Invariance and first integrals of continuous and discrete Hamiltonian equations

Дородницын, В. А.; Kozlov, Roman

doi:10.1007/s10665-009-9312-0

Cited by 53 publications

(49 citation statements)

References 40 publications

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“…Hamiltonian symmetries in evolutionary or canonical form have been considered ( [18]). Furthermore, symmetry properties of the Hamiltonian action have been investigated in the space (t, q, p) by [19] and [20]. In the latter, the authors considered the general form of the symmetries (7) and provided a Hamiltonian version of Noether's theorem.…”

Section: A Hamiltonian Version Of the Noether-type Theoremmentioning

confidence: 99%

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A partial Hamiltonian approach for current value Hamiltonian systems

Naz

Mahomed

Chaudhry

2014

Communications in Nonlinear Science and Numerical Simulation

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We develop a partial Hamiltonian framework to obtain reductions and closedform solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.keyword: Current value Hamiltonian, partial Hamiltonian approach, economic growth models

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Section: A Hamiltonian Version Of the Noether-type Theoremmentioning

confidence: 99%

“…The following important results which are analogs of Noether symmetries and the Noether theorem (see [18,21,22,20] for a discussion) were established.…”

Section: A Hamiltonian Version Of the Noether-type Theoremmentioning

confidence: 99%

A partial Hamiltonian approach for current value Hamiltonian systems

Naz

Mahomed

Chaudhry

2014

Communications in Nonlinear Science and Numerical Simulation

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“…This corresponds of course to a completely standard case (cf. [5,35]; see also [15] for a different approach to the searching for first integrals and their relation with symmetry properties a ).…”

Section: Symmetries and First Integrals Of Hamiltonian Equations Of Mmentioning

confidence: 99%

“…a It can be noted that all examples of symmetries given in [15] which admit a first integral belong to case (i) and those with no first integral belong to case (ii) below.…”

Section: Symmetries and First Integrals Of Hamiltonian Equations Of Mmentioning

confidence: 99%

Symmetries of Hamiltonian Equations and Λ-Constants of Motion

Cicogna¹

2021

JNMP

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We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Λ-symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence of first integrals, we recall the definition and the properties of Λ-symmetries. We show that in the presence of a Λ-symmetry for the Hamiltonian equations, one can introduce the notion of "Λ-constant of motion". The presence of a Λ-symmetry leads also to a nice and useful reduction of the form of the equations. We then consider the case in which the Hamiltonian problem is deduced from a Λ-invariant Lagrangian. We illustrate how the Lagrangian Λ-invariance is transferred into the Hamiltonian context and show that the Hamiltonian equations are Λ-symmetric. We also compare the "partial" (Lagrangian) reduction of the Euler-Lagrange equations with the reduction which can be obtained for the Hamiltonian equations. Several examples illustrate and clarify the various situations.

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“…In [92], there is consideration of the relationship between symmetries and first integrals (conservation laws) for both continuous and discrete Hamiltonian equations. It is shown that canonical Hamiltonian equations can be obtained through a variational principle from an action functional.…”

Section: Invariance and First Integrals Of Continuous And Discrete Hamentioning

confidence: 99%