Perturbation confusion

I have two examples in real use cases that can trigger perturbation confusion error in ForwardDiff.jl.

The first example (Lagrangian):

using ForwardDiff
m,g = 1, 9.8
t = 1
q = [1,2]
q̇ = [3,4]
L(t,q,q̇) = m/2 * dot(q̇,q̇) - m*g*q[2]

∂L∂q̇(L, t, q, q̇) = ForwardDiff.gradient(a->L(t,q,a), q̇)
Dqq̇(L, t, q, q̇) = ForwardDiff.jacobian(a->∂L∂q̇(L,t,a,q̇), q)
Dqq̇(L, t, q, q̇)  # ERROR: potential perturbation confusion...

The second one (Hamiltonian):

using ForwardDiff
m,g = 1, 9.8
t = 1
p = [5,6]
q = [1,2]

function Legendre_transformation(F, w)
    wv = a->ForwardDiff.gradient(F, a)
    z = zeros(w)
    M = ForwardDiff.jacobian(wv, z)
    b = wv(z)
    v = Symmetric(M)\(w-b)
    w'v - F(v)
end

function Lagrangian2Hamiltonian(Lagrangian, t, q, p)
    L = q̇ -> Lagrangian(t, q, q̇)
    Legendre_transformation(L, p)
end
Lagrangian2Hamiltonian(L, t, q, p)  # This is OK
ForwardDiff.gradient(a->Lagrangian2Hamiltonian(L, t, a, p), q)
# ERROR: potential perturbation confusion...

P.S.: I am on ForwardDiff master branch

julia> Pkg.status("ForwardDiff")
 - ForwardDiff                   0.5.0+             master

Start by running the two Julia reproductions in the issue with ForwardDiff.gradient and ForwardDiff.jacobian, focusing on the nested differentiation that triggers perturbation confusion. Trace how the perturbation-confusion error arises in these examples; done means both reported use cases differentiate without that error and retain their expected results.