Relativistic Equations of Motion within Nambu's Formalism of Dynamics

Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures

“…In Equation (9), the symbol M stands for a semi-positive definite n × n matrix, the so-called mobility matrix [38]. Semi-positive definite means that for any vector Z = (Z 1 , .…”

“…The diffusion term, i.e., the third expression on the right-hand side of the equal sign in Equation (11), is nonlinear with respect to P. This type of nonlinear diffusion coefficient has been introduced by Plastino and Plastino in the context of Fokker-Planck equations associated with the non-extensive Tsallis entropy [45] and is a benchmark nonlinearity of nonlinear diffusion equations in material physics [46][47][48]. Due to the fact that in general, the model (9) is nonlinear with respect to P (e.g., see Equation (11)) and in view of the fact that the structure of the model (9) is similar to a Fokker-Planck equation, it has been suggested to refer to Equation (9) as the nonlinear Fokker-Planck equation. That is, from a mathematical point of view, the proposed model (9) can be regarded as a nonlinear partial differential equation or a nonlinear Fokker-Planck equation.…”

“…A variety of oscillatory systems [8][9][10][11][12][13][14] have been studied within the framework of Nambu mechanics. Particles moving on curved surfaces correspond under appropriate conditions to system that live in n-dimensional spaces and satisfy n − 1 invariants, such that Nambu mechanics applies [15][16][17][18][19].…”

Mapping between the dynamic and mechanical properties of the relativistic oscillator and Euler free rigid body

“…In this section we will consider a simplified Nambu system based on a three dimensional phase space coordinatised by the triplet {x, p, q} within the following physical interpretation [16,17,18]: we take the physical system as formed by a real (corporeal) particle of mass m which is localised in a given point by the coordinates (x, P ) of two dimensional phase space, hence the position of the real particle is given by the coordinate x while its momentum is given by the coordinate P . This system will be considered as a composed system, formed by two subparticles with momenta p and q, and masses m p and m q , respectively; this situation is directly generalised to three dimensional triplet configuration space ( x, p, q).…”

“…However, first attempts to understand the kind of mechanics involved within such formalism were not fruitful leaving the question to the exploration of some similarities and interrelations between Hamiltonian formalism and Newton mechanics. It is only by considering Yamaleev's construction that we obtain a complete realisation of Nambu formalism into real mechanics [16,17,18].…”