Infinitesimal Time Reparametrisation and Its Applications

Cariñena, José F.; Martínez, Eduardo; Muñoz-Lecanda, Miguel C.

doi:10.1007/s44198-022-00037-w

Cited by 7 publications

(13 citation statements)

References 42 publications

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“…Particularly interesting cases are the geodesic Lagrangians defined through a Riemann structure g by L = 1 2 g(v, v) (see, e.g., [7,28,29]) and the so-called natural Lagrangians defined by means of a Riemann structure g and a function V in Q as follows: L = 1 2 g(v, v) − V(q). A remarkable fact is the possibility of the existence of alternative compatible structures of the same type.…”

Section: Invariant Symplectic Structuresmentioning

confidence: 99%

A Geometric Approach to the Sundman Transformation and Its Applications to Integrability

Cariñena

2024

Symmetry

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A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed.

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Section: Invariant Symplectic Structuresmentioning

confidence: 99%

A Geometric Approach to the Sundman Transformation and Its Applications to Integrability

Cariñena

2024

Symmetry

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“…We first review the geometric generalisation of Sundman transformations given in [18]. The classical Sundman transformation [3] is an infinitesimal scaling of time from t to a new fictitious time τ given by dt…”

Section: A Geometric Approach To Generalised Sundman Transformationmentioning

confidence: 99%

“…and the solutions of the system of equations (2.2) provide us the integral curves of the vector field X. It has been proved in [18] that if the curve γ(t) is an integral curve of X, and we carry out the reparametrisation for which the new parameter τ is defined by the relation…”

Section: A Geometric Approach To Generalised Sundman Transformationmentioning

confidence: 99%

Sundman transformation and alternative tangent structures

Cariñena

Martínez

Muñoz³

2023

J. Phys. A: Math. Theor.

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A geometric approach to Sundman transformation deﬁned by basic functions for systems of second-order diﬀerential equations is developed and the necessity of a change of the tangent structure by means of the function deﬁning the Sundman transformation is shown. Among other applications of such theory we study the linearisability of a system of second-order diﬀerential equations and in particular the simplest case of a second-order diﬀerential equation. The theory is illustrated with several examples.

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“…Their integral curves are geodesic curves for the Levi-Civita connection ∇, they are solutions to the equation ∇ γ(t) γ(t) = 0. Some interesting properties of these vector field are in [13] and references therein.…”

Section: Hamilton-jacobi Vector Fieldsmentioning

confidence: 99%

“…It is well known that the solutions γ of the Newton equation ∇ γ γ = −grad V • γ with fixed energy E 0 are, under convenient reparametrization, the geodesic lines of g 0 . In [24] there are interesting comments on this topic, and you can see a nice new approach of the same question in [13].…”

Section: Jacobi Metricmentioning

confidence: 99%

Newtonian mechanics in a Riemannian manifold

Muñoz-Lecanda¹

2021

Preprint

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The work done by Isaac Newton more than three hundred years ago, continues being a path to increase our knowledge of Nature. To better understand all the ideas behind it, one of the finest ways is to generalize them to wider situations. In this report we make a review of one of these enlargements, the one that bears the mechanical systems from the elementary homogeneous three dimensional Euclidean space to the more abstract geometry of a Riemannian manifold.

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Infinitesimal Time Reparametrisation and Its Applications

Cited by 7 publications

References 42 publications

A Geometric Approach to the Sundman Transformation and Its Applications to Integrability

A Geometric Approach to the Sundman Transformation and Its Applications to Integrability

Sundman transformation and alternative tangent structures

Newtonian mechanics in a Riemannian manifold

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