It seems Hamiltonian systems handle conservative systems because of invariant Hamiltonian and Lagrangian mechanics does so for it is equivalent to Hamiltonian mechanics. Is there anything a like incorporating dissipative forces such as friction or damping?
Here it is a recent PRL by a colleague of mine:
Phys. Rev. Lett. 110, 174301 (2013) [5 pages]
Classical Mechanics of Nonconservative Systems
http://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:1210.2745
http://prl.aps.org/abstract/PRL/v110/i17/e174301
Thank you. Does non-Hamiltonian system coincide with non-conservative system?
Dear Shu-Ch'ang Chang.
As usual, we are study the dissipative processes on the basis of Lagrangian and Hamiltonian mechanics in an empirical manner, adding dissipative forces to the Lagrange equation. It is often give a good results, although the strictness of the description is broken, because the Lagrange equation obtained under the condition of holonomic constraints. For non-equilibrium systems with dissipation the condition of the connections holonomicity is not correct. Without going into details, I will say that these restrictions of the mechanics due to the fact that Newton's laws are valid only for the material points, while the real body are the system. I make attempt to build the mechanics by taking into account the structuring of the interacting bodies by replacing the material point on the system. This allows in the framework of Newton's laws to find the dissipative forces, which are determined by a nonlinear transformation of the energy of motion of the system in its internal energy. More detail your can get acquainted, for example, here you can get acquainted, for example, here: SOMSIKOV V. M. NON-EQUILIBRIUM SYSTEMS AND MECHANICS OF THE STRUCTURED PARTICLES. Elsever. Chaos and Complex system. 2013, XV, 581 p. 31-40
@Shuchang Zhang: You asked:
Does non-Hamiltonian system coincide with non-conservative system?
One should be careful with the definition one uses but e.g. a Hamiltonian system with explicitly time-dependent Hamiltonian will, by construction, be Hamiltonian but there will be no energy conservation.
Basing on the Newton's motion equation for MP, the Lagrange and Hamilton formalisms for the dynamics of the systems from MP were created. But these formalisms convenient only to reversible processes in the systems near to the equilibrium and can’t describe the dissipative processes of equilibration. It is one of the great difficult of physics.
To come to expansion of the classical mechanics, necessary to take into account that all natural bodies are systems and have the internal energy. Thanks to their internal energy and possibilities of its changes in the interactions, the irreversibility is exists. It is because the systems’ dynamics determined both as the symmetry of a system and the symmetry of space.
Hamiltonian (and/or Lagrangian) systems are based on the least action principle, which states that the path integral of the Lagrangian is stationary. In short, it leads to the equality between the time-derivative of the 6-dimensional state vector (p,q) and the symplectic gradient of the Hamiltonian. Dissipative forces usually are not the gradient of a scalar vector field (i.e of a potential or of a generalized potential), therefore their treatment in the context of Hamiltonian of Lagrangian paradigms is impossible.
But there are several other perturbations methods...
Dear Shuchang Zhang,
I would like to mention a tool, or 'method' that was not mentioned in the previous answers, for the treatment of dissipative systems in classical mechanics. The two papers that seem important to me are by Fred Riewe :
Mechanics with fractional derivatives, PRE 55, 3581 (1997)
Nonconservative Lagrangian and Hamiltonian mechanics, PRE53, 1890 (1996)
(of course there are many more.) In these works, the Lagrangian and Hamiltonian mechanics is formulated to include derivatives of fractional order (!)
Lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction. These equations, or equivalently the corresponding Hamilton's equations, are derived using generalised classical mechanics with fractional and higher-order derivatives!
Best,
Nikos Lazarides
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