Read-back: Def_HryniewiczCriterion_Basic

All declarations live in the namespace HryniewiczCriterion. Throughout, $H:\mathbb R^4\to\mathbb R$ is an arbitrary function (no regularity is assumed unless a definition says so).

Ambient space $\mathbb R^4$ (R4)

R4 is the space of functions $\{0,1,2,3\}\to\mathbb R$, i.e. $\mathbb R^4$ with coordinates $x=(x_0,x_1,x_2,x_3)$, indexed from $0$. Its built-in norm is the sup norm $\max_i |x_i|$; this norm is used implicitly by every derivative (fderiv) below, but the Euclidean norm is defined separately (see euclidNorm). Derivatives on $\mathbb R^4$ do not depend on the choice of norm.

liouvilleForm

For $x,v\in\mathbb R^4$:

$$\lambda_x(v)=\tfrac12\bigl(x_0v_1-x_1v_0+x_2v_3-x_3v_2\bigr).$$

The coordinate pairs are $(x_0,x_1)$ and $(x_2,x_3)$, with the first member of each pair carrying the $+$ sign in front of $x_\cdot v_\cdot$ of the second member. Note $\lambda_x(x)=0$ for every $x$.

omega0

For $u,v\in\mathbb R^4$:

$$\omega_0(u,v)=u_0v_1-u_1v_0+u_2v_3-u_3v_2 ,$$

i.e. $\omega_0=dx_0\wedge dx_1+dx_2\wedge dx_3$ in this sign convention. One has $\omega_0(x,v)=2\lambda_x(v)$.

dot4, euclidNorm

$\langle x,y\rangle=\sum_{i=0}^{3}x_iy_i$ (standard Euclidean inner product) and $|x|=\sqrt{\langle x,x\rangle}$ (Euclidean norm; $|0|=0$).

partialDeriv

$$\partial_iH(x)=DH_x(e_i),\qquad i\in\{0,1,2,3\},$$

where $e_i$ is the $i$-th standard basis vector and $DH_x$ is the Fréchet derivative of $H$ at $x$. Junk value: if $H$ is not differentiable at $x$, then $DH_x=0$ and $\partial_iH(x)=0$ for all $i$.

hamiltonianVectorField

$$X_H(x)=\bigl(-\partial_1H(x),\ \partial_0H(x),\ -\partial_3H(x),\ \partial_2H(x)\bigr).$$

With the conventions above this satisfies $\omega_0(X_H(x),v)=-DH_x(v)$ for all $v$ (wherever $H$ is differentiable), and

$$\lambda_x\bigl(X_H(x)\bigr)=\tfrac12\sum_{i}x_i\,\partial_iH(x)=\tfrac12\,DH_x(x).$$

At points where $H$ is not differentiable, $X_H(x)=0$.

energySurface

$$\Sigma_H=\{x\in\mathbb R^4: H(x)=1\}.$$

The level is fixed to $1$. No regularity or nonemptiness is assumed.

IsStrictlyStarShapedLevel

$H$ satisfies this property iff all three hold:

1. $H$ is $C^\infty$ on all of $\mathbb R^4$;
2. for every $u\neq0$ there is exactly one $r>0$ with $H(ru)=1$ (each open ray from the origin meets $\Sigma_H$ in exactly one point);
3. for every $x$ with $H(x)=1$: $DH_x(x)>0$.

Nothing is required about $H$ off the level set, about $H(0)$ beyond what follows (condition 3 excludes $H(0)=1$, since $DH_0(0)=0$), or about boundedness/compactness of $\Sigma_H$ beyond what conditions 1–2 imply.

IsStrictlyConvexLevel

For every $x$ with $H(x)=1$ and every $v\neq0$ with $DH_x(v)=0$:

$$D^2H_x(v,v)>0,$$

where $D^2H_x(v,v)$ is the Fréchet derivative of $y\mapsto DH_y$ at $x$, applied to $v$ and then to $v$. No smoothness is assumed by this definition itself. Junk values: if $H$ is not differentiable at $x$ then $DH_x=0$, so every $v\neq0$ satisfies $DH_x(v)=0$; if $y\mapsto DH_y$ is not differentiable at $x$ then $D^2H_x=0$ and the inequality $0>0$ fails. Since $\ker DH_x$ is always nonzero in $\mathbb R^4$, the condition can only hold at points of $\Sigma_H$ where $DH$ is differentiable. If $\Sigma_H=\varnothing$ the condition holds vacuously.

IsTrajectory

A curve $y:\mathbb R\to\mathbb R^4$ is a trajectory of $H$ iff

• for every $t\in\mathbb R$, $y$ is differentiable at $t$ with $y'(t)=X_H(y(t))$, and
• $H(y(t))=1$ for every $t\in\mathbb R$.

So trajectories are defined for all real time and lie on $\Sigma_H$. If $X_H$ vanishes at a point $p\in\Sigma_H$ (including any point where $H$ is not differentiable), the constant curve $y\equiv p$ is a trajectory.

PeriodicOrbit (structure)

A periodic orbit $P$ of $H$ consists of data:

• a curve $x_P:\mathbb R\to\mathbb R^4$,
• a real number $T_P$,

subject to: $T_P>0$; $x_P$ is a trajectory of $H$ (as above); and $x_P(t+T_P)=x_P(t)$ for all $t\in\mathbb R$.

$T_P$ is any positive period, not necessarily the minimal one; the same curve with periods $T$, $2T$, $3T,\dots$ gives distinct periodic orbits (multiple covers are included). Two orbits differing by a time shift of the curve are also distinct objects. Constant (stationary) trajectories are allowed if they exist.

PeriodicOrbit.IsPrime

$$\forall t\in(0,T_P):\ x_P(t)\neq x_P(0).$$

The curve does not return to its starting point strictly before time $T_P$.

PeriodicOrbit.image

$\operatorname{im}P=\{x_P(t):t\in\mathbb R\}$, the full range of the curve (a set, carrying no parametrisation, orientation or multiplicity).

quatQ1, quatQ2

Linear maps $\mathbb R^4\to\mathbb R^4$:

$$Q_1(x)=(-x_2,\ x_3,\ x_0,\ -x_1),\qquad Q_2(x)=(x_3,\ x_2,\ -x_1,\ -x_0).$$

By direct computation: $\langle Q_ix,x\rangle=0$, $\langle Q_1x,Q_2x\rangle=0$, $|Q_ix|=|x|$, $\lambda_x(Q_ix)=0$, and $\omega_0(Q_2x,Q_1x)=|x|^2$.

xiFrameRaw

For a map $Q:\mathbb R^4\to\mathbb R^4$ and $x\in\mathbb R^4$:

$$\tilde\xi_Q(x)=Q(x)-\frac{DH_x(Q(x))}{DH_x(x)}\,x .$$

Junk value: if $DH_x(x)=0$ (in particular if $H$ is not differentiable at $x$, or $x=0$), the quotient is $0$ and $\tilde\xi_Q(x)=Q(x)$. When $DH_x(x)\neq0$, $DH_x(\tilde\xi_Q(x))=0$.

xiFrame1, xiFrame2

$$\xi_1(x)=|x|^{-1}\,\tilde\xi_{Q_2}(x),\qquad \xi_2(x)=|x|^{-1}\,\tilde\xi_{Q_1}(x).$$

Note the index swap: $\xi_1$ is built from $Q_2$ and $\xi_2$ from $Q_1$. The normalising factor is the Euclidean norm of the base point $x$, not of the vector itself. At $x=0$ the inverse is taken as $0$, so $\xi_1(0)=\xi_2(0)=0$.

Consequences of the formulas (for $x\neq0$): $\lambda_x(\xi_i(x))=0$; if $DH_x(x)\neq0$ then $DH_x(\xi_i(x))=0$; and, since $\omega_0(x,\cdot)=2\lambda_x=0$ on $x$ and on $Q_ix$,

$$\omega_0\bigl(\xi_1(x),\xi_2(x)\bigr)=|x|^{-2}\,\omega_0(Q_2x,Q_1x)=1 .$$

The vectors $\xi_i(x)$ are generally not unit vectors and not Euclidean-orthogonal (the $x$-correction terms change their lengths and inner product).