Vertical extension of Noether theorem for scaling symmetries

Abstract

The aim of this paper is to present a new approach to construct constants of motion associated with scaling symmetries of dynamical systems. Scaling maps could be symmetries of the equations of motion but not of its associated Lagrangian action. We have constructed a Noether inspired theorem in a vertical extended space that can be used to obtain constants of motion for these symmetries. Noether theorem can be obtained as a particular case of our construction. To illustrate how the procedure works, we present two interesting examples, (a) the Schwarzian Mechanics based on Schwarzian derivative operator and (b) the Korteweg–de Vries nonlinear partial differential equation in the context of the asymptotic dynamics of General Relativity on $\hbox {AdS}_3$. We also study the inverse of Noether theorem for scaling symmetries and show how we can construct and identify the generator of the scaling transformation, and how it works for the vertical extended constant of motion that we are able to construct. We find an interesting contribution to the symmetry associated with the fact that the scaling symmetry is not a Noether symmetry of the action. Finally, we have contrasted our results with recent analysis and previous attempts to find constants of motion associated with these beautiful scaling laws.

Appendix

1.1 A1: Symmetries of the equations of motion

The infinitesimal symmetries of the equations of motion obey a very interesting relation known as the Jacobi equation, or the second variation of the Lagrangian action, if the system admits a variational formulation. But these symmetries are defined just as symmetries of the equation of motion, independently of whether the equations of motion admit a variational formulation or not. They are defined as infinitesimal transformations that map solutions of the dynamical system into solutions of the dynamical system. First, we can associate with every infinitesimal symmetry that also moves the time

$$\begin{aligned} \delta q^i={\bar{q}}^i({\bar{t}})-q^i(t),\qquad \delta t={\bar{t}}-t, \end{aligned}$$

an infinitesimal symmetry that does not move time, given by

$$\begin{aligned} \Delta q^i=\delta q^i-\dot{q}^i \delta t. \end{aligned}$$

Now, if we define $\Delta q^i=\eta ^i(q,\dot{q},t),$ it will be a symmetry of the equations of motion iff

$$\begin{aligned} \ddot{q}^i-F^i(q^j,\dot{q}^j,t)=0 \quad \Longleftrightarrow \quad \ddot{{\bar{q}}}^i-F^i({{\bar{q}}}^j,\dot{{\bar{q}}}^j,t)=0, \end{aligned}$$

which in turn implies [6]

$$\begin{aligned} \frac{{\bar{\mathrm{d}}}}{\mathrm{d}t}\frac{{\bar{\mathrm{d}}}}{\mathrm{d}t} \eta ^i-\frac{\partial F^i}{\partial \dot{q}^j}\frac{{\bar{\mathrm{d}}}}{\mathrm{d}t}\eta ^j- \frac{\partial F^i}{\partial q^j} \eta ^j=0. \end{aligned}$$

(106)

So, every symmetry of the equations of motion (including Noether symmetries) are solutions of this equation. In particular, for the scaling transformation $\eta ^i=aq^i-b\dot{q}^i t$, the Jacobi equation gives

$$\begin{aligned} \frac{{\bar{\mathrm{d}}}}{\mathrm{d}t}\frac{{\bar{\mathrm{d}}}}{\mathrm{d}t} (a q^i-b\dot{q}^i t)-\frac{\partial F^i}{\partial \dot{q}^j}\frac{{\bar{\mathrm{d}}}}{\mathrm{d}t}(a q^j-b\dot{q}^j t)- \frac{\partial F^i}{\partial q^j} (a q^j-b\dot{q}^j t)=0, \end{aligned}$$

(107)

which in turn implies the following condition for the forces of the dynamical system

$$\begin{aligned} -bt\frac{\partial F^i}{\partial t}+(a-2b)F^i-(a-b) \frac{\partial F^i}{\partial \dot{q}^j}\dot{q}^j-\frac{\partial F^i}{\partial q^j} a q^j=0. \end{aligned}$$

(108)

For all the examples that we have considered, these equations are fulfilled. We are unaware if these relations were considered previously in the literature. In the case of AFF conformal mechanics, $F\sim \frac{1}{q^3}$, and $a=1,b=2$, so the equation (108) reduces to

$$\begin{aligned} \frac{\partial F}{\partial q} q=-3 F, \end{aligned}$$

thus, F must be a homogeneous function of degree $-3$ as expected.

1.2 A2: The Jacobi equation in terms of the Lagrangian

The coefficients of the Jacobi equation in terms of the Lagrangian function L are

$$\begin{aligned} W_{ij}= & {} \frac{\partial ^2 L}{\partial \dot{q}^i\partial \dot{q}^j}, \end{aligned}$$

(109)

$$\begin{aligned} N_{ij}= & {} \frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\partial ^2 L}{\partial \dot{q}^i\partial \dot{q}^j}\right) +\frac{\partial ^2 L}{\partial \dot{q}^i\partial q^j}-\frac{\partial ^2 L}{\partial q^i\partial \dot{q}^j}, \end{aligned}$$

(110)

and

$$\begin{aligned} M_{ij}=\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\partial ^2 L}{\partial \dot{q}^i\partial q^j}\right) -\frac{\partial ^2 L}{\partial q^i\partial q^j}. \end{aligned}$$

(111)