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Huan-hang Chi - One of the best experts on this subject based on the ideXlab platform.

  • single valued hamiltonian via Legendre fenchel Transformation and time translation symmetry
    Nuclear Physics, 2014
    Co-Authors: Huan-hang Chi
    Abstract:

    Abstract Under conventional Legendre Transformation, systems with a non-convex Lagrangian will result in a multi-valued Hamiltonian as a function of conjugate momentum. This causes problems such as non-unitary time evolution of quantum state and non-determined motion of classical particles, and is physically unacceptable. In this work, we propose a new construction of single-valued Hamiltonian by applying Legendre–Fenchel Transformation, which is a mathematically rigorous generalization of conventional Legendre Transformation, valid for non-convex Lagrangian systems, but not yet widely known to the physics community. With the new single-valued Hamiltonian, we study spontaneous breaking of time translation symmetry and derive its vacuum state. Applications to theories of cosmology and gravitation are discussed.

  • Single-valued Hamiltonian via Legendre–Fenchel Transformation and time translation symmetry
    Nuclear Physics B, 2014
    Co-Authors: Huan-hang Chi
    Abstract:

    Abstract Under conventional Legendre Transformation, systems with a non-convex Lagrangian will result in a multi-valued Hamiltonian as a function of conjugate momentum. This causes problems such as non-unitary time evolution of quantum state and non-determined motion of classical particles, and is physically unacceptable. In this work, we propose a new construction of single-valued Hamiltonian by applying Legendre–Fenchel Transformation, which is a mathematically rigorous generalization of conventional Legendre Transformation, valid for non-convex Lagrangian systems, but not yet widely known to the physics community. With the new single-valued Hamiltonian, we study spontaneous breaking of time translation symmetry and derive its vacuum state. Applications to theories of cosmology and gravitation are discussed.

Dana Smetanová - One of the best experts on this subject based on the ideXlab platform.

  • Higher Order Hamiltonian Systems with Generalized Legendre Transformation
    Mathematics, 2018
    Co-Authors: Dana Smetanová
    Abstract:

    The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre Transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre Transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre Transformation does not exist; (c) the Legendre Transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition.

  • Legendre Transformation for Regularizable Lagrangians in Field Theory
    Letters in Mathematical Physics, 2001
    Co-Authors: Olga Krupková, Dana Smetanová
    Abstract:

    Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre Transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.

  • Legendre Transformation for regularizable Lagrangians in field theory
    arXiv: Mathematical Physics, 2001
    Co-Authors: Olga Krupková, Dana Smetanová
    Abstract:

    Hamilton equations based not only upon the Poincare--Cartan equivalent of a first-order Lagrangian, but rather upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton--De Donder theory, but regularizable in this generalized sense are studied. Legendre Transformation for regularizable Lagrangians is proposed, and Hamilton equations, equivalent with the Euler--Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.

  • Legendre Transformation for Regularizable Lagrangians in Field Theory
    Letters in Mathematical Physics, 2001
    Co-Authors: Olga Krupková, Dana Smetanová
    Abstract:

    Hamilton equations based not only upon the Poincare–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre Transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.

Olga Krupková - One of the best experts on this subject based on the ideXlab platform.

  • euler lagrange and hamilton equations for non holonomic systems in field theory
    Journal of Physics A, 2005
    Co-Authors: Olga Krupková, Petr Volný
    Abstract:

    A generalization of the concept of a system of non-holonomic constraints to fibred manifolds with n-dimensional bases is considered. Motion equations in both Lagrangian and Hamiltonian settings for systems subjected to such constraints are investigated. Regularity conditions for the existence of a non-holonomic Legendre Transformation, and the corresponding formulae for Hamiltonian and momenta are found. In particular, Lagrangian constraints and semi-holonomic constraints, and simplifications arising in this case are discussed.

  • Recent results in the geometry of constrained systems
    Reports on Mathematical Physics, 2002
    Co-Authors: Olga Krupková
    Abstract:

    Abstract Nonholonomic mechanical systems on fibered manifolds are investigated from a geometrical point of view. Regularity, variationality and existence of Legendre Transformation is studied. A solution to the variational inverse problem for constrained systems is presented.

  • Legendre Transformation for Regularizable Lagrangians in Field Theory
    Letters in Mathematical Physics, 2001
    Co-Authors: Olga Krupková, Dana Smetanová
    Abstract:

    Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre Transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.

  • Legendre Transformation for regularizable Lagrangians in field theory
    arXiv: Mathematical Physics, 2001
    Co-Authors: Olga Krupková, Dana Smetanová
    Abstract:

    Hamilton equations based not only upon the Poincare--Cartan equivalent of a first-order Lagrangian, but rather upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton--De Donder theory, but regularizable in this generalized sense are studied. Legendre Transformation for regularizable Lagrangians is proposed, and Hamilton equations, equivalent with the Euler--Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.

  • Legendre Transformation for Regularizable Lagrangians in Field Theory
    Letters in Mathematical Physics, 2001
    Co-Authors: Olga Krupková, Dana Smetanová
    Abstract:

    Hamilton equations based not only upon the Poincare–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre Transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.

Freddy Van Oystaeyen - One of the best experts on this subject based on the ideXlab platform.

  • q Legendre Transformation partition functions and quantization of the boltzmann constant
    Journal of Physics A, 2010
    Co-Authors: Arthur E Ruuge, Freddy Van Oystaeyen
    Abstract:

    In this paper we construct a q-analogue of the Legendre Transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wavefunctions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton–Jacobi equation in mechanics to the q-Legendre Transformation in thermodynamics. It is shown that this requires a non-commutative analogue of the Planck–Boltzmann constants ( and kB) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the quasithermodynamic parameter corresponding to the Boltzmann constant with an infinite collection of generators of the so-called epoche (bracketing) algebra. The latter is an infinite-dimensional non-commutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of 'deformations of deformations' of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way that the trees with a single leaf correspond to the observables of the limiting thermodynamic system.

Arthur E Ruuge - One of the best experts on this subject based on the ideXlab platform.

  • q Legendre Transformation partition functions and quantization of the boltzmann constant
    Journal of Physics A, 2010
    Co-Authors: Arthur E Ruuge, Freddy Van Oystaeyen
    Abstract:

    In this paper we construct a q-analogue of the Legendre Transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wavefunctions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton–Jacobi equation in mechanics to the q-Legendre Transformation in thermodynamics. It is shown that this requires a non-commutative analogue of the Planck–Boltzmann constants ( and kB) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the quasithermodynamic parameter corresponding to the Boltzmann constant with an infinite collection of generators of the so-called epoche (bracketing) algebra. The latter is an infinite-dimensional non-commutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of 'deformations of deformations' of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way that the trees with a single leaf correspond to the observables of the limiting thermodynamic system.

  • q-Legendre Transformation: Partition Functions and Quantization of the Boltzmann Constant
    Journal of Physics A: Mathematical and Theoretical, 2010
    Co-Authors: Arthur E Ruuge, Freddy Van Oystaeyen
    Abstract:

    In this paper we construct a q-analogue of the Legendre Transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wave functions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton-Jacobi equation in mechanics to the q-Legendre Transformation in thermodynamics. It is shown, that this requires a non-commutative analogue of the Planck-Boltzmann constants (hbar and k_B) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the Boltzmann constant with an infinite collection of generators of the so-called epoch\'e (bracketing) algebra. The latter is an infinite dimensional noncommutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of "deformations of deformations" of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way, that the trees with a single leaf correspond to the observables of the limiting thermodynamic system.