10,207 research outputs found
Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints
We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian)
systems, a generalized formulation of Lagrangian mechanics that can incorporate
degenerate Lagrangians as well as holonomic and nonholonomic constraints. We
refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi
equation. For non-degenerate Lagrangian systems with nonholonomic constraints,
the theory specializes to the recently developed nonholonomic Hamilton-Jacobi
theory. We are particularly interested in applications to a certain class of
degenerate nonholonomic Lagrangian systems with symmetries, which we refer to
as weakly degenerate Chaplygin systems, that arise as simplified models of
nonholonomic mechanical systems; these systems are shown to reduce to
non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian
systems defined with non-closed two-forms. Accordingly, the
Dirac-Hamilton-Jacobi equation reduces to a variant of the nonholonomic
Hamilton-Jacobi equation associated with the reduced system. We illustrate
through a few examples how the Dirac-Hamilton-Jacobi equation can be used to
exactly integrate the equations of motion.Comment: 44 pages, 3 figure
A junction condition by specified homogenization and application to traffic lights
Given a coercive Hamiltonian which is quasi-convex with respect to the
gradient variable and periodic with respect to time and space at least "far
away from the origin", we consider the solution of the Cauchy problem of the
corresponding Hamilton-Jacobi equation posed on the real line. Compact
perturbations of coercive periodic quasi-convex Hamiltonians enter into this
framework for example. We prove that the rescaled solution converges towards
the solution of the expected effective Hamilton-Jacobi equation, but whose
"flux" at the origin is "limited" in a sense made precise by the authors in
\cite{im}. In other words, the homogenization of such a Hamilton-Jacobi
equation yields to supplement the expected homogenized Hamilton-Jacobi equation
with a junction condition at the single discontinuous point of the effective
Hamiltonian. We also illustrate possible applications of such a result by
deriving, for a traffic flow problem, the effective flux limiter generated by
the presence of a finite number of traffic lights on an ideal road. We also
provide meaningful qualitative properties of the effective limiter.Comment: 41 page
The Hamilton-Jacobi Equations for Strings and p-Branes
Simple derivation of the Hamilton-Jacobi equation for bosonic strings and
p-branes is given. The motion of classical strings and p-branes is described by
two and p+1 local fields, respectively. A variety of local field equations
which reduce to the Hamilton-Jacobi equation in the classical limit are given.
They are essentially nonlinear, having no linear term.Comment: 7 page
Quantum Hamilton-Jacobi equation
The nontrivial transformation of the phase space path integral measure under
certain discretized analogues of canonical transformations is computed. This
Jacobian is used to derive a quantum analogue of the Hamilton-Jacobi equation
for the generating function of a canonical transformation that maps any quantum
system to a system with a vanishing Hamiltonian. A formal perturbative solution
of the quantum Hamilton-Jacobi equation is given.Comment: 4 pages, RevTe
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