Method of normal coordinates in the formulation of a system with dissipation: the harmonic oscillator

The Harmonic Approximation in Heavy-Ion Reaction Study

“…This model explains the mechanism for energy dissipation in a physical system, based on the coupling of intrinsic and collective degrees of freedom. The model can be extended to nuclear fission and heavy-ion reactions, where the collective degree of freedom is the relative coordinates of the two heavy-ions and the intrinsic degrees of freedom are the single-particle degrees of freedom [22].…”

“…these off-diagonal terms give rise to the coupling of the collective and intrinsic motions and the coupling of the intrinsic oscillators to each other. By a transformation to normal coordinates the quadratic forms of the kinetic and potential energies in Equation (9), can be reduced simultaneously to sums of squares in these coordinates and their derivatives and hence make the coupled-oscillator problem separable into independent motions, each with a particular normal frequencies [16]- [22].…”

“…where i r are the intrinsic normal coordinates defined in terms of the intrinsic coordinates given in Equation (22).…”

“…It should be noted that the derivation of Equation (34) demonstrates that the form of the collective amplitude, ( ) 1 2 3 4 5 6 1 2 3 4 5 6 , , , , , , , , , , , , n n n n n n t t t t t t f x y z satisfy the normalization condition for the total wave function. The collective amplitude is the expansion coefficient of the total wave function ( ) , , , , , , t t t t t t x y z ξ ξ ξ Φ  , and its form can be obtained by using Equations (12), (16), (17), (18) and (22), which then leads to the probability distribution ( ) 1 2 3 4 5 6 1 2 3 4 5 6 2 , , , , , 3 , , , , , 8π , , n n n n n n t t t t t t f x y z as functions of interesting physical parameters for example, energy, intrinsic and collective quantum numbers, etc.…”

Non‐Leibniz Hamiltonian and Lagrangian formalisms for certain class of dissipative systems

“…Mshelia evaluated the probability amplitude for the transfer of collective excitation energy in coupled dissipative systems. The normalization and orthogonalization of eigenvectors corresponding to degenerate eigenvalues were carefully considered.…”