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Federico Talamucci

Publications and source records attributed to Federico Talamucci.

15 recordsLinked to original sources
arXiv2026

On the commutation of variation and differentiation in nonholonomic Systems: A Chetaev-based approach

The derivation of the equations of motion for nonholonomic systems remains a central issue in analytical mechanics, primarily due to the tension between the d'Alembert-Lagrange differential principle and integral variational approaches. This study investigates the validity of the commutation relation between the variational operator and the time derivative, which is a geometric identity in holonomic manifolds but becomes problematic when dealing with velocity-dependent constraints. By analyzing the transposition rule, we define a formal relationship between the Chetaev variation and the total variation of the constraints. We show that the simultaneous requirement of kinematically admissible variations and the fulfillment of the Chetaev condition is generally incompatible with the standard commutation rule, unless a specific geometric condition - encoded through a skew-symmetric algebraic structure and the Lagrangian derivative of the constraints - is satisfied. Furthermore, this work extends the analysis to systems with multiple constraints introducing the concept of dynamic compensation. While Frobenius' Theorem provides a static criterion for integrability based on individual vector fields, our results suggest that dynamic consistency according to Chetaev's principle emerges as a collective phenomenon. We demonstrate that even when individual constraints are intrinsically non-integrable, their interactions can cancel out deviations from holonomy, maintaining global consistency. Notably, we show that for systems with high constraints this property is satisfied regardless of the constraints' form. These findings broaden the class of analyzable physical systems, suggesting that dynamic consistency is a resilient property that persists even in the absence of simple geometric integrability.

physics.class-ph↗
arXiv2025

Transpositional rule for constrained systems

This paper investigates the dynamics of nonholonomic mechanical systems, focusing on fundamental variational assumptions and the role of the transpositional rule. We analyze how the Cetaev condition and the first variation of constraints define compatible virtual displacements for systems subject to kinematic constraints, including those nonlinear in generalized velocities. The study explores the necessary conditions for commutation relations to hold, clarifying their impact on the consistency of the derived equations of motion. By detailing the interplay between these variational identities and the Lagrangian derivatives of constraint functions, we elucidate the differences between equations of motion formulated via the d'Alembert--Lagrange principle and those obtained from extended time-integral variational principles. This work aims to provide a clearer theoretical framework for applying these core principles to nonholonomic dynamics.

physics.class-ph↗
arXiv2025

On the transpositional relation for nonholonomic systems

This paper investigates the dynamics of nonholonomic mechanical systems, with a particular focus on the fundamental variational assumptions and the role of the transpositional rule. We analyze how the $\check Cetaev condition and the first variation of constraints define compatible virtual displacements for systems subject to kinematic constraints, which can be both linear and nonlinear in generalized velocities. The study meticulously explores the necessary conditions for the commutation relations to hold, clarifying their impact on the consistency of the derived equations of motion. By detailing the interplay between these variational identities and the Lagrangian derivatives of the constraint functions, we shed light on the differences between equations of motion formulated via d'Alembert--Lagrange principle and those obtained from extended time-integral variational principles. This work aims to provide a clearer theoretical framework for understanding and applying these core principles in the complex domain of nonholonomic dynamics.

physics.class-ph↗
arXiv2024

Equations of motion for general nonholonomic systems from the d'Alembert principle via an algebraic method

The aim of this study is to present an alternative way to deduce the equations of motion of general (i.e., also nonlinear) nonholonomic constrained systems starting from the d'Alembert principle and proceeding by an algebraic procedure. The two classical approaches in nonholonomic mechanics -- Cetaev method and vakonomic method -- are treated on equal terms, avoiding integrations or other steps outside algebraic operations. In the second part of the work we compare our results with the standard forms of the equations of motion associated to the two method and we discuss the role of the transpositional relation and of the commutation rule within the question of equivalence and compatibility of the Cetaev and vakonomic methods for general nonholonomic systems.

physics.class-ph↗
arXiv2024

Is the Cetaev condition inferable?

In the context of holonomic constrained systems the identification of virtual displacements is clear and consolidated: this gives the possibility, once the class of displacements have been combined with Newton's equations, to write the correct equations of motion for the constrained system. The method combines d'Alembert principle with the Lagrange formalism. As far as nonholonomic constraints are concerned, the conjecture that dates back to Ceteav actually defines a class of virtual displacements through which the method d'Alebert-Lagrange can be applied again. Much literature is dedicated to the Cetaev rule from both the theoretical and experimental points of view. The absence of a rigorous (mathematical) validation of the rule inferable from the constraint equations has been declared expired in a recent publication: our main objective is a critical investigation of the stated result.

physics.class-ph↗
arXiv2023

A simple approach to nonlinear nonholonomic systems with several examples

The main theme of the article is the study of discrete systems of material points subjected to constraints not only of a geometric type (holonomic constraints) but also of a kinematic type (nonholonomic constraints). The setting up of the equations of motion follows a simple principle which generalizes the holonomic case. Furthermore, attention is paid to the fact that the kinematic variables retain their velocity meaning, without resorting to the pseudo-velocity technique. Particular situations are examined in which the modeling of the constraints can be carried out in several ways to evaluate their effective equivalence. Numerous examples, many of which taken from the most recurring ones in the literature, are provided in order to illustrate the proposed theory.

physics.class-ph↗
arXiv2023

Cetaev condition for nonlinear nonholonomic systems and homogeneous constraints

We first present a way to formulate the equations of motion for a nonholonomic system with nonlinear constraints with respect to the velocities. The formulation is based on the Cetaev condition which aims to extend the practical method of virtual displacements from the holonomic case to the nonlinear nonholonomic one. The condition may appear in a certain sense artificial and motivated only to coherently generalize that concerning the holonomic case. In the second part we show that for a specific category of nonholonomic constraints (homogeneous functions with respect to the generalized velocities) the Cetaev condition reveals the same physical meaning that emerges in systems with holonomic constraints. In particular the aspect of the mechanical energy associable to the system is analysed.

physics.class-ph↗
arXiv2023

The energy equation for nonholonomic systems with nonlinear kinematic restrictions: a special category of constraints

The main topic of this work concerns the formulation of the equations of motion and the consequent energy balance that they imply for this type of systems, In particular, the analytical development that we will carry out on the equations of motion has as its objective the energy balance of the system. the delicate question of defining the displacements admitted by the system leads, as we shall see, to a non-univocal definition of the energy of the system, which finds coherence and unity for a particular class of nonholonomic constraints.

physics.class-ph↗
arXiv2022

La matematica armonia dei suoni naturali. Ovvero, l'armonica matematica dei suoni naturali

The collaboration of mathematics in the two musical systems of Pythagorean sounds and of equal sounds is evident and opportune for generating the elements and managing their relationships. The only essential notion for a rational intervention in the two sound worlds is that of distance between sounds. The scale of natural sounds does not seem to dialogue directly with a purely mathematical module that articulates its definition and is generally presented as a manual adjustment of the ancient Pythagorean sounds. The model of proportions that formed and inspired the architecture of the sixteenth century, an era in which the natural scale emerges, provides a clear idea for systematically adjusting natural sounds; on the other hand, the generation of sounds based and started exclusively on the respect of the proportions proposes interesting natural sound systems. The object is not so much to illustrate the harmonic device of the already formed natural scale, as to implement the formation of sound elements that respond to the unitary principle of proportion and combine harmoniously.

math.HO↗
arXiv2021

Energy balance for nonholonomic nonlinear systems

We consider nonholonomic systems with nonlinear restrictions with respect to the velocities. The mathematical problem is formulated by means of the Voronec equations extended to the nonlinear case. The main point of the paper is the balance of the mechanical energy induced by the equations of motion; the conservation of the energy on the basis of the tipology of the constraint equations is discussed. Several examples are performed.

math.CA↗
arXiv2020

Rheonomic systems with nonlinear nonholonomic constraints: the Voronec equations

One of the earliest formulations of dynamics of nonholonomic systems traces back to 1895 and it is due to Caplygin, who developed his analysis under the assumption that a certain number of the generalized coordinates do not occur neither in the kinematic constraints nor in the Lagrange function. A few years later Voronec derived the equations of motion for nonholonomic systems removing the restrictions demanded by the Caplygin systems. Although the methods encountered in the following years favour the use of the quasi-coordinates, we will pursue the Voronec method which deals with the generalized coordinates directly. The aim is to establish a procedure for extending the equations of motion to nonlinear nonholonomic systems, even in the rheonomic case.

physics.class-ph↗
arXiv2019

Nonlinear nonholonomic constraints

One of the founders of the mechanics of nonoholonomic systems is Voronec who published in 1901 a significant generalization of the Caplygin's equations, by removing some restrictive assumptions. In the frame of nonholonomic systems, the Voronec equations are probably less frequent and common with respect to the prevalent methods of quasi--coordinates (Hamel--Boltzmann equations) and of the acceleration energy (Gibbs--Appell equations). In this paper we start from the case of linear nonholonomic constraints, in order to extend the Voronec equations to nonlinear nonholonomic systems. The comparison between two ways of expressing the equations of motion is performed. We finally comment that the adopted procedure is appropriated to implement further extensions.

math-ph↗
arXiv2018

An extended Lagrangian formalism

A simple formal procedure makes the main properties of the lagrangian binomial extendable to functions depending to any kind of order of the time--derivatives of the lagrangian coordinates. Such a broadly formulated binomial can provide the lagrangian components, in the classical sense of the Newton's law, for a quite general class of forces. At the same time, the generalized equations of motions recover some of the classical alternative formulations of the Lagrangian equations.

physics.class-ph↗
arXiv2018

Synchronization of a double pendulum with moving pivots: a study of the spectrum

The model we consider consists in a double pendulum set, where the pivot points are free to shift along a horizontal line. Moreover, the two pendula are coupled by means of a spring whose extremities connect two points of each pendulum, at a fixed distance from the corresponding pivot. The mathematical model is first written encompassing a large class of setting for the device (different sizes, different physical properties, ...). In order to carry on the problem of synchronization via analytical me\-thods, we focus on the circumstance of identical pendula: in that case, some classical theorems concerning the zeroes of polynomial equations are used in order to locate the eigenvalues governing the process, so that the possibility of synchronization of the device can be better understood.

physics.class-ph↗