Volume 110A, number 5 PHYSICS LETTERS 29 luty 1985 SCALE-INVARIANT LYAPUNOV EXPONENTS FOR CLASSICAL HAMILTONIAN SYSTEMS T H SELIGMAN 1 j j M VERBAARSCHOT and M R ZIRNBAUER Ma,;- Plato k- Instltut fur Kernphyszk and lnstUut fur Theorettsche Phvstk der Unwersltat HeMelberg HeMelberg, Fed Rep Germany Received 18 January 1985, accepted for publication 16 May 1985 The Lyapunov exponent for classical hamdtonlan systems is made dimensionless b) introducing a characteristic time K This modification yields an energy-independent exponent for systems with scale m~arlance At present, much theoretical activity focuses on the connection between classical non-hnear dynamics, and the spectral statistics of the underlying quantum hamfltoman For systems exlubltlng a transition from classical regular to classical chaotic motion, there have been recent attempts [1-5] to characterize the quan- tum level statistics m terms of a classical "order pa- rameter" It was suggested that the change m level statistics IS governed by the fraction of classical phase space covered by chaotic trajectories On the other hand, we know that the degree of m- stablhty of a classical system is described by its Lyapunov exponents [6] (For a hamfltonmn system with two degrees of freedom, there exists only one independent positive exponent for each connected chaotic region and the average over phase space equals the Kolmogorov entropy ) It Is therefore natural to ask whether the degree of mstabthty of the classical trajectories, as measured by the Lyapunov exponent, has any influence on the quantum level statistics In the present short note we wish to point out that the usual definition of the Lyapunov exponent IS unsatis- factory for the purpose of such mqmry We also pro- pose an amended defimtlon In standard textbooks [6] the Lyapunov exponent is defined as 1 Permanent address Instituto de Flsica, University of Mexico m Cuernavaca (UNAM), Apdo post 20364, 01000 Mexico DF, Mexico 0 375-9601/85/$ 03 30 © Elsevier Science Publishers B V (North-Holland Physms Pubhshlng Dwlsxon) A = hm t -1 log[d(t)/d(O)] , (la) t-~oo d(O)-~ 0 where d(t) measures the divergence of two mltlaUy close trajectories For a system with two degrees of freedom tins definition yields the positive exponent of a pair of exponents satisfying A 1 = - A 2 For sys- tems with more than two degrees of freedom, the def- 1ration (la) yields the largest exponent (Other expo- nents can be obtained using the procedure of Benettm et al [7] ) Numerically, d(t) is calculated from d(t) = ([Ax(t)] 2 + [Ap(t)] 2}1/2, (lb) although the precise mathematical defimtlon of d(t) makes use of an appropriate tangent space From eqs (1) we observe that (1) A is a dnnenslonal quantity (It has the reverse dimension of time ) (u) The euclidean phase space metric used m eq (lb) mixes quantities with different dimension We will argue m the following that features (1) and (u) are undestrable and should be removed Consider the class of hamfltomans wlth a potential homogeneous of degree 2m m the coordinates x,, N H = 1_~ p2 + v(2m)(xl,x2, ,XN) (2) 2Mt=l For these hamdtonlans the classical motion is mechan- Ically slmdar [8] at all energies, due to the existence of scale transformations 231