Table 3.

For the polytropic equations of state (2.14).

Hamiltonian
point symmetry Xdescriptionexponentcorrespondence
tt+x⌋∂xdilationγ arbitrarynil
Embedded Imagescalingγ≠1nil
Embedded Imagesimilarity scalingγ=1+2/nsimilarity energy
t2t+tx⌋∂x+(xtu)⌋∂untρ∂ρGalilean dilationγ=1+2/ndilational energy