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The two-body problem consists of determining the motion of two gravitationally interacting bodies with given masses and initial velocities. The problem was first solved by Isaac Newton in 1687 using geometric arguments. In this thesis, we present selected parts of Newton's solution together with an alternative geometric solution by Richard Feynman and a modern solution using differential calculus. All three solutions rely on the three laws of Newton and treat the two bodies as point masses; they differ in their approach to the the three laws of Kepler and to the inverse-square force law. Whereas the geometric solutions aim to prove some of these laws, the modern solution provides a method for calculating the positions and velocities given their initial values. It is notable that Newton in his most famous work Principia, where the general law of gravity and the solution to the two-body problem are presented, used mathematics that is not widely studied today. One might ask if today's low emphasis on classical geometry and conic sections affects our understanding of classical mechanics and calculus.

The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.

Nas pesquisas que buscam a inserção da História e Filosofia da Ciência no Ensino de Física aparecem dois papeis principais para a Filosofia, a saber, o de atuar como estratégia didática para a compreensão conceitual e o de ser necessária para a compreensão da Natureza da Ciência. Com respeito ao primeiro, investigamos se a Filosofia pode ser mais do que apenas uma estratégia e passar a contribuir de maneira essencial, uma vez que identificamos nas propostas mencionadas que elas não trouxeram um modo de compreensão diferente dos que antes se criticava. Começamos analisando as relações entre Física e Filosofia e encontramos que o pensamento físico possui intersecções constitutivas com o pensamento filosófico, tanto na atribuição de significações, ao conectar as teorias formalizadas com o mundo físico, quanto na crítica de suas próprias teorias. Em seguida, exploramos a concepção de compreensão na Filosofia da Linguagem tardia de Wittgenstein. Para ele, a compreensão de um conceito não está em processos ou estados mentais, mas sim nos usos que fazemos das palavras em diferentes contextos, fundamentadas em uma normatividade presente na própria linguagem. O pensamento filosófico, como parte desta normatividade é também condição para a compreensão e formação de conceitos físicos em um grau suficiente para permitir a autonomia do sujeito. Exemplificamos nossa defesa a partir de um estudo teórico da mecânica newtoniana, explorando as questões metafísicas relacionadas ao problema do espaço e do movimento e através da análise de discussões entre professores em formação inicial, procurando observar o papel dos pressupostos filosóficos para a significação. Concluímos que, sem eles, havia grandes lacunas de significação que eram preenchidas por conceitos pertencentes a contextos não físicos, além de que a mecânica clássica era sinônima de um conjunto de pressupostos para a resolução de exercícios. Estes permitiram que os futuros professores fizessem descrições, deduções e classificações de fenômenos físicos, mas apenas em conjunto com as proposições filosóficas, passaram a permitir um grau mais alto de compreensão, bem como a formação de novas habilidades, tais como a elaboração de hipóteses, argumentações, deduções e críticas. / Most proposals that advocate the inclusion of history and philosophy of science in physics curriculum award two main roles to philosophy: as a teaching strategy for conceptual understanding and as a method to understand the Nature of Science. With respect to the first role, we inquire whether philosophy can be more than just a teaching strategy and to contribute in an essential way instead, as we find that the above-mentioned proposals do not yield better results in terms of conceptual understanding then the methods criticized by them. We begin by analysing the relationship between physics and philosophy and we find that physical thinking has constitutive intersections with philosophical thought, so much in assigning meanings in order to connect formalized theories to the world as in the criticism of its own theories. Then we explore the conception of understanding in the philosophy of language of the late Wittgenstein. For him, understanding of a concept is not a mental process or state, but it rather consists in the use we make of the words in different contexts, based on the normativity present in language itself. Philosophical thought, as a part of this normativity is also a condition for the comprehension and development of physicals concepts in a level sufficient for the subject\'s autonomy. We exemplify our conclusions in theoretical study of newtonian mechanics, exploring the metaphysical questions related to the problem of space and movement and analyzing discussions among teachers in initial training, trying to observe the role of philosophical assumptions in shaping meanings. We conclude that, without them, there were great significance gaps, which were filled with concepts belonging to non-physical contexts, and that classical mechanics was synonymous to a set of assumptions for solving exercises. These assumptions enabled the prospective teachers to make descriptions, deductions and classifications of physical phenomena, but only together with philosophical propositions, did they increased the degree of understanding and enable formation of new skills, such as development of hypothesis, argumentation and criticism.

This report provides an introduction to geometric mechanics, which seeks to model the behavior of physical mechanical systems using differential geometric objects. In addition to its elegance as a method of representation, this formulation also admits the application of powerful analytical techniques from geometry as an aid to understanding these systems. In particular, it reveals the fundamental role that symplectic geometry plays in mechanics (something which is not at all obvious from the traditional Newtonian formulation), and in the case of systems exhibiting symmetry, leads to an elucidation of conservation and reduction laws which can be used to simplify the analysis of these systems. The contribution here is primarily one of exposition. Geometric mechanics was developed as an aid to understanding physics, and we have endeavored throughout to highlight the physical principles at work behind the mathematical formalism. In particular, we show quite explicitly the entire development of mechanics from first principles, beginning with Newton's laws of motion and culminating in the geometric reformulation of Lagrangian and Hamiltonian mechanics. Self-contained presentations of this entire range of material do not appear to be common in either the physics or the mathematics literature, but we feel very strongly that this is essential in order to understand how the more abstract mathematical developments that follow actually relate to the real world. We have also attempted to make many of the proofs contained herein more explicit than they appear in the standard references, both as an aid in understanding and simply to make them easier to follow, and several of them are original where we feel that their presentation in the literature was unacceptably opaque (this occurs primarily in the presentation of the geometric formulation of Lagrangian mechanics and the appendix on symplectic geometry). Finally, we point out that the fields of geometric mechanics and symplectic geometry are vast, and one could not hope to get more than a fragmentary glimpse of them in a single work, which necessiates some parsimony in the presentation of material. The subject matter covered herein was chosen because it is of particular interest from an applied or engineering perspective in addition to its mathematical appeal. / text

Nas pesquisas que buscam a inserção da História e Filosofia da Ciência no Ensino de Física aparecem dois papeis principais para a Filosofia, a saber, o de atuar como estratégia didática para a compreensão conceitual e o de ser necessária para a compreensão da Natureza da Ciência. Com respeito ao primeiro, investigamos se a Filosofia pode ser mais do que apenas uma estratégia e passar a contribuir de maneira essencial, uma vez que identificamos nas propostas mencionadas que elas não trouxeram um modo de compreensão diferente dos que antes se criticava. Começamos analisando as relações entre Física e Filosofia e encontramos que o pensamento físico possui intersecções constitutivas com o pensamento filosófico, tanto na atribuição de significações, ao conectar as teorias formalizadas com o mundo físico, quanto na crítica de suas próprias teorias. Em seguida, exploramos a concepção de compreensão na Filosofia da Linguagem tardia de Wittgenstein. Para ele, a compreensão de um conceito não está em processos ou estados mentais, mas sim nos usos que fazemos das palavras em diferentes contextos, fundamentadas em uma normatividade presente na própria linguagem. O pensamento filosófico, como parte desta normatividade é também condição para a compreensão e formação de conceitos físicos em um grau suficiente para permitir a autonomia do sujeito. Exemplificamos nossa defesa a partir de um estudo teórico da mecânica newtoniana, explorando as questões metafísicas relacionadas ao problema do espaço e do movimento e através da análise de discussões entre professores em formação inicial, procurando observar o papel dos pressupostos filosóficos para a significação. Concluímos que, sem eles, havia grandes lacunas de significação que eram preenchidas por conceitos pertencentes a contextos não físicos, além de que a mecânica clássica era sinônima de um conjunto de pressupostos para a resolução de exercícios. Estes permitiram que os futuros professores fizessem descrições, deduções e classificações de fenômenos físicos, mas apenas em conjunto com as proposições filosóficas, passaram a permitir um grau mais alto de compreensão, bem como a formação de novas habilidades, tais como a elaboração de hipóteses, argumentações, deduções e críticas. / Most proposals that advocate the inclusion of history and philosophy of science in physics curriculum award two main roles to philosophy: as a teaching strategy for conceptual understanding and as a method to understand the Nature of Science. With respect to the first role, we inquire whether philosophy can be more than just a teaching strategy and to contribute in an essential way instead, as we find that the above-mentioned proposals do not yield better results in terms of conceptual understanding then the methods criticized by them. We begin by analysing the relationship between physics and philosophy and we find that physical thinking has constitutive intersections with philosophical thought, so much in assigning meanings in order to connect formalized theories to the world as in the criticism of its own theories. Then we explore the conception of understanding in the philosophy of language of the late Wittgenstein. For him, understanding of a concept is not a mental process or state, but it rather consists in the use we make of the words in different contexts, based on the normativity present in language itself. Philosophical thought, as a part of this normativity is also a condition for the comprehension and development of physicals concepts in a level sufficient for the subject\'s autonomy. We exemplify our conclusions in theoretical study of newtonian mechanics, exploring the metaphysical questions related to the problem of space and movement and analyzing discussions among teachers in initial training, trying to observe the role of philosophical assumptions in shaping meanings. We conclude that, without them, there were great significance gaps, which were filled with concepts belonging to non-physical contexts, and that classical mechanics was synonymous to a set of assumptions for solving exercises. These assumptions enabled the prospective teachers to make descriptions, deductions and classifications of physical phenomena, but only together with philosophical propositions, did they increased the degree of understanding and enable formation of new skills, such as development of hypothesis, argumentation and criticism.

abstract: This dissertation describes Space Vector 1 and Space Vector 2, two video games that introduce Newtonian mechanics concepts. Space Vector 1 is a side-scrolling game, in which players choose to drop bombs or supplies. Players had to identify if the physics was correct during a mission, or they had to plot the trajectory of a falling object, which was then simulated. In Space Vector 2, players were given velocity and acceleration values and had to plot the trajectory of a spaceship across a grid, or players were given a trajectory of a spaceship on a grid and had to program the velocity and acceleration values to produce the trajectory. Space Vector 1 was evaluated with 65 college undergraduates. Space Vector 2 was evaluated with 18 high school students. All participants were given a subset of the Force Concept Inventory, a standard assessment tool in physics education, as a pretest and posttest. Space Vector 1 was evaluated with a single group pretest-posttest design. Space Vector 2 was evaluated with a 2 x 2 ANOVA, where the factors were game mechanic (prediction mechanic or programming mechanic) and bonus questions (bonus question after a mission or no bonus question). Bayesian statistical methods were used for the data analysis. The best estimate for the average change in test scores for Space Vector 1 was a score gain of 1.042 (95% Highest Density Interval (HDI) [0.613, 1.487]) with an effect size of 0.611 (95% HDI [0.327, 0.937]). The best estimate for the grand mean of change scores in Space Vector 2 was an increase of 0.78 (95% HDI [-0.3, 1.85]) with an effect size of 0.379 (95% HDI [-0.112, 0.905]). The prediction/no bonus question version produced the largest change in score, where the best estimate for the mean change score was an increase of 1.2. The estimation intervals for the Space Vector 2 results were wide, and all included zero as a credible value. / Dissertation/Thesis / Doctoral Dissertation Educational Technology 2014

Modern physics is based on Newton’s laws of motion, which describe interaction via forces. In this paper, we argue that interaction needs to be described in terms of processes. By introducing the momentum work and the associated momentum energy in mechanics, we present a coherent formulation of the process equations for mechanics and thermodynamics. This naturally leads to a simple derivation of the Lorentz-transformed mass, according to which any object changes its mass in real terms when its velocity is changed. Momentum work requires a revision of Newton’s laws of motion. For the first time in the history of physics, the elastic collision between objects, such as particles, can be described as a temporal process, not as interaction via force = counter-force. The mechanism of energy conversion during the elastic collision and other mechanical processes, such as free fall, becomes clear and demonstrates the validity of the principle of energy conservation on microscale at any point in time. The results suggest that physics can be rebuilt on a more coherent footing of dynamic processes up to quantum-process thermodynamics.

Explicações científicas são um dos elementos mais importantes do ensino de ciências. Elas estão relacionadas com a noção de natureza de ciência que reproduzimos, à compreensão adequada de conceitos e ao compartilhamento de hipóteses e princípios científicos. Embora não haja uma definição única de explicação científica que seja aceita de forma consensual por parte dos pesquisadores desse tema, nas últimas décadas ele tem sido abordado em um número crescente de trabalhos publicados em periódicos internacionais especializados em pesquisas sobre Ensino de Ciências. Apesar disso, consideramos que este tema tem recebido pouca atenção da comunidade de pesquisa em ensino de Ciências no Brasil, em particular no que diz respeito a atividades relacionadas à construção de explicações científicas durante a formação de professores. Neste trabalho investigamos elementos que fazem parte de explicações produzidas por futuros professores e a forma como estes elementos são relacionados entre si na organização da explicação. Para isso, propomos uma forma de estrutura organizacional de explicações científicas e a utilizamos para analisar seminários apresentados por alunos de um curso de licenciatura em Física durante uma das disciplinas deste curso, referente a tópicos relacionados ao conteúdo de mecânica. Nossa estrutura organizacional foi elaborada a partir de trabalhos realizados por James Wertsch (2008) sobre narrativas, a partir dos quais adaptamos para o estudo de explicações científicas duas estruturas que Wertsch propôs com respeito a narrativas: explicações específicas e moldes esquemáticos explicativos. Relacionamos as explicações específicas aos momentos explicativos propostos por Ogborn, Kress, Martins e Mcgillicuddy (1996) e os moldes esquemáticos explicativos a tipos de explicações propostos por filósofos das explicações – em particular o modelo Mecânico-Causal, as explicações de Unificação e explicações Funcionais. Em nossa análise empírica buscamos identificar objetivos pedagógicos que os diferentes tipos de explicações científicas identificadas se destinaram a atender e observamos uma (inesperada) baixa ocorrência de explicações científicas nos seminários que acompanhamos, acompanhada de uma forte ênfase no treinamento da habilidade de resolução de problemas matemáticos. Utilizamos estes resultados para problematizar questões relacionadas à formação de professores para a educação básica à luz do uso que é feito de explicações científicas. / Scientific explanations are among the most important elements in science teaching. They relate to the notion of nature of science we reproduce, to the better understanding of concepts and to share scientific principles and hypotheses. Although there is no single scientific explanation's definition consensually accept by researchers in this theme, in the last decades it has been addressed in an increasing number of papers published in international journals on science teaching. However, we believe that this issue has received little attention from the research community in science teaching in Brazil, specially about activities related to constructing scientific explanations during teacher training. In this work we investigate elements related to scientific explanations elaborated by future teachers, and the way those elements are related to each others in explanation organization. For this, we propose an organizational structure of scientific explanation and we use it to analyze seminars presented by students of a Brazilian physics' teacher graduation during one of its disciplines, about topics related to mechanics. Our organizational structure was drawn from works done by James Wertsch (2008) on narratives, from which we adapted two structures that Wertsch proposed with respect to narratives for the study of scientific explanation: specific explanations and explanatory schematic templates. We associated specific explanations with explicative moments proposed by Ogborn, Kress, Martins e Mcgillicuddy (1996) and the schematic explanatory templates to explanation types proposed by explanation philosophers – especially the Causal Mechanical model, the Unification approach and the functional explanations. In our empirical analyze we seek to identify teaching goals that the different types of scientific explanations identified intended to meet and We noticed a (unexpected) low occurrence of scientific explanation in the seminars we follow, beside a strong emphasis on education to provide training in mathematical problem solving abilities. We used those results to discuss issues related to training teachers to basic education in light of the way scientific explanations are used.

Neste trabalho, propomos um conjunto de estratégias para enfrentar os obstáculos epistemológicos à aprendizagem da cinemática na Teoria da Relatividade Restrita. Tais estratégias são baseadas na proposta da noção de perfil conceitual de referencial (Ayala & Frezza, 2007) e visam promover uma interação cognitiva entre a Mecânica Newtoniana e Relativística. O perfil conceitual (Mortimer, 1995-2000) é um modelo para compreender a evolução conceitual em sala de aula. Para tais objetivos, foi desenvolvido um conjunto de ações em uma seqüência de aplicações de pré-testes e pós-testes com a finalidade de explicitar a concepção dos alunos sobre o aspecto a ser investigado e avaliar a evolução conceitual destes. A diferenciação entre as concepções dos alunos e aquelas cientificamente aceitas deu-se através da interpretação das concepções apresentadas, de discussões teóricas acerca do tema e da apresentação de animações construídas com o software Modellus. O público-alvo deste trabalho é formado por alunos de Licenciatura em Física, e o objetivo do mesmo é contribuir para a formação destes estudantes, futuros professores de Física. / In this work, we propose a set of strategies to face the epistemological obstacles to learning the kinematics in the Restricted Theory of Relativity. Such strategies are based on the proposal of the notion of conceptual profile of reference (Ayala & Frezza, 2007) and promote an interaction between cognitive Newtonian Mechanics and Relativistic Mechanics. The conceptual profile (Mortimer, 1995-2000) is a model for understanding the conceptual evolution of students in classroom. For these goals, we developed a set of actions in a sequence of applications of pre-tests and post-tests in order to explain the conception of students on the aspect to be investigated, and to assess your conceptual evolution. The differentiation between the students' conceptions and scientifically acceptable ones came through the interpretation of the concepts presented, theoretical discussions about the topic and the presentation of animations built with the software Modellus. The target of this work is formed by graduation students in Physics in order to contribute to your formation as future physics teachers.