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By using the theory of analytic vectors and manifolds modeled on normed spaces, we provide a rigorous symplectic differential geometric approach to $ t $-dependent Schrödinger equations on separable (possibly infinite-dimensional) Hilbert spaces determined by families of unbounded self-adjoint Hamiltonians admitting a common domain of analytic vectors. This allows one to cope with the lack of smoothness of structures appearing in quantum mechanical problems while using differential geometric techniques. Our techniques also allow for the analysis of problems related to unbounded operators that are not self-adjoint. As an application, the Marsden-Weinstein reduction procedure was employed to map the above-mentioned $ t $-dependent Schrödinger equations onto their projective spaces. We also analyzed other physically and mathematically relevant applications, demonstrating the usefulness of our techniques.

By using the theory of analytic vectors and manifolds modeled on normed spaces, we provide a rigorous symplectic differential geometric approach to t -dependent Schrödinger equations on separable (possibly infinite-dimensional) Hilbert spaces determined by families of unbounded self-adjoint Hamiltonians admitting a common domain of analytic vectors. This allows one to cope with the lack of smoothness of structures appearing in quantum mechanical problems while using differential geometric techniques. Our techniques also allow for the analysis of problems related to unbounded operators that are not self-adjoint. As an application, the Marsden-Weinstein reduction procedure was employed to map the above-mentioned t -dependent Schrödinger equations onto their projective spaces. We also analyzed other physically and mathematically relevant applications, demonstrating the usefulness of our techniques. [ABSTRACT FROM AUTHOR]

Both statistical phase space (SPS), which is Γ = T * R 3 N of N -body particle system, and kinetic theory phase space (KTPS), which is the cotangent bundle T * P (Γ) of the probability space P (Γ) thereon, carry canonical symplectic structures. Starting from this first principle, we provide a canonical derivation of thermodynamic phase space (TPS) of nonequilibrium thermodynamics as a contact manifold in two steps. First, regarding the collective observation of observables in SPS as a moment map defined on KTPS, we apply the Marsden–Weinstein reduction and obtain a mesoscopic phase space in between KTPS and TPS as a (infinite-dimensional) symplectic fibration. Then we show that the reduced relative information entropy defines a generating function that provides a covariant construction of a thermodynamic equilibrium as a Legen-drian submanifold. This Legendrian submanifold is not necessarily graph-like. We interpret the Maxwell construction of equal-area law as the procedure of finding a continuous, not necessarily differentiable, thermodynamic potential and explain the associated phase transition by identifying the procedure with that of finding a graph selector in symplecto-contact geometry and in the Aubry-Mather theory of dynamical system. [ABSTRACT FROM AUTHOR]

We discuss, in all generality, the reduction of a Hamilton — Jacobi theory for systems subject to nonholonomic constraints and invariant under the action of a group of symmetries. We consider nonholonomic systems subject to both linear and nonlinear constraints and with different positioning of such constraints with respect to the symmetries. [ABSTRACT FROM AUTHOR]

The investigation of symmetries of b -symplectic manifolds and folded-symplectic manifolds is well-understood when the group under consideration is a torus (see, for instance, [25,21,28] for b -symplectic manifolds and [12,14] for folded symplectic manifolds). However, reduction theory has not been set in this realm in full generality. This is fundamental, among other reasons, to advance in the "quantization commutes with reduction" programme for these manifolds initiated in [29,30]. In this article, we fill in this gap and investigate the Marsden-Weinstein reduction theory under general symmetries for general b m -symplectic manifolds and other singular symplectic manifolds, including certain folded symplectic manifolds. In this new framework, the set of admissible Hamiltonian functions is larger than the category of smooth functions as it takes the singularities of the differential forms into account. The quasi-Hamiltonian set-up is also considered and brand-new constructions of (singular) quasi-Hamiltonian spaces are obtained via a reduction procedure and the fusion product. [ABSTRACT FROM AUTHOR]

We review and slightly improve the known k -polysymplectic Marsden–Weinstein reduction theory by removing some technical conditions on k -polysymplectic momentum maps by developing a theory of affine Lie group actions for k -polysymplectic momentum maps, removing the necessity of their co-adjoint equivariance. Then, we focus on the analysis of a particular case of k -polysymplectic manifolds, the so-called fibred ones, and we study their k -polysymplectic Marsden–Weinstein reductions. Previous results allow us to devise a k -polycosymplectic Marsden–Weinstein reduction theory, which represents one of our main results. Our findings are applied to study coupled vibrating strings and, more generally, k -polycosymplectic Hamiltonian systems with field symmetries. We show that k -polycosymplectic geometry can be understood as a particular type of k -polysymplectic geometry. Finally, a k -cosymplectic to ℓ -cosymplectic geometric reduction theory is presented, which reduces, geometrically, the space-time variables in a k -cosymplectic framework. An application of this latter result to a vibrating membrane with symmetries is given. [ABSTRACT FROM AUTHOR]

Wong's equations for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are derived. The equations obtained are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these equations, we obtain Wong's equations in a pure Yang-Mills gauge theory with Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures performed in a finite-dimensional dynamical system and the reduction procedure in Yang-Mills gauge fields. [ABSTRACT FROM AUTHOR]

An approach to describing nonlinear Lax type integrable dynamical systems of modern mathematical and theoretical physics, based on the Marsden-Weinstein reduction method on canonically symplectic manifolds with group symmetry, is proposed. Its natural relationship with the well-known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated R-matrix approach is analyzed. A new generalized exactly solvable spatially one-dimensional quantum superradiance model, describing a charged fermionic medium interacting with external electromagnetic field, is suggested. The Lax type operator spectral problem is presented, the related R-structure is calculated. The Hamilton operator renormalization procedure subject to a physically stable vacuum is described, the quantum excitations and quantum solitons, related with the thermodynamical equilibrity of the model, are discussed. [ABSTRACT FROM AUTHOR]

Abstract: The geometrical representation of the path integral reduction Jacobian obtained in the problem of the path integral quantization of a scalar particle motion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group has been found for the case when the local reduced motion is described by means of dependent coordinates. The result is based on the scalar curvature formula for the original manifold which is viewed as a total space of the principal fiber bundle. [Copyright &y& Elsevier]

Abstract: By using the formula for the scalar curvature of the manifold with the Kaluza–Klein metric we obtain the geometrical representation of the Jacobian resulted from the path integral reduction problem in Wiener path integrals for a scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group. [Copyright &y& Elsevier]

We consider the optimal control problem for null curves in de Sitter 3-space defined by a functional which is linear in the curvature of the trajectory. We show how techniques based on the method of moving frames and exterior differential systems, coupled with the reduction procedure for systems with a Lie group of symmetries, lead to the integration by quadratures of the extremals. Explicit solutions are found in terms of elliptic functions and integrals. [ABSTRACT FROM AUTHOR]

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown. [ABSTRACT FROM AUTHOR]