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Invariant relationships deriving from classical scaling transformations. (English) Zbl 1316.85007

Summary: Because scaling symmetries of the Euler-Lagrange equations are generally not variational symmetries of the action, they do not lead to conservation laws. Instead, an extension of Noether’s theorem reduces the equations of motion to evolutionary laws that prove useful, even if the transformations are not symmetries of the equations of motion. In the case of scaling, symmetry leads to a scaling evolutionary law, a first-order equation in terms of scale invariants, linearly relating kinematic and dynamic degrees of freedom. This scaling evolutionary law appears in dynamical and in static systems. Applied to dynamical central-force systems, the scaling evolutionary equation leads to generalized virial laws, which linearly connect the kinetic and potential energies. Applied to barotropic hydrostatic spheres, the scaling evolutionary equation linearly connects the gravitational and internal energy densities. This implies well-known properties of polytropes, describing degenerate stars and chemically homogeneous nondegenerate stellar cores.{
©2011 American Institute of Physics}

MSC:

85A15 Galactic and stellar structure
76E20 Stability and instability of geophysical and astrophysical flows
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References:

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[13] This evolutionary equation is important elsewhere in mathematical physics. In population dynamics, with log r replaced by time t, it becomes the Lotka-Volterra equation for predator/prey evolution. The \(uv_n\) cross-terms lead to growth of the predator \(v_n\) at the expense of the prey u, so that a population that is exclusively prey initially \((v_n = 0\), u = 3) is ultimately devoured u → 0. For the weakest predator/prey interaction (n = 5), the predator takes an infinite time to reach only the finite value \(v_n = 1\). For stronger predator/prey interaction (n < 5), the predator grows infinitely \(v_n\) → ∞ in a finite time. See W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 2001), and D. W. Jordon and P. Smith, Nonlinear Ordinary Differential Equations, 3rd ed. (Oxford University Press, New York, 1999).
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