scholarly journals Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’s Quantum Calculus

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Frédéric Pierret ◽  
Delfim F. M. Torres
Keyword(s):  
Hamiltonian Systems ◽  
Hamiltonian Formulation ◽  
Quantum Calculus ◽  
Helmholtz Theorem

We derive the Helmholtz theorem for nondifferentiable Hamiltonian systems in the framework of Cresson’s quantum calculus. Precisely, we give a theorem characterizing nondifferentiable equations, admitting a Hamiltonian formulation. Moreover, in the affirmative case, we give the associated Hamiltonian.

scholarly journals ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS

1995 ◽  
Vol 10 (04) ◽  
pp. 579-610 ◽  
Author(s):  
V. MUKHANOV ◽  
A. WIPF
Keyword(s):  
String Theory ◽  
Hamiltonian Systems ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Linear Constraints ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Diffeomorphism Invariance ◽  
Symmetry Transformations ◽  
Local Symmetries

In this paper we show how the well-known local symmetries of Lagrangian systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangian system. The non-linear constraints (which we have, for instance, in gravity, supergravity and string theory) generate the dynamics of the corresponding Lagrangian system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We show the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems, in particular those which are diffeomorphism-invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian and Lagrangian formalisms is found. The possible applications of our results are discussed.

Hamiltonian Formulation

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Gauge Theory ◽  
Hamiltonian Systems ◽  
Hamiltonian Formulation ◽  
Particle Dynamics ◽  
Standard Formulation ◽  
Canonical Formulation ◽  
Constrained Hamiltonian Systems

The Hamiltonian formulation of relational particle dynamics unveils its equivalence with modern gauge theory, which admits exactly the same canonical formulation. Both are constrained Hamiltonian systems with nonhonolomic constraints, for which Dirac’s analysis, made popular by his lectures, is necessary. Dirac’s analysis is briefly summarized in this chapter for readers unfamiliar with it. The Hamiltonian formulation of the kind of systems we’re interested in is nontrivial. In fact the standard formulation fails to be predictive, precisely because of the relational nature of our dynamics.

Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation

2006 ◽  
Vol 59 (4) ◽  
pp. 230-248 ◽  
Author(s):  
W. Q. Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Stochastic Dynamics ◽  
First Passage Time ◽  
Hamiltonian Formulation ◽  
Stochastic Averaging ◽  
Passage Time ◽  
Stochastic Bifurcation ◽  
Dynamics And Control ◽  
Nonlinear Stochastic Dynamics ◽  
And Control

The significant advances in nonlinear stochastic dynamics and control in Hamiltonian formulation during the past decade are reviewed. The exact stationary solutions and equivalent nonlinear system method of Gaussian-white -noises excited and dissipated Hamiltonian systems, the stochastic averaging method for quasi Hamiltonian systems, the stochastic stability, stochastic bifurcation, first-passage time and nonlinear stochastic optimal control of quasi Hamiltonian systems are summarized. Possible extension and applications of the theory are pointed out. This review article cites 158 references.

Constrained Hamiltonian systems on Lie groups, moment map reductions, and central extensions

1994 ◽  
Vol 72 (7-8) ◽  
pp. 375-388 ◽  
Author(s):  
J. Harnad
Keyword(s):  
Hamiltonian Systems ◽  
Vector Bundles ◽  
Geometric Quantization ◽  
Hamiltonian Formulation ◽  
Coadjoint Orbit ◽  
Loop Group ◽  
Moment Map ◽  
Central Extensions ◽  
Constrained Hamiltonian Systems ◽  
Holomorphic Vector Bundles

The canonical symplectic structure on the cotangent bundle T*G of a Lie group, whose algebra gc admits a central extension gc, is modified so that the moment map associated to right translations generates a representation of gc. Zero moment map constraints are applied to a class of Hamiltonian systems on T*G that are not necessarily right invariant to yield reduced systems on coadjoint orbits in gc*. The Hamiltonian formulation of the WZW model is developed as an example in which G = LG0 is a loop group and the reduced space is the group of based loops ΩG0, identified as a coadjoint orbit of the affine Kac–Moody algebra [Formula: see text]. An extension of the geometric quantization method leads to representations of [Formula: see text] on Hilbert spaces of sections of holomorphic vector bundles over ΩG0.

scholarly journals EXACT SOLUTIONS OF (1 + 1)-DIMENSIONAL DILATON GRAVITY COUPLED TO MATTER

1996 ◽  
Vol 11 (21) ◽  
pp. 1691-1704 ◽  
Author(s):  
A.T. FILIPPOV
Keyword(s):  
Exact Solutions ◽  
Electromagnetic Fields ◽  
Hamiltonian Systems ◽  
Integrable Systems ◽  
Hamiltonian Formulation ◽  
Scalar Fields ◽  
Einstein Gravity ◽  
Dilaton Gravity ◽  
Spacetime Dimension ◽  
Special Case

A class of integrable models of (1 + 1)-dimensional dilaton gravity coupled to scalar and electromagnetic fields is obtained and explicitly solved. More general models are reduced to (0 + 1)-dimensional Hamiltonian systems, for which two integrable classes (called s-integrable) are found and explicitly solved. As a special case, static spherical solutions of the Einstein gravity coupled to electromagnetic and scalar fields in any real spacetime dimension are derived. A generalization of the “no-hair” theorem is pointed out and the Hamiltonian formulation that enables an exact quantization of the s-integrable systems is outlined.

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

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