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We consider the geometrization of quantum mechanics. We then focus on the pull-back of the Fubini-Study metric tensor field from the projective Hibert space to the orbits of the local unitary groups. An inner product on these tensor fields…
The geometrical formulation of the quantum Hamilton-Jacobi theory shows that the quantum potential is never trivial, so that it plays the r\^ole of intrinsic energy. Such a key property selects the Wheeler-DeWitt (WDW) quantum potential…
The Separation of Variables theory for the Hamilton-Jacobi equation is 'by definition' related to the use of special kinds of coordinates, for example Jacobi coordinates on the ellipsoid or St\"ackel systems in the Euclidean space. However,…
In this paper we propose a Hamilton-Jacobi theory for implicit contact Hamiltonian systems in two different ways. One is the understanding of implicit contact Hamiltonian dynamics as a Legendrian submanifold of the tangent contact space,…
The Hamilton-Jacobi equation (HJE) is one of the most elegant approach to Lagrangian systems such as geometrical optics and classical mechanics, establishing the duality between trajectories and waves and paving the way naturally for the…
Studying the behaviour of a quantum field in a classical, curved, spacetime is an extraordinary task which nobody is able to take on at present time. Independently by the fact that such problem is not likely to be solved soon, still we…
A necessary and sufficient condition for a parameter transformation that leaves invariant the energy of a one dimensional autonomous system is obtained. Using a parameter transformation the Hamilton-Jacobi equation is solved by a…
We establish the quantum stationary Hamilton-Jacobi equation in 3-D and its solutions for three symmetrical potentials, Cartesian symmetry potential, spherical symmetry potential and cylindrical symmetry potential. For the two last…
We make use of the Quantum Hamilton-Jacobi (QHJ) theory to investigate conditional quasi-solvability of the quantum symmetric top subject to combined electric fields (symmetric top pendulum). We derive the conditions of quasi-solvability of…
We establish a framework that allows to prove Gamma-converge of functionals of Lagrangian form on spaces of trajectories based on convergence of viscosity solutions of associated Hamilton-Jacobi equations. Gamma convergence follows from a:…
In this Thesis we develop the geometric formulations for higher-order autonomous and non-autonomous dynamical systems, and second-order field theories. In all cases, the physical information of the system is given in terms of a Lagrangian…
In this article, the quantum Hamilton- Jacobi theory based on the position dependent mass model is studied. Two effective mass functions having different singularity structures are used to examine the Morse and Poschl- Teller potentials.…
Let $(M,g)$ be a closed, connected and orientable Riemannian manifold with nonnegative Ricci curvature. Consider a Lagrangian $L(x,v):TM\to\R$ defined by $L(x,v):=\frac 12g_x(v,v)-\omega(v)+c$, where $c\in\R$ and $\omega$ is a closed…
We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is $\geq 0$,…
Quadratic Lagrangians are introduced adding surface terms to a free particle Lagrangian. Geodesic equations are used in the context of the Hamilton-Jacobi formulation of constrained sysytem. Manifold structure induced by the quadratic…
Hamilton-Jacobi theory provides a natural starting point for a covariant description of the gravitational field. Using a spatial gradient expansion, one may solve for the phase of the wavefunction by using a line-integral in superspace.…
This paper gives a technically elementary treatment of some aspects of Hamilton-Jacobi theory, especially in relation to the calculus of variations. The second half of the paper describes the application to geometric optics, the…
A geometric approach to integrability and reduction of dynamical system is developed from a modern perspective. The main ingredients in such analysis are the infinitesimal symmetries and the tensor fields that are invariant under the given…
PT-/non-PT-symmetric and non-Hermitian deformed Morse and Poschl-Teller potentials are studied first time by quantum Hamilton-Jacobi approach. Energy eigenvalues and eigenfunctions are obtained by solving 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…