# Dynamics of a Circular Foil and Two Pairs of Point Vortices: New Relative Equilibria and a Generalization of Helmholtz Leapfrogging

^{1}

^{2}

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*Symmetry*)

## Abstract

**:**

## 1. Introduction

## 2. Equations of Motion

**Main assumptions.**Consider the plane-parallel motion of a cylindrical body (cylinder) in an infinite volume of an ideal fluid with two vortex pairs. Each vortex pair consists of two point vortices with strengths equal in magnitude and opposite in sign. Let the following conditions be satisfied:

- The foil of the body is a circle of radius a with its center of mass at the geometric center of the circle C;
- The circulation of the velocity of the fluid around the foil is zero.

- A fixed (inertial) coordinate system $OXY$;
- A moving coordinate system $Cxy$ attached to the foil, with its origin at the center of the circle.

**Reduced system.**The equations of motion do not explicitly depend on the choice of the origin and the direction of the axes of the fixed coordinate system $OXY$, so that the vector field of the system possesses symmetry fields. An explicit form of these symmetry fields is presented in the paper [1], which also gives a detailed description of the reduction procedure for the problem of the motion of n point vortices and a circular foil in an ideal fluid.

**Reconstruction of dynamics.**In order to define the motion of the foil in the fixed coordinate system from the known solution of the reduced system in (1), it is necessary to use the Noether integrals [1]:

**Discrete symmetries and invariant submanifold.**In the analysis of the dynamics of the system in (1), an important role is played not only by the above-mentioned conservation laws (the first integrals and the Poisson tensor), but also discrete symmetries. Let us consider them in more detail.

**Remark**

**1.**

**Restriction of the system to the invariant submanifold.**Let us parameterize manifold $\mathcal{N}$ using the dimensionless variables $\mathit{z}=({x}_{1},{y}_{1},{x}_{2},{y}_{2})$ and f as follows:

- One of the vortices lies inside the foil$${D}_{i}=\left\{\mathit{z}\phantom{\rule{4pt}{0ex}}\right|\phantom{\rule{4pt}{0ex}}{x}_{i}^{2}+{y}_{i}^{2}<1\},\phantom{\rule{4pt}{0ex}}i=1,2;$$
- Vortices in one of the pairs collide$${P}_{i}=\left\{\mathit{z}\phantom{\rule{4pt}{0ex}}\right|\phantom{\rule{4pt}{0ex}}{y}_{i}=0\},\phantom{\rule{4pt}{0ex}}i=1,2;$$
- Vortices from different pairs collide$$W=\left\{\mathit{z}\phantom{\rule{4pt}{0ex}}\right|\phantom{\rule{4pt}{0ex}}{x}_{1}={x}_{2},{y}_{1}={y}_{2}\}.$$

## 3. Relative Equilibria and Their Stability

- $\mathbf{A}=\parallel \frac{\partial {\dot{z}}_{i}}{\partial {z}_{j}}{\parallel}_{\mathit{z}={\mathit{z}}_{0}}$ is the linearization matrix of the vector field in a neighborhood of the fixed point;
- $\mathbf{H}=\parallel \frac{{\partial}^{2}H}{\partial {z}_{i}\partial {z}_{j}}{\parallel}_{\mathit{z}={\mathit{z}}_{0}}$ is the matrix of the quadratic part of the Hamiltonian (Hessian).

- (1)
- The index of the symmetric matrix $\mathbf{H}$ (i.e., the number of its negative eigenvalues) is denoted by $ind$ in what follows. It can take the values $ind=0,1,2,3,4$.
- (2)
- The type of the fixed point defined by the eigenvalues ${\lambda}_{i}$, $i=1,\cdots ,4$ of the matrix $\mathbf{A}$ is as follows:
- –
- Center–center (${\lambda}_{1,2}=\pm i\alpha $, ${\lambda}_{3,4}=\pm i\beta $);
- –
- Saddle–center (${\lambda}_{1,2}=\pm i\alpha $, ${\lambda}_{3,4}=\pm \beta $);
- –
- Saddle–saddle (${\lambda}_{1,2}=\pm \alpha $, ${\lambda}_{3,4}=\pm \beta $);
- –
- Focus–focus (${\lambda}_{1,2,3,4}=\pm \alpha \pm i\beta $),

where $\alpha $ and $\beta $ are real numbers. Only a fixed point that is of center–center type is Lyapunov stable.

- (1)
- If $ind=0,4$, then the fixed point is stable and of center–center type;
- (2)
- If $ind=1,3$, then the fixed point is unstable and of saddle–center type;
- (3)
- If $ind=2$, then the fixed point is of one of three types: center–center, saddle–saddle, or focus–focus. For it to be unstable, it suffices that the matrix $\mathbf{A}$ has eigenvalues with a nonzero real part. To prove the stability of such a point, it is necessary to use the KAM theorem and it requires that all eigenvalues of the matrix $\mathbf{A}$ are imaginary numbers and additional nonlinear conditions hold [16].

- 1.
- We plot the values of $f,h$ corresponding to critical points of $H\left(\mathit{z}\right)$; they form bifurcation curves, and when such a curve is crossed, the topological type of the isoenergetic manifolds changes [13].
- 2.
- We put the index of the corresponding critical point on the bifurcation curves, in [13] it was shown that the index does not change along a smooth branch of the bifurcation curve.
- 3.
- For each branch of the bifurcation curves with the index $ind=2$, we first plot the curves $({a}_{\lambda}\left(f\right)$ and ${b}_{\lambda}\left(f\right))$ on the plane of coefficients of the characteristic polynomial$$\chi \left(\lambda \right)=det(\mathbf{A}-\lambda \mathbf{E})={\lambda}^{4}+{a}_{\lambda}{\lambda}^{2}+{b}_{\lambda}=0$$

**Relative equilibria.**We note that the fixed points of the reduced systems in (8) and (9) are given by solutions of four polynomial equations of sufficiently high order. This makes an explicit solution impossible, so we solve this system numerically.

- On each bifurcation curve ${\mathrm{\Sigma}}_{i}$, $i=0,1,2$, a critical point of the cusp type is visible. The values of f and h for each of these cusps correspond to the birth of two isolated relative equilibria. Then, as f increases, each of these relative equilibria persists and forms a branch (smooth part) of a bifurcation curve with the same index. Only the curve ${\mathrm{\Sigma}}_{2}$ has a branch with $ind=4$ and, thus, corresponds to stable relative equilibria. Both branches of the bifurcation curves with $ind=2$ correspond to fixed points of the saddle–saddle type (see, e.g., Figure 7).
- The branches of the curves ${\mathrm{\Sigma}}_{0}$ and ${\mathrm{\Sigma}}_{1}$ with $ind=1$ and $ind=2$, respectively, correspond to relative equilibria where vortices become more distant from the foil as f increases. By contrast, for the other relative equilibria, the distance between vortices and the foil decreases as f increases.

**Remark**

**2.**

## 4. Bounded Trajectories: Generalization of the Helmholtz Leapfrogging

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Coordinates and parameters describing a cylindrical foil and two pairs of point vortices in an ideal fluid.

**Figure 2.**A typical curve $\Delta ({y}_{1},{y}_{2})=0$ and dependences ${y}_{1}\left(f\right)$ and ${y}_{2}\left(f\right)$ for collinear configurations with $\gamma =-0.4$, $\mu =0.6$. The red and blue lines denote parts of the curve ${\mathrm{\Sigma}}_{0}$ with the indices $ind=1$ and $ind=2$, respectively.

**Figure 4.**Typical dependences ${r}_{1}\left(f\right)$, ${r}_{2}\left(f\right)$ and ${x}_{1}\left(f\right)-{x}_{2}\left(f\right)$ for relative equilibria corresponding to the bifurcation curve ${\mathrm{\Sigma}}_{1}$ with $\gamma =-0.4$, $\mu =0.6$. The red and blue lines denote parts of the curve ${\mathrm{\Sigma}}_{1}$ with the indices $ind=2$ and $ind=3$, respectively.

**Figure 5.**Typical dependences ${r}_{1}\left(f\right)$, ${r}_{2}\left(f\right)$ and ${x}_{1}\left(f\right)-{x}_{2}\left(f\right)$ for the relative equilibria corresponding to the bifurcation curve ${\mathrm{\Sigma}}_{2}$ with $\gamma =-0.4$, $\mu =0.6$. The red and blue lines denote parts of the curve ${\mathrm{\Sigma}}_{2}$ with the indices $ind=3$ and $ind=4$, respectively.

**Figure 6.**Bifurcation diagram (

**a**) and the dependence $\Delta h\left(f\right)$, which is the difference between the values of h on the upper and lower branches of the bifurcation curve ${\mathrm{\Sigma}}_{2}$ (

**b**).

**Figure 7.**Dependence of the coefficients of the characteristic polynomial for a portion of the bifurcation curve ${\mathrm{\Sigma}}_{1}$ (black) with the index $ind=2$.

**Figure 8.**Poincaré map (

**a**) and the corresponding cross-section ${\mathcal{M}}_{{\phi}_{2}}^{2}$ (

**b**) for ${\phi}_{2}=4.7624$ and the parameters $h=1.51$, $f=4.6$, $\gamma =-0.4$, $\mu =0.6$.

**Figure 9.**Poincaré map (

**a**) and the corresponding cross-section ${\mathcal{M}}_{{\phi}_{2}}^{2}$ (

**b**) for ${\phi}_{2}=4.65$ and the parameters $h=1.5101$, $f=4.6$, $\gamma =-0.4$, $\mu =0.6$.

**Figure 10.**Trajectories of vortices with initial conditions ${r}_{1}\left(0\right)=1.25$, ${r}_{2}\left(0\right)=1.06$, ${\phi}_{2}\left(0\right)=4.7624$; parameters $f=4.6$, $\gamma =-0.4$, $\mu =0.6$; and different ${\phi}_{1}\left(0\right)$.

**Figure 11.**Dependences ${x}_{1}\left(\tau \right)$ and ${x}_{2}\left(\tau \right)$ (red and blue, respectively) (

**a**), vortex trajectories in the coordinate system $Cxy$ (

**b**), and the dependence $\frac{\pi}{a}\frac{dX}{d\tau}\left(\tau \right)$ (

**c**), which characterizes the translational velocity of the foil in the fixed coordinate system. All graphs have been plotted for the fixed initial conditions ${r}_{1}\left(0\right)=1.47$, ${r}_{2}\left(0\right)=1.85$, ${\phi}_{2}\left(0\right)=4.65$, ${\phi}_{1}\left(0\right)=4.8649$ and the parameters $f=4.6$, $\gamma =-0.4$, $\mu =0.6$.

**Table 1.**An example of numerical values of the fixed points of the system in (8) for $\gamma =-0.4$, $\mu =0.6$, and $f=5$.

${\mathit{x}}_{1}$ | ${\mathit{y}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{y}}_{2}$ | |
---|---|---|---|---|

${\mathrm{\Sigma}}_{1}$ | −3.938836165 | 2.661361030 | −3.940909660 | 0.3738183364 |

−1.151164395 | 0.2241720364 | −1.159436579 | 0.03680779387 | |

${\mathrm{\Sigma}}_{2}$ | −1.214662769 | 0.1785458502 | −1.046342147 | 0.04920780747 |

−1.302316693 | 0.1573333876 | −1.064157855 | 0.09971334570 |

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**MDPI and ACS Style**

Bizyaev, I.A.; Mamaev, I.S.
Dynamics of a Circular Foil and Two Pairs of Point Vortices: New Relative Equilibria and a Generalization of Helmholtz Leapfrogging. *Symmetry* **2023**, *15*, 698.
https://doi.org/10.3390/sym15030698

**AMA Style**

Bizyaev IA, Mamaev IS.
Dynamics of a Circular Foil and Two Pairs of Point Vortices: New Relative Equilibria and a Generalization of Helmholtz Leapfrogging. *Symmetry*. 2023; 15(3):698.
https://doi.org/10.3390/sym15030698

**Chicago/Turabian Style**

Bizyaev, Ivan A., and Ivan S. Mamaev.
2023. "Dynamics of a Circular Foil and Two Pairs of Point Vortices: New Relative Equilibria and a Generalization of Helmholtz Leapfrogging" *Symmetry* 15, no. 3: 698.
https://doi.org/10.3390/sym15030698