# Noether Symmetries Quantization and Superintegrability of Biological Models

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## Abstract

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## 1. Introduction

“It is frequently claimed that—like Newton’s invention of calculus—biological theory will require ‘new mathematics’.... There are, however, many areas of mathematics that have been neglected by theoretical biology that could prove to be of great value. Einstein’s work on general relativity, for instance, made good use of mathematical ideas, in particular differential geometry that had previously been developed with completely different motivation. More likely than not, the formal structures have been set forth in some context, and await their discovery and subsequent development in representing biological theory.”

## 2. Quantizing with Noether Symmetries

**Step****I.**- Find the Lie symmetries of the Lagrange equations$$\mathrm{{\rm Y}}=W(t,\mathbf{x}){\partial}_{t}+\sum _{k=1}^{N}{W}_{k}(t,\mathbf{x}){\partial}_{{x}_{k}}.$$
**Step****II.**- Among them, find the Noether symmetries$$\mathrm{\Gamma}=V(t,\mathbf{x}){\partial}_{t}+\sum _{k=1}^{N}{V}_{k}(t,\mathbf{x}){\partial}_{{x}_{k}}.$$
**Step****III.**- Construct the Schrödinger equation, where we assume $\u0127=1$ without loss of generality, admitting these Noether symmetries as Lie symmetries, namely$$2i{\psi}_{t}+\sum _{k,j=1}^{N}{f}_{kj}\left(\mathbf{x}\right){\psi}_{{x}_{j}{x}_{k}}+\sum _{k=1}^{N}{h}_{k}\left(\mathbf{x}\right){\psi}_{{x}_{k}}+{f}_{0}\left(\mathbf{x}\right)\psi =0$$$$\mathsf{\Omega}=V(t,\mathbf{x}){\partial}_{t}+\sum _{k=1}^{N}{V}_{k}(t,\mathbf{x}){\partial}_{{x}_{k}}+G(t,\mathbf{x},\psi ){\partial}_{\psi},$$$$\psi {\partial}_{\psi},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\alpha (t,\mathbf{x}){\partial}_{\psi},$$

**Step****I.**- We have found the three Lie symmetries, i.e., (9).
**Step****II.**- Those symmetries are also the Noether symmetries of the Lagrangian (11).
**Step****III.**- We consider the general equation$$2i{\mathrm{\Psi}}_{t}+{f}_{11}\left(u\right){\mathrm{\Psi}}_{uu}+{h}_{1}\left(u\right){\mathrm{\Psi}}_{u}+{f}_{0}\left(u\right)\mathrm{\Psi}=0$$$${\mathsf{\Omega}}_{1}={\mathrm{\Gamma}}_{1}+{G}_{1}(t,u,\mathrm{\Psi}){\partial}_{\mathrm{\Psi}},\phantom{\rule{0.277778em}{0ex}}{\mathsf{\Omega}}_{2}={\mathrm{\Gamma}}_{2}+{G}_{2}(t,u,\mathrm{\Psi}){\partial}_{\mathrm{\Psi}},\phantom{\rule{0.277778em}{0ex}}{\mathsf{\Omega}}_{3}={\mathrm{\Gamma}}_{3}+{G}_{3}(t,u,\mathrm{\Psi}){\partial}_{\mathrm{\Psi}},$$$$\mathrm{\Psi}{\partial}_{\mathrm{\Psi}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\alpha (t,u){\partial}_{\mathrm{\Psi}},$$

## 3. In the Wake of Volterra: A Superintegrable System

“I have been able to show that the equations of the struggle for existence depend on a question of Calculus of Variations (omissis). In order to obtain this result, I have replaced the notion of population by that of quantity of life [14]. In this manner I have also obtained some results by which dynamics is brought into relation to problems of the struggle for existence.”

## 4. Discussion and Final Remarks

“Only rarely does one find mention, at post-graduate level, of any problem in connection with the process of actually solving such equations. The electronic computer may perhaps be partly to blame for this, since the impression prevails in many quarters that almost any differential equation problem can be merely put on the machine, so that finding an analytical solution is largely a waste of time. This, however, is only a small part of the truth, for at higher levels there are generally so many parameters or boundary conditions involved that numerical solutions, even if practicable, give no real idea of the properties of the equation. Moreover, any analyst of sensibility will feel that to fall back on numerical techniques savours somewhat of breaking a door with a hammer when one could, with a little trouble, find the key”.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Nucci, M.C.; Sanchini, G.
Noether Symmetries Quantization and Superintegrability of Biological Models. *Symmetry* **2016**, *8*, 155.
https://doi.org/10.3390/sym8120155

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Nucci MC, Sanchini G.
Noether Symmetries Quantization and Superintegrability of Biological Models. *Symmetry*. 2016; 8(12):155.
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**Chicago/Turabian Style**

Nucci, Maria Clara, and Giampaolo Sanchini.
2016. "Noether Symmetries Quantization and Superintegrability of Biological Models" *Symmetry* 8, no. 12: 155.
https://doi.org/10.3390/sym8120155