35,616 research outputs found
Is it possible to formulate least action principle for dissipative systems?
A longstanding open question in classical mechanics is to formulate the least
action principle for dissipative systems. In this work, we give a general
formulation of this principle by considering a whole conservative system
including the damped moving body and its environment receiving the dissipated
energy. This composite system has the conservative Hamiltonian
where is the kinetic energy of the moving body, its potential
energy and the energy of the environment. The Lagrangian can be derived
by using the usual Legendre transformation where is the
total kinetic energy of the environment. An equivalent expression of this
Lagrangian is where is the energy dissipated by the
friction from the moving body into the environment from the beginning of the
motion. The usual variation calculus of least action leads to the correct
equation of the damped motion. We also show that this general formulation is a
natural consequence of the virtual work principle.Comment: 11 pages, no figur
Local equilibrium and the second law of thermodynamics for irreversible systems with thermodynamic inertia
Validity of local equilibrium has been questioned for non-equilibrium systems
which are characterized by delayed response. In particular, for systems with
non-zero thermodynamic inertia, the assumption of local equilibrium leads to
negative values of the entropy production, which is in contradiction with the
second law of thermodynamics. In this paper we address this question by
suggesting a variational formulation of irreversible evolution of a system with
non-zero thermodynamic inertia. We introduce the Lagrangian, which depends on
the properties of the normal and the so-called "mirror-image" systems. We show
that the standard evolution equations, in particular the
Maxwell-Cattaneo-Vernotte equation, can be derived from the variational
procedure without going beyond the assumption of local equilibrium. We also
argue, that the second law of thermodynamics should be understood as a
consequence of the variational procedure and the property of local equilibrium.
For systems with instantaneous response this leads to the standard requirement
of the local instantaneous entropy production being always positive. However,
if a system is characterized by delayed response, the formulation of the second
law of thermodynamics should be altered. In particular, the quantity, which is
always positive, is not the instantaneous entropy production, but the entropy
production averaged over the period of the heat wave.Comment: 12 pages, 7 figure
Generalized Lagrangians and spinning particles
The use of generalized Lagrangians for describing elementary particles was
already claimed by Ostrogradskii. It is shown how the spin structure of
elementary particles arises if one allows the Lagrangian to depend on higher
order derivatives. One part is related to the rotation of the particle and the
other, which is coming from the dependence of the Lagrangian on the
acceleration, is known as the zitterbewegung part of spin.Comment: Contribution to special issue in the 200th Ostrogradskii anniversary
by the Journal of Ukrainian Mathematical Societ
The "Symplectic Camel Principle" and Semiclassical Mechanics
Gromov's nonsqueezing theorem, aka the property of the symplectic camel,
leads to a very simple semiclassical quantiuzation scheme by imposing that the
only "physically admissible" semiclassical phase space states are those whose
symplectic capacity (in a sense to be precised) is nh + (1/2)h where h is
Planck's constant. We the construct semiclassical waveforms on Lagrangian
submanifolds using the properties of the Leray-Maslov index, which allows us to
define the argument of the square root of a de Rham form.Comment: no figures. to appear in J. Phys. Math A. (2002
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