When I am plotting two quadratic surfaces $y^2/2 + z^2/2 - 16.7 z = -30$ and $-(x^2/2) + 16.7 z = 200$, then its intersecting orbit is found to be a single orbit (see attached image).
But, when I am changing the RHS value of both the expressions, then the shape of the intersecting orbit and the number of intersecting orbit changes i.e. after a certain value of RHS in both the equations, the number of intersecting orbit= 2. But, before that, we are getting only one/single intersecting orbit. Example: Intersecting orbit of $y^2/2 + z^2/2 - 16.7 z = -90$ and $-(x^2/2) + 16.7 z = 70$ is shown below.
I know these RHS values are only numbers or constant values. Due to these values, the nature of the surfaces remains the same and the shape and number of intersecting orbits change. So, It is confirmed that something is happening (I mean some kind of changes in the dynamics of both the surfaces are taking place). I have attached the code below that clearly shows how orbits are changing w.r.t RHS values.
Manipulate[
ContourPlot3D[{y^2/2 + z^2/2 - 16.7 z == -30 + k, -(x^2/2) +
16.7 z == 10 + m}, {x, -40, 40}, {y, -40, 40}, {z, -10, 40},
Mesh -> None,
ContourStyle ->
Directive[SkyBlue, Opacity[0.4], Specularity[White, 30]],
BoundaryStyle -> {1 -> None,
2 -> None, {1, _} -> Directive[Thick, Red]},
PlotTheme ->(*"Classic"*)"Scientific", PlotRange -> Full,
AxesLabel -> {Style[x, 15, Bold, Red], Style[y, 15, Bold, Red],
Style[z, 15, Bold(*Plain*), Red]}], {k, -100, 100}, {m, -100,
100}]
Though these surfaces are conserved surfaces, but something is changing due to RHS values. How can I put these changes in words? My question is, what exactly is happening here? What is the physical significance of these changes?
Since the question is asked on an explicit example, I am not sure on how much topological background I may assume. Hence, I try to keep the answer explicit and accessible — which implies that it is rather vague at times. I can elaborate further / point to literature, if this is wished.
We are given two (quadratic polynomial) functions $f,g:\mathbb R^3\to\mathbb R$ and two values $a,b\in\mathbb R$ such that $f^{-1}(a)$ and $g^{-1}(b)$ are (quadratic) surfaces. We would like to described how the number of components of $$ f^{-1}(a)\cap g^{-1}(b)=\left\{P\in\mathbb R^3~\middle|~f(P)-a=g(P)-b=0\right\}$$ depends on $a$ and $b$. By assuming that $b$ was already part of $g$, we only need to consider the number of components of $$ I_t:=f^{-1}(t)\cap g^{-1}(0)=\left\{P\in\mathbb R^3~\middle|~f(P)-t=g(P)=0\right\} $$ for $t\in\mathbb R$. For most values “regular” of $t$ nothing interesting happens with $I_t$. But there are some “critical” value where something happens. In your example, this is the case when the bottom of the cylinder barely touches the bottom of the paraboloid:
Here, $I_t$ is still connected but only at a single point. The number of components changes from 1 to 2. Another critical value would be where the cylinder first enters the paraboloid from below. Then the number of components changes from 0 to 1.
The branch of mathematics that deals with this type of behaviour (at least from the topological point of view), is Morse theory. To apply it, we only want to consider a single function, hence we define $$\begin{array}{rcl} \Phi:\mathbb R^3\times\mathbb R&\to&\mathbb R\\ (P,t)&\mapsto&(f(P)-t)^2+(g(P))^2 \end{array} $$ and notice that — since we are dealing with real functions — $I_t=\Phi^{-1}(0)\cap\mathbb R^3\times\{t\}$. Morse Theory now provides a condition on the differential of $\Phi$ that detects the critical values. Roughly, $t\in\mathbb R$ is critical if the rank of the derivative of $\Phi$ on $\mathbb R^3\times\{t\}$ is not always 3. A more careful study of this rank in critical values would also describe how the number of components changes but exhibiting this requires some knowledge of differential topology.