In this paper we proposed a proposition: for any nonconservative classical mechanical system and any initial condition, there exists a conservative one; the two systems share one and only one common phase curve; the Hamiltonian of the conservative system is the sum of the total energy of the nonconservative system on the aforementioned phase curve and a constant depending on the initial condition. Hence, this approach entails substituting an infinite number of conservative systems for a dissipative mechanical system corresponding to varied initial conditions. One key way we use to demonstrate these viewpoints is that by the Newton-Laplace principle the nonconservative force can be reasonably assumed to be equal to a function of a component of generalized coordinates $q_i$ along a phase curve, such that a nonconservative mechanical system can be reformulated as countless conservative systems. Utilizing the proposition, one can apply the method of Hamiltonian mechanics or Lagrangian mechanics to dissipative mechanical system. The advantage of this approach is that there is no need to change the definition of canonical momentum and the motion is identical to that of the original system.
We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively determine if a given time dispersive system can be extended to a conservative one; (ii) to construct that very conservative system -- which we show is essentially unique. We illustrate the method by applying it to the spectral analysis of time dispersive dielectrics and the damped oscillator with retarded friction. In particular, we obtain a conservative extensio…
Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems have opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturb…
We study the long-time behavior of conservative interacting particle systems in $Z$: the activated random walk model for reaction-diffusion systems and the stochastic sandpile. We prove that both systems undergo an absorbing-state phase transition.
The dynamics of some non-conservative and dissipative systems can be derived by calculating the first variation of an action-dependent action, according to the variational principle of Herglotz. This is directly analogous to the variational principle of Hamilton commonly used to derive the dynamics of conservative systems. In a similar fashion, just as the second variation of a conservative system's action can be used to infer whether that system's possible trajectories are dynamically stable, so too can the second variation of the action-dependent action be used to infer whether the possible …
We formulate the dynamics of an infinitely extended open dissipative quantum system, ${Σ]$,in the Schroedinger picture.The generic model on which this is based comprises a C*-algebra,$[\cal A}$,of observables, a folium, ${\cal F}$, of states on this algebra and a one parameter semigroup,$τ$, of linear transformations of ${\cal F}$ that represents its dynamics and is given by a natural infinite volume limit of the corresponding semigroup for a finite system. On this basis, we establish that the dynamic of $Σ$ is given by a one parameter semigroup of completely positive transformations of the W*…
Hamilton--Jacobi theory for non-conservative field theories in the $k$-contact framework
Centre de Recherches Mathématiques, Université de Montréal（蒙特利尔大学）；University of Warsaw（华沙大学）；Universitat Rovira i Virgili（罗维拉-伊维尔吉利大学）；Universidad Politécnica de Madrid（马德里理工大学）
General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that mixes with the usual velocity viscosity. While the Poisson bracket of the ideal version of this model have already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view for deriving dissipative brackets, a means of derivation that is natural for understanding…
