Area of the symmetric difference of two disks

This post goes back to high school: the area \delta_d of the symmetric difference of two d-dimensional disks when one center is shifted a little bit. Let’s start with d=1. So we have two intervals [-r,r] and [x-r,x+r]. It is easy to see the symmetric difference is of length \delta_1(x)=2x.

Then we move to d=2: two disks L and R of radius r and center distance x=2a<r. So the angle \theta(x) satisfies \cos\theta=\frac{a}{r}.

difference

The symmetric difference is the union of R\backslash L and L\backslash R, which have the same area: \displaystyle (\pi-\theta)r^2+2x\sqrt{r^2-x^2}-\theta r^2=2(\frac{\pi}{2}-\arccos\frac{x}{r})r^2+2x\sqrt{r^2-x^2}. Note that the limit
\displaystyle \lim_{x\to0}\frac{\text{area}(\triangle)}{2a}=\lim_{a\to0}2\left(\frac{r^2}{\sqrt{1-\frac{a^2}{r^2}}}\cdot\frac{1}{r}+\sqrt{r^2-a^2}\right)=4r.
So \delta_2(x)\sim 4rx.

I didn’t try for d\ge3. Looks like it will start with a linear term 2d r^{d-1}x.

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Now let {\bf r}(t)=(a\cos t,\sin t) be an ellipse with a>1, and {\bf r}'(t)=(-a\sin t,\cos t) be the tangent vector at {\bf r}(t). Let \omega be the angle from {\bf j}=(0,1) to {\bf r}'(t).
Let s(t)=\int_0^t |{\bf r}'(u)|du be the arc-length parameter and K(s)=|{\bf l}''(s)| be the curvature at {\bf l}(s)={\bf r}(t(s)). Alternatively we have \displaystyle K(t)=\frac{a}{|{\bf r}'(t)|^{3}}.

ellipse

The following explains the geometric meaning of curvature:

\displaystyle K(s)=\frac{d\omega}{ds}, or equivalently, K(s)\cdot ds=d\omega. (\star).

Proof. Viewed as functions of t, it is easy to see that (\star) is equivalent to K(t)\cdot \frac{ds}{dt}=\frac{d\omega}{dt}.

Note that \displaystyle \cos\omega=\frac{{\bf r}'(t)\cdot {\bf j}}{|{\bf r}'(t)|}=\frac{\cos t}{|{\bf r}'(t)|}. Taking derivatives with respect to t, we get
\displaystyle -\sin\omega\cdot\frac{d\omega}{dt}=-\frac{a^2\sin t}{|{\bf r}'(t)|^3}. Then (\star) is equivalent to

\displaystyle \frac{a}{|{\bf r}'(t)|^{3}}\cdot |{\bf r}'(t)|=\frac{a^2\sin t}{\sin\omega\cdot |{\bf r}'(t)|^3}, or
\displaystyle \sin\omega\cdot |{\bf r}'(t)|=a\sin t. Note that \displaystyle \sin^2\omega=1-\cos^2\omega=1-\frac{\cos^2 t}{|{\bf r}'(t)|^2}. Therefore \displaystyle \sin^2\omega\cdot |{\bf r}'(t)|^2= |{\bf r}'(t)|^2-\cos^2 t=a^2\sin^2 t, which completes the verification.


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More generally let’s start with a plane curve {\bf r}(x)=(x,y(x)). Then \displaystyle  K(t)=\frac{|\ddot{y}|}{(1+\dot{y}^2)^{3/2}} (there should be some convention to choose the signature of the curvature. We set it as \displaystyle \text{sgn}(\frac{d\omega}{ds})),
\displaystyle \cos\omega=\frac{{\bf r}'(x)\cdot {\bf j}}{|{\bf r}'(x)|}=\frac{\dot{y}}{\sqrt{1+\dot{y}^2}}, and
\displaystyle -\sin\omega\cdot\frac{d\omega}{dt}=\frac{\ddot{y}}{(1+\dot{y}^2)^{3/2}}. Then we can check that (\star) \displaystyle K(s)=\frac{d\omega}{ds} also holds.

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Let (M,\omega) be a symplectic manifold and H:M\to\mathbb{R} be a smooth function, X be the vector field satisfying dH=\imath_X\omega. Then H is called the Hamiltonian function and X the Hamiltonian vector field. It is well known that induced Hamiltonian flow \phi_t preserves \omega.
Proof in Arnold’s book. To show \phi^\ast_t\omega=\omega, it suffices to show that \int_{\phi_t D}\omega=\int_D \phi^\ast_t\omega=\int_D \omega for all small disks D and all small t>0.

To this end let’s consider the track D_\tau of D under the flow \phi_t, 0\le t\le \tau. It is easy to see \partial D_\tau=\phi_\tau D-D+(\partial D)_\tau. So we only need to show \int_{C_\tau}\omega=0 for any smooth circle C:[0,1]\to M, C(0)=C(1).

Let \xi=\frac{\partial}{\partial s}\phi_tC(s) and \eta=\frac{\partial}{\partial t}\phi_tC(s)=X(\phi_tC(s)). Then \int_{C_\tau}\omega=\int_0^\tau\int_0^1\omega(\xi,\eta)ds dt=\int_0^\tau\int_0^1 dH_{\phi_tC(s)}(\xi)dsdt =\int_0^\tau\int_{\phi_t C} dH dt=0, since C is closed. This ends the proof.

(From the same book) Let (\mathbb{R}^{2n},\omega) be the standard symplectic space and S be a symplectic isomorphism, that is, \omega(S\xi,S\eta)=\omega(\xi,\eta) for all \xi,\eta\in\mathbb{R}^{2n}. The eigenvalues of S are symmetric with respect to the real axis and with respect to the unit cirle. For example \lbrace\lambda,\overline{\lambda},\frac{1}{\lambda},\frac{1}{\overline{\lambda}}\rbrace forms a quadruple of ev’s.

Note that the ev’s can leave S^1 only by colliding with another ev’s and from two pairs of ev’s on S^1 we obtain one quadruple. Conversely, the conjugate pairs \lbrace\lambda,\frac{1}{\lambda}\rbrace,\lbrace\overline{\lambda},\frac{1}{\overline{\lambda}}\rbrace may collide and slide on S^1. In particular S is (stably) Lyapunov-stable if all the ev’s are distinct and lie on the unit circle.

Krein noticed that not every collision will result in one of the previous bifurcations. In fact he defined the signs of complex ev’s, and proved that the complex ev’s with same sign will ‘go through one another’ after the collision and can’t leave S^1.

\star Sign of a simple complex ev pair \lambda,\overline{\lambda}\in S^1. Let \pi be the corresponding two-dimensional invariant plane. It can be shown that \omega(\xi,S\xi)\neq0 for all unit vectors v\in\pi and hence have the same sign. Then \lambda is said to be positive if \omega(\xi,S\xi)>0 (so well defined). Or equivalently, the quadratic form \omega(\cdot,S\cdot) is positive-definite.

(Note that \omega(\xi,S\xi)\to 0 as \lambda\in S^1\to \pm 1. So the sign of \lambda may be changed after crossing the real axis).

\star Sign of a complex ev pair \lambda,\overline{\lambda}\in S^1 with multiplicity k\ge2. Let \pi be the corresponding 2k-dimensional invariant subspace. Then \lambda is said to be positive definite sign if \omega(\cdot,S\cdot) is positive-definite on \pi.

Then it follows that, S is (stably) Lyapunov-stable if and only if all the ev’s lie on the unit circle and are of definite signs.

————————

Let \omega be a 2-form on \mathbb{R}^{2n+1}, which can be represented by \omega(\xi,\eta)=\langle A\xi,\eta\rangle for some asymmetric matrix A. Since A^t=-A, \det(A)=\det(-A)=-\det(A) and hence \det(A)=0. So there always exists \xi such that \imath_\xi\omega=0.

A 2-form \omega on \mathbb{R}^{2n+1} is said to be nonsingular if the \xi‘s with \imath_\xi\omega=0 form a 1-dimensional subspace \langle\xi\rangle. For example \omega=dp\wedge dq-\alpha\wedge dt is nonsingular (for arbitrary 1-form \alpha).

Now let M be a (2n+1)-dimensional manifold, \alpha be a 1-form on M such that \omega=d\alpha is nonsingular. Then the zero-direction V_x\langle\xi_x\rangle of \omega is called the vortex direction of \alpha. The integral curves tangent to V are called the vortex lines of \alpha. Pick arbitrary closed curve \gamma\subset m. The vortex lines passing \gamma form a vortex tube. Then

Multidimensional Stokes Lemma. The integrals of \alpha along any two curves are the same if they encircle the same vortex tube. That is, \int_\gamma \alpha=\int_\beta \alpha if \gamma-\beta=\partial \sigma for some piece \sigma of the vortex tube.
Proof. \int_\gamma \alpha-\int_\beta \alpha=\int_{\partial \sigma} \alpha=\int_\sigma d\alpha=\int_\sigma \omega=0, since \sigma contains the vortex direction.

Consider a Hamilton function H=H(p,q,t) and 1-form \alpha=pdq-Hdt. Then \omega=d\alpha=dp\wedge dq-H_p dp\wedge dt-H_q dq\wedge dt is nonsingular and corresponds to the matrix A=\begin{pmatrix}0 & -I & H_p\\ I & 0 & H_q\\ -H_p & -H_q & 0\end{pmatrix}. Now it is easy to see that \xi=(-H_q,H_p,1) gives the vortex direction and the vortex line satisfies \dot{p}=-H_q, \dot{q}=H_p. In other words, the vortex lines coincide with the flow lines along the Hamiltonian vector field.

Consider a closed curve \gamma consisting of simultaneous states (that is, lying in \{t=t_0\}\simeq\mathbb{R}^{2n}). Note that \phi_{t_0,t_1}\gamma\subset \{t=t_1\}. Along such curves dt=0 and hence \displaystyle \int_\gamma p dq=\int_{\phi_{t_0,t_1}\gamma}pdq. In particular Hamiltionian flow preserves the integral \int pdq on closed curves in \mathbb{R}^{2n}.

Let \omega=dp\wedge dq. Then for any 2-disk D\subset \mathbb{R}^{2n}, \int_D \omega=\int_\gamma pdq=\int_{\phi_{t_0,t_1}\gamma}pdq =\int_{\phi_{t_0,t_1}D}\omega. So even the time-dependent Hamiltonian flow also preserves this 2-form \omega.

$latex $

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