Abordamos a questão do movimento de uma partícula com massa variável observada a partir de um referencial inercial. Consideramos duas situações diferentes de corpos com massa variável: (i) o valor de massa intrínseca da partícula varia ao longo do tempo; (ii) consideramos um sistema de partículas cujo o valor da massa do centro de massa (CM) varia, sendo que o conjunto é formado por partículas, essas com massa constante, mas cujo o número de partículas que o constituem varia no tempo. Mostramos que a Segunda Lei de Newton distingue o caso em que a massa intrínseca da partícula varia ao longo tempo da situação em que temos sistemas compostos de partículas cuja massa total varia com o tempo. No primeiro caso, estudamos as consequências da equação de movimento de um partícula com massa variável não ser covariante em referenciais inerciais sob as transformações galileanas. Nós também mostramos que a equação que descreve a dinâmica do CM de um sistema com número variável de partículas preserva a equivalência de todos os referenciais inerciais sob as transformações galileanas. Verificamos a não conservação do vetor de momento linear do CM de um conjunto de partículas livres durante o tempo em que uma partícula sai ou entra no sistema.

We approach the question of the movement of a particle with variable mass observed from an inertial frame. We consider two different situations: (i) a particle whose intrinsic mass value varies over time; (ii) the center of mass (CM) of a set of particles with constant mass but with a variable number of particles belonging to it. We show that Newton’s Second Law distinguishes the case in which the intrinsic mass of the particle varies over time from systems composed of particles, with constant mass, whose total mass varies over time. In the first case, we study the consequences of the equation of motion of a particle with variable mass is not covariant in inertial references under Galilean transformations. We also show that the equation that drives the dynamics of the CM of a system with variable number of particles preserves the equivalence of all inertial frames under the Galilean transformations. We verify the non-conservation of the linear momentum vector of the CM of a set of free particles during the time that one particle leaves or comes into the system.

Resumo Quando estudamos o movimento de rotação de corpos rígidos precisamos de uma cinemática de rotação que descreva este tipo de deslocamento que os corpos podem ter. É dito nos livros textos que: “… não podemos identificar rotações finitas com vetores.” Esta nota usa uma demonstração algébrica simples para mostrar a validade dessa afirmação no caso do movimento angular finito de uma partícula e entender porque apesar desta impossibilidade podemos associar um vetor a uma variação angular infinitesimal, sem qualquer conflito.

Abstract When we study the rotation movement of rigid bodies we need a rotation kinematics that describes this type of displacement that the bodies can have. It is said in the textbooks that: “… we cannot identify finite rotations with vectors.” This note uses a simple algebraic demonstration to show the validity of this statement in the case of finite angular movement of a particle and to understand why despite this impossibility we can associate a vector with an infinitesimal angular variation, without any conflict.

Abstract We study the dynamics of the center of mass (CM) of a set of particles when the number of particles in the system varies during a short interval of time, or when the value of the mass of one of the particles depends on time. We show that in both situations, while the population or mass of the system varies, the linear momentum of the CM differs from the total linear momentum of the system. During this time interval, the dynamics of the CM is driven by the external forces that act on the system plus transient forces that depend on the position and velocity of the CM. This dynamics is applied to study the time evolution of the vector position of the CM of a system of free particles moving on an air rail in which the mass of one the particles varies in a short time interval. We want to show it to students that with the knowledge and tools they have at an elementary Mechanics course they can formulate and answer questions which are not in the textbooks.

We show that the difference of adiabatic phases, that are basis-dependent, in noncyclic evolution of non-degenerate quantum systems have to be taken into account to give the correct interference result in the calculation of physical quantities in states that are a superposition of instantaneous eigenstates of energy. To verify the contribution of those adiabatic phases in the interference phenomena, we consider the spin-1/2 model coupled to a precessing external magnetic field. In the model, the adiabatic phase increases in time up to reach the difference of the Berry's phases of the model when the external magnetic field completes a period.

We study the adiabatic evolution of a two-level model in the presence of an external classical electric field. The coupling between the quantum model and the classical field is taken in the electric dipole approximation. In this regime, we show the absence of geometric phases in the interacting two-level model in the presence of any periodic real time-dependent classical electric field. We obtain a conservative scalar potential in the calculation of Berry's phases of the instantaneous eigenstates of the model. For complex electric fields, we recover the existence of geometric phases. In particular, the geometric phases of the instantaneous eigenstates of the model in the presence of a positive or of a negative frequency component of the monochromatic electric field differ by an overall sign. As a check on our results, we map this interacting two-level model onto a spin-1/2 model under the action of a classical magnetic field. We confirm that the first one acquires Berry's phase only in the rotating wave approximation [RWA].

We calculate the thermodynamics of the one-dimensional spin-1/2 Ising model in the presence of a constant skew magnetic field. We obtain the high-temperature expansion of its Helmholtz free energy (HFE), for the ferromagnetic and antiferromagnetic cases, up to order beta7. This expansion permits us to obtain the behaviour of the model for $ |J| \beta \stackrel {<}{_\sim}} 1 $, when it cannot be described by its classical version. Among the calculated thermodynamical functions of the model, we have the diagonal elements of the magnetic susceptibility tensor for the transverse and logitudinal Ising models, obtained by taking the limits h z -> 0 and h y -> 0, respectively, of the beta-expansion of the HFE. The y-component of the magnetization and the chiyy component of the magnetic susceptibility tensor are almost the same for the antiferro- and ferromagnetic models, at least for $ |J| \beta \stackrel {<}{_\sim}} 1 $; and, <FONT FACE=Symbol>c yy</FONT> is practically independent of the direction of the external magnetic. We also show that, in this region of temperature, the thermodynamics of the Ising model with skew magnetic field and that of an XXZ model with longitudinal magnetic field are not similar.

The Heisenberg XXZ model is a chain model with nearest-neighbor interactions. Its thermodynamics is exactly obtained via Bethe ansatz. Recently, we developed a method to derive the high-temperature expansion of the grand potential per site of translationally invariant chain models, with periodic boundary conditions. Here we apply this approach to the XXZ model with periodic boundary conditions for the Ising limit case (t = 0) and the free fermion case (delta = 0 and h = 0), obtaining results in agreement with the literature. In this new way of obtaining the coefficients of the high-temperature expansion of the grand potential, the coefficients are derived from an auxiliary function written only in terms of open connected sub-chains.

Charret et al. applied the properties of Grassmann generators to develop a new method to calculate the coefficientes of the high temperature expansion of the grand canonical partition function of self-interacting fermionic models on d-dimensions (d > or = 1). The method explores the anti-commuting nature of fermionic fields and avoids the calculation of the fermionic path integral. We apply this new method to the relativistic free Dirac fermions and recover the known results in the literature without the beta-independent and mu-independent infinities that plague the continuum path integral formulation.