# On the Quadratization of the Integrals for the Many-Body Problem

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## Abstract

**:**

## 1. Introduction

**Problem**

**1.**

- 1.
- It possesses a sufficient number of algebraic integrals of motion in order to express additional variables in terms of the coordinates of the bodies;
- 2.
- With some choice of constant values in these integrals, its solutions coincide with those of the original system;
- 3.
- It has integrals of motion, which, if one takes into account the relationship between additional variables and the coordinates of the bodies, are transformed into 10 classical integrals of the many-body problem;
- 4.
- All integrals of motion of the system are quadratic in the coordinates and velocities of bodies, as well as in additional variables.

## 2. Rationalization of the n-Body Problem

- The momentum conservation (three scalar integrals):$$\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}=\mathrm{const},$$
- The angular momentum conservation (three scalar integrals):$$\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}\times {\overrightarrow{r}}_{i}=\mathrm{const},$$
- The center-of-mass inertial motion (three scalar integrals):$$\sum _{i=1}^{n}{m}_{i}{\overrightarrow{r}}_{i}-t\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}=\mathrm{const},$$
- The energy conservation (one scalar integral):$$\sum _{i=1}^{n}\frac{{m}_{i}}{2}{\left|{\overrightarrow{v}}_{i}\right|}^{2}-\gamma \sum _{i,j}\frac{{m}_{i}{m}_{j}}{{r}_{ij}}=\mathrm{const}.$$

**Theorem**

**1.**

**Proof.**

- The momentum conservation:$$\frac{d}{dt}\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}=\sum _{i=1}^{n}{m}_{i}{\dot{\overrightarrow{v}}}_{i}=\sum _{i=1}^{n}\sum _{j=1}^{n}\gamma \frac{{m}_{i}{m}_{j}}{{r}_{ij}^{3}}\left(\right)open="("\; close=")">{\overrightarrow{r}}_{j}-{\overrightarrow{r}}_{i}$$
- The angular momentum conservation:$$\frac{d}{dt}\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}\times {\overrightarrow{r}}_{i}=\sum _{i=1}^{n}{m}_{i}{\dot{\overrightarrow{v}}}_{i}\times {\overrightarrow{r}}_{i}=\sum _{i=1}^{n}\sum _{j=1}^{n}\gamma \frac{{m}_{i}{m}_{j}}{{r}_{ij}^{3}}\left(\right)open="("\; close=")">{\overrightarrow{r}}_{j}-{\overrightarrow{r}}_{i}$$
- The center-of-mass inertial motion:$$\frac{d}{dt}\sum _{i=1}^{n}{m}_{i}{\overrightarrow{r}}_{i}-t\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}=\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}-\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}=0.$$
- The energy conservation:$$\begin{array}{cc}\hfill \frac{d}{dt}\sum _{i=1}^{n}\frac{{m}_{i}}{2}({u}_{i}^{2}+{v}_{i}^{2}+{w}_{i}^{2})& =\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}\xb7{\dot{\overrightarrow{v}}}_{i}=\sum _{i=1}^{n}\sum _{j=1}^{n}\gamma \frac{{m}_{i}{m}_{j}}{{r}_{ij}^{3}}{\overrightarrow{v}}_{i}\xb7\left(\right)open="("\; close=")">{\overrightarrow{r}}_{j}-{\overrightarrow{r}}_{i}\hfill \end{array}$$$$\begin{array}{c}\hfill \frac{d}{dt}\sum _{i,j}\frac{{m}_{i}{m}_{j}}{{r}_{ij}}=-\sum _{i,j}\frac{{m}_{i}{m}_{j}}{{r}_{ij}^{2}}{\dot{r}}_{ij}=-\sum _{i,j}\frac{{m}_{i}{m}_{j}}{{r}_{ij}^{3}}({\overrightarrow{r}}_{i}-{\overrightarrow{r}}_{j})\xb7({\overrightarrow{v}}_{i}-{\overrightarrow{v}}_{j}).\end{array}$$
- The additional conservation laws:$${r}_{ij}^{2}-{({x}_{i}-{x}_{j})}^{2}-{({y}_{i}-{y}_{j})}^{2}-{({z}_{i}-{z}_{j})}^{2}=\mathrm{const}$$$$\frac{d}{dt}\left(\right)open="("\; close=")">{r}_{ij}^{2}-{({x}_{i}-{x}_{j})}^{2}-{({y}_{i}-{y}_{j})}^{2}-{({z}_{i}-{z}_{j})}^{2}$$$$2{r}_{ij}{\dot{r}}_{ij}-2({\overrightarrow{r}}_{i}-{\overrightarrow{r}}_{j})({\overrightarrow{v}}_{i}-\overrightarrow{j})=0.$$

## 3. System with Quadratic Polynomial Integrals

**Theorem**

**2.**

- The momentum conservation:$$\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}=const,$$
- The angular momentum conservation:$$\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}\times {\overrightarrow{r}}_{i}=const,$$
- The center-of-mass inertial motion:$$\sum _{i=1}^{n}{m}_{i}{\overrightarrow{r}}_{i}-t\sum _{i=1}^{n}{m}_{i}{\overrightarrow{v}}_{i}=const,$$
- The energy conservation in the form:$$\sum _{i=1}^{n}\frac{{m}_{i}}{2}({u}_{i}^{2}+{v}_{i}^{2}+{w}_{i}^{2})-\gamma \sum _{i,j}{m}_{i}{m}_{j}{\rho}_{ij}=const,$$

**Proof.**

- It has the quadratic integrals of motion:$${r}_{ij}^{2}-{({x}_{i}-{x}_{j})}^{2}-{({y}_{i}-{y}_{j})}^{2}-{({z}_{i}-{z}_{j})}^{2}=\mathrm{const}$$$${r}_{ij}{\rho}_{ij}=\mathrm{const},$$
- If the constants in these integrals are chosen in such a way that:$${r}_{ij}^{2}-{({x}_{i}-{x}_{j})}^{2}-{({y}_{i}-{y}_{j})}^{2}-{({z}_{i}-{z}_{j})}^{2}=0$$$${r}_{ij}{\rho}_{ij}=1,$$
- The new system has quadratic integrals of motion, which, taking into account the relationship between additional variables and the coordinates of the bodies, turn into 10 classical integrals of the many-body problem.

## 4. Conservative Schemes for N Body Problem

## 5. Choreographic Test

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Choreographic test, step $dt=0.01$. Dependence of energy H on time for the approximate solutions found using the midpoint scheme (blue) and the rk4 scheme (red).

**Figure 2.**Choreographic test, step $dt=0.01$, 10 iterations. Dependence of energy increment $\Delta H$ on time for approximate solution found by midpoint scheme with (

**above**) and without (

**below**) auxiliary variables.

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**MDPI and ACS Style**

Ying, Y.; Baddour, A.; Gerdt, V.P.; Malykh, M.; Sevastianov, L.
On the Quadratization of the Integrals for the Many-Body Problem. *Mathematics* **2021**, *9*, 3208.
https://doi.org/10.3390/math9243208

**AMA Style**

Ying Y, Baddour A, Gerdt VP, Malykh M, Sevastianov L.
On the Quadratization of the Integrals for the Many-Body Problem. *Mathematics*. 2021; 9(24):3208.
https://doi.org/10.3390/math9243208

**Chicago/Turabian Style**

Ying, Yu, Ali Baddour, Vladimir P. Gerdt, Mikhail Malykh, and Leonid Sevastianov.
2021. "On the Quadratization of the Integrals for the Many-Body Problem" *Mathematics* 9, no. 24: 3208.
https://doi.org/10.3390/math9243208