152 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
Progress in Classical and Quantum Variational Principles
We review the development and practical uses of a generalized Maupertuis
least action principle in classical mechanics, in which the action is varied
under the constraint of fixed mean energy for the trial trajectory. The
original Maupertuis (Euler-Lagrange) principle constrains the energy at every
point along the trajectory. The generalized Maupertuis principle is equivalent
to Hamilton's principle. Reciprocal principles are also derived for both the
generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis
Principle is the classical limit of Schr\"{o}dinger's variational principle of
wave mechanics, and is also very useful to solve practical problems in both
classical and semiclassical mechanics, in complete analogy with the quantum
Rayleigh-Ritz method. Classical, semiclassical and quantum variational
calculations are carried out for a number of systems, and the results are
compared. Pedagogical as well as research problems are used as examples, which
include nonconservative as well as relativistic systems
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