running optimizers (LBFGS, augmented Lagrangian) should be confirmed on full runs.

[rogeriojorge]

Done with review of this PR.

[rogeriojorge]

Hi @EstevaoMGomes, @eduardolneto, please review this today or tomorrow. If it looks good (check the new examples locally), then please approve the review or add your comments.

[EstevaoMGomes]

There are some changes needed. I'm not sure if in this or another PR, but I can help with understanding PyTrees and tracers. Nevertheless, the changes are appreciated! Thanks for fixing the examples :)

I'll try to join the next meeting to give you some input on losses.py and how we actually should differentiate over PyTrees. When I started coding ESSOS, I also did not know exactly how this worked, so the repo may still be confusing when mixing the 2 approaches.

[eduardolneto]

There was an open pull request here #30 that has conflicts with some parts of this pull request. I think the changes in that pull request should be merged before as they were already fixing a lot of these issues including coil_perturbation, coils, surfaces and augmented_lagragian/losses iteraction. The exampels were also working . @EstevaoMGomes can you recheck that one first please.

[eduardolneto]

I left my comments, I am finsihed with the review of the PR.