New methods of isochrone mechanics

Formulas (2.3) are true, in general, and explicit formulas such as the rhs of (2.2) are not necessary to derive (2.3) from $J=12π∮prdr$⁠. Fundamentally, this can be understood from the fact that, locally around the equilibrium (circular orbit), the pair (H, t) itself defines symplectic coordinates (see Ref. 26 for more details).

There is a subtlety in the case a₂ < 0, since then the function $u(x)=c2x+c3$ is decreasing. This is resolved by keeping track of sign(a₂) = sign(b), which results in the ±|b| in (2.17).

We focus on elliptic equilibria since we want to describe a periodic motion of the particle.

In the general case $(q,p)∈Rn×Rn$⁠, then $ρ∈Rn$⁠, and accordingly, the Birkhoff invariants $b$ and $B$ are linear and bilinear forms, respectively.

We assume that F, G behave nicely: they can be differentiated several times and inverted on their domain of definition. We know this will be the case as we know their explicit form (1.2) and (1.3).

In the 4D phase space, this change of variable is rendered symplectic by changing the angle accordingly. For example, the mapping (x, θ, p_x, Λ) ↦ [x − x_c(Λ), $θ̂$⁠, $p̂x$⁠, Λ] is symplectic if we take $θ̂=θ−xc′(Λ)px$⁠.

We would also need to change the angle $θ̂↦θ̂+η′(Λ)η(Λ)x̂$ to ensure symplecticity in the 4D phase space.