safer Kepler

docs/src/examples/kepler_problem.md

@@ -10,7 +10,7 @@ Also, we know that
 $${\displaystyle {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}\quad ,\quad {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}=+{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}}}$$
 
 ```@example kepler
-using OrdinaryDiffEq, LinearAlgebra, ForwardDiff, Plots
+using OrdinaryDiffEq, LinearAlgebra, ForwardDiff, NonlinearSolve, Plots
 H(q, p) = norm(p)^2 / 2 - inv(norm(q))
 L(q, p) = q[1] * p[2] - p[1] * q[2]
 
@@ -26,6 +26,9 @@ prob = DynamicalODEProblem(pdot, qdot, initial_velocity, initial_position, tspan
 sol = solve(prob, KahanLi6(), dt = 1 // 10);
 ```
 
+!!! note
+    Note that NonlinearSolve.jl is required to be imported for ManifoldProjection
+
 Let's plot the orbit and check the energy and angular momentum variation. We know that energy and angular momentum should be constant, and they are also called first integrals.
 
 ```@example kepler