1. The document presents a new method for finding complete observables in classical mechanics, which are gauge invariant quantities.
2. The method starts with partial observables and clocks, which are non-gauge invariant phase space functions. Using constants of motion, the partial observables can be written in terms of the clocks to obtain complete observables.
3. As an example, the method is applied to a particle in a gravitational field, where the Hamiltonian is used as a constant of motion to write the position variable as a function of the momentum and time.
This document contains Homer Reid's solutions to problems from Goldstein's Classical Mechanics textbook. The first problem solved involves a radioactive nucleus decaying and emitting an electron and neutrino. The solution finds the direction and momentum of the recoiling nucleus. The next problems solved include deriving the escape velocity of Earth, the equation of motion for a rocket, relating kinetic energy to momentum for varying mass systems, and several other kinematic and constraint problems.
1. The document discusses various kinematics problems involving motion under uniform acceleration. It provides solutions using graphical, analytical and vector methods.
2. Methods include calculating time taken to cross a river based on velocities and angles, determining average and instantaneous velocities from distance-time graphs, resolving velocities into components, and finding the distance between particles moving with different initial velocities.
3. One problem involves three particles moving in a circle such that they are always at the vertices of an equilateral triangle, and calculates the distance traveled by one particle before they meet.
Aurelian Isar - Decoherence And Transition From Quantum To Classical In Open ...SEENET-MTP
Lecture by Prof. Dr. Aurelian Isar (Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania) on October 20, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
This document discusses the emergence of chimera states, a unique collective state where coherent and incoherent dynamics coexist, in a system of non-locally coupled phase oscillators with propagation delays. It presents the complex Ginzburg-Landau equation (CGLE) model of reaction-diffusion systems and derives a phase reduction with propagation time delays. Numerical simulations and solutions to the self-consistency equation validate the existence of chimera clusters induced by time delays.
This document discusses the emergence of chimera states, a unique collective state where coherent and incoherent dynamics coexist, in a system of non-locally coupled phase oscillators with propagation delays. It presents the complex Ginzburg-Landau equation (CGLE) model of reaction-diffusion systems and derives a phase reduction with propagation time delays. Numerical simulations and solutions to the self-consistency equation reveal chimera cluster states induced by the time delays.
This document provides an overview of queueing theory and Markov processes. It begins with a review of Poisson processes and their properties. It then introduces discrete-time and continuous-time Markov processes. Key concepts covered include state transition probabilities, n-step transition matrices, and the Chapman-Kolmogorov equations. Examples are provided to illustrate Markov chains and how to calculate limiting and stationary distributions. The document concludes with definitions of state classes in Markov processes.
The document discusses the calculus of variations and derives Euler's equation. It explains that the fundamental problem is to find an unknown function that makes a functional (function of a function) an extremum. Euler's equation provides the necessary condition for a functional to be extremized. As an example, it derives the Euler-Lagrange equation for the shortest distance between two points in Cartesian coordinates, showing the path is a straight line.
This document summarizes research on quantum chaos, including the principle of uniform semiclassical condensation of Wigner functions, spectral statistics in mixed systems, and dynamical localization of chaotic eigenstates. It discusses how in the semiclassical limit, Wigner functions condense uniformly on classical invariant components. For mixed systems, the spectrum can be seen as a superposition of regular and chaotic level sequences. Localization effects can be observed if the Heisenberg time is shorter than the classical diffusion time. The document presents an analytical formula called BRB that describes the transition between Poisson and random matrix statistics. An example is given of applying this to analyze the level spacing distribution for a billiard system.
This document contains Homer Reid's solutions to problems from Goldstein's Classical Mechanics textbook. The first problem solved involves a radioactive nucleus decaying and emitting an electron and neutrino. The solution finds the direction and momentum of the recoiling nucleus. The next problems solved include deriving the escape velocity of Earth, the equation of motion for a rocket, relating kinetic energy to momentum for varying mass systems, and several other kinematic and constraint problems.
1. The document discusses various kinematics problems involving motion under uniform acceleration. It provides solutions using graphical, analytical and vector methods.
2. Methods include calculating time taken to cross a river based on velocities and angles, determining average and instantaneous velocities from distance-time graphs, resolving velocities into components, and finding the distance between particles moving with different initial velocities.
3. One problem involves three particles moving in a circle such that they are always at the vertices of an equilateral triangle, and calculates the distance traveled by one particle before they meet.
Aurelian Isar - Decoherence And Transition From Quantum To Classical In Open ...SEENET-MTP
Lecture by Prof. Dr. Aurelian Isar (Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania) on October 20, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
This document discusses the emergence of chimera states, a unique collective state where coherent and incoherent dynamics coexist, in a system of non-locally coupled phase oscillators with propagation delays. It presents the complex Ginzburg-Landau equation (CGLE) model of reaction-diffusion systems and derives a phase reduction with propagation time delays. Numerical simulations and solutions to the self-consistency equation validate the existence of chimera clusters induced by time delays.
This document discusses the emergence of chimera states, a unique collective state where coherent and incoherent dynamics coexist, in a system of non-locally coupled phase oscillators with propagation delays. It presents the complex Ginzburg-Landau equation (CGLE) model of reaction-diffusion systems and derives a phase reduction with propagation time delays. Numerical simulations and solutions to the self-consistency equation reveal chimera cluster states induced by the time delays.
This document provides an overview of queueing theory and Markov processes. It begins with a review of Poisson processes and their properties. It then introduces discrete-time and continuous-time Markov processes. Key concepts covered include state transition probabilities, n-step transition matrices, and the Chapman-Kolmogorov equations. Examples are provided to illustrate Markov chains and how to calculate limiting and stationary distributions. The document concludes with definitions of state classes in Markov processes.
The document discusses the calculus of variations and derives Euler's equation. It explains that the fundamental problem is to find an unknown function that makes a functional (function of a function) an extremum. Euler's equation provides the necessary condition for a functional to be extremized. As an example, it derives the Euler-Lagrange equation for the shortest distance between two points in Cartesian coordinates, showing the path is a straight line.
This document summarizes research on quantum chaos, including the principle of uniform semiclassical condensation of Wigner functions, spectral statistics in mixed systems, and dynamical localization of chaotic eigenstates. It discusses how in the semiclassical limit, Wigner functions condense uniformly on classical invariant components. For mixed systems, the spectrum can be seen as a superposition of regular and chaotic level sequences. Localization effects can be observed if the Heisenberg time is shorter than the classical diffusion time. The document presents an analytical formula called BRB that describes the transition between Poisson and random matrix statistics. An example is given of applying this to analyze the level spacing distribution for a billiard system.
This document provides an outline for a lecture on complex dynamics in Hamiltonian systems. Some key points:
1) Simple periodic orbits called nonlinear normal modes exist and can destabilize, leading to weak or strong chaos depending on their properties.
2) Dynamical indicators like Lyapunov exponents and the Generalized Alignment Index (GALI) can identify regions of order and chaos. The Lyapunov spectrum indicates when orbits explore the same chaotic region.
3) GALI rapidly detects chaos as deviation vectors become aligned, and identifies quasiperiodic motion by vectors remaining independent. It distinguishes weak and strong chaos based on exponential decay rates.
1) The document discusses exact solutions of nonequilibrium steady states in many-body quantum systems using the matrix product ansatz.
2) It presents an exactly solvable model of a boundary driven XXZ spin chain, where the nonequilibrium steady state can be represented as a matrix product operator with a quadratic algebra.
3) Transport properties like the spin conductivity can be computed exactly using conservation laws that emerge from the integrability of the model.
Fourier Series for Continuous Time & Discrete Time SignalsJayanshu Gundaniya
- Fourier introduced Fourier series in 1807 to solve the heat equation in a metal plate. The heat equation is a partial differential equation describing the distribution of heat in a body over time.
- Prior to Fourier's work, there was no known solution to the heat equation in the general case. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves.
- This superposition or linear combination of sine and cosine waves is called the Fourier series. It allows any periodic function to be decomposed into the sum of simple oscillating functions. Although originally introduced for heat problems, Fourier series have wide applications in mathematics and physics.
Geometric and viscosity solutions for the Cauchy problem of first orderJuliho Castillo
This document summarizes a doctoral dissertation on geometric and viscosity solutions to first order Cauchy problems. It introduces two types of solutions - viscosity solutions and minimax solutions - which are generally different. The aim is to show that iterating the minimax procedure over shorter time intervals approaches the viscosity solution. This extends previous work relating geometric and viscosity solutions in the symplectic case. The document outlines characteristics methods, generating families, Clarke calculus tools, and a proof constructing generating families to relate iterated minimax solutions to viscosity solutions.
The document discusses the Schrodinger equation and its applications in quantum mechanics. It covers:
1. The postulates of quantum mechanics including that systems are described by wavefunctions and observables are represented by Hermitian operators.
2. Examples of operators for observables like position, momentum, energy.
3. The time-independent Schrodinger equation for a time-independent potential and its solution for an infinite square well potential.
4. Other examples like an infinite square well potential trapping an electron and calculating its energy levels and wavefunctions.
This document discusses nonlocal cosmology and modifications to Einstein's theory of gravity. It presents three cases of nonlocal modified gravity models:
1. When P(R)=R and Q(R)=R, nonsingular bounce cosmological solutions were found with scale factor a(t)=a0(σeλt+τe-λt).
2. When P(R)=R-1 and Q(R)=R, several power-law cosmological solutions were obtained, including a(t)=a0|t-t0|α.
3. For the case P(R)=Rp and Q(R)=Rq, the trace and 00 equations of motion were transformed into an equivalent
Serway, raymond a physics for scientists and engineers (6e) solutionsTatiani Andressa
This document contains a chapter outline and sample questions and solutions for a physics and measurement chapter. The chapter outline lists topics like standards of length, mass and time, density and atomic mass, and dimensional analysis. The questions and solutions provide examples of calculations involving converting between units, determining densities, and applying dimensional analysis.
1. This document contains solutions to problems from Chapter 1 of the textbook "The Physics of Vibrations and Waves".
2. It provides detailed calculations and explanations for problems related to simple harmonic motion, including determining restoring forces, stiffness, frequencies, and solving differential equations of motion.
3. Examples include a simple pendulum, a mass on a spring, and vibrations of strings, membranes, and gas columns.
This document discusses dynamical systems. It defines a dynamical system as a system that changes over time according to fixed rules determining how its state changes from one time to the next. It then covers:
- The two parts of a dynamical system: state space and function determining next state.
- Classification of systems as deterministic/stochastic, discrete/continuous, linear/nonlinear, and autonomous/nonautonomous.
- Examples of discrete and continuous models, differential equations, and linear vs nonlinear systems.
- Terminology including phase space, phase curve, phase portrait, and attractors.
- Analysis methods including fixed points, stability, and perturbation analysis.
- Examples of harmonic oscillator,
This document provides an introduction and overview of quantum Monte Carlo methods. It begins by reviewing the Metropolis algorithm and how it can be used to evaluate integrals and quantum mechanical operators. It then outlines the key topics which will be covered, including the path integral formulation of quantum mechanics, diffusion Monte Carlo, and calculating the one-body density matrix and excitation energies. The document proceeds to explain how the path integral formulation leads to the Schrodinger equation in the limit of small time steps, and how imaginary time evolution can be used to project out the ground state wavefunction. It concludes by providing examples of applying these methods to calculate properties of hydrogen, molecular hydrogen, and the one-body density matrix of silicon.
This document provides an overview of classical and quantum mechanics concepts relevant to statistical thermodynamics. It begins with an introduction to classical mechanics using Lagrangian and Hamiltonian formulations. It then discusses limitations of classical mechanics and introduces key concepts in quantum mechanics, including the Schrodinger equation. Examples are provided of a particle in a box and harmonic oscillator to illustrate differences between classical and quantum descriptions of particle behavior.
1) Markov models and hidden Markov models describe systems that transition between states based on probabilities, where the next state depends only on the current state.
2) Markov models assume each state corresponds to a directly observable event, while hidden Markov models allow states to be hidden and observations to depend probabilistically on the current state.
3) Transition and initial state probabilities can be described using a transition matrix in Markov models to calculate the probability of state sequences.
This document provides an introduction to Hidden Markov Models (HMMs). It begins by explaining the key differences between Markov Models and HMMs, noting that in HMMs the states are hidden and can only be indirectly observed through observations. It then outlines the main elements of an HMM - the number of states, observations, state transition probabilities, observation probabilities, and initial state distribution. An example HMM is provided. Finally, it briefly introduces three common problems in HMMs - determining the most likely model given observations, determining the most likely state sequence, and determining the model parameters that are most likely to have generated the observations.
The document provides an overview of quantum computation and algorithms. It introduces concepts from quantum physics like quantum states, observables, and measurement. It discusses how classical computation can be done using reversible gates. It also covers quantum gates, universal quantum gate sets, and the quantum complexity class BQP. A key example covered is Grover's search algorithm for searching an unstructured database on a quantum computer. The document aims to give a quick introduction to the foundations and applications of quantum computation.
This document discusses quantum chaos in clean many-body systems. It begins by outlining the topic and noting that quantum chaos fits into many-body physics and statistical mechanics. It then discusses how the quantum chaos conjecture relates semiclassical physics to many-body systems. Specifically, it discusses how quantum ergodicity, decay of correlations, and Loschmidt echo relate to the integrability-breaking phase transition in spin chains. It also briefly mentions how quantum chaos appears in non-equilibrium steady states of open many-body systems.
This document provides an introduction to quantum Monte Carlo methods. It discusses using Monte Carlo integration to evaluate multi-dimensional integrals that arise in quantum mechanical problems. Variational Monte Carlo is introduced as using a trial wavefunction to sample configuration space and estimate observables, like the energy. The Metropolis algorithm is described as a way to generate Markov chains that sample a given probability distribution. This allows using Monte Carlo methods to solve the electronic structure problem by approximating many-body wavefunctions and integrals over configuration space.
The document introduces second order differential equations and their solutions. It defines an initial value problem for a second order equation as consisting of the equation and two initial conditions. Linear equations are introduced, which can be written in standard form or with constant or variable coefficients. The dynamical system formulation converts a second order equation to a system of first order equations. Undamped and damped free vibrations are discussed. Examples are provided, including finding the solution to an initial value problem, and determining the quasi-frequency, quasi-period, and equilibrium crossing time.
This document contains sample problems and questions related to thermodynamic processes and the first law of thermodynamics. It defines key terms like work (w), heat (q), internal energy change (ΔU), and enthalpy change (ΔH) for various thermodynamic processes including isobaric, isochoric, isothermal, reversible adiabatic, and irreversible processes. It then provides examples of calculating w, q, ΔU, and ΔH for gas expansion/compression processes under different conditions. Finally, it includes some multiple choice questions testing understanding of concepts like signs of w and q and properties of closed, open, and isolated systems.
This document presents two theorems about repunit Lehmer numbers. Theorem 1 states that for any fixed base g > 1, there are only finitely many positive integers n such that the repunit number un = (gn - 1)/(g - 1) is a Lehmer number, and these can all be effectively computed. Theorem 2 states that there are no Lehmer numbers of the form un when 2 ≤ g ≤ 1000. The document provides background on Lehmer numbers and repunit numbers, establishes some preliminary results, and gives the proof of Theorem 1 by considering primitive divisors of the repunit numbers.
This chapter introduces discrete and continuous dynamical systems through examples. Discrete examples include rotations and expanding maps of the circle, as well as endomorphisms and automorphisms of the torus. Continuous examples include flows generated by autonomous differential equations. Periodic points are also defined and analyzed for specific examples. Basic constructions for building new dynamical systems from existing ones are described.
This document provides an outline for a lecture on complex dynamics in Hamiltonian systems. Some key points:
1) Simple periodic orbits called nonlinear normal modes exist and can destabilize, leading to weak or strong chaos depending on their properties.
2) Dynamical indicators like Lyapunov exponents and the Generalized Alignment Index (GALI) can identify regions of order and chaos. The Lyapunov spectrum indicates when orbits explore the same chaotic region.
3) GALI rapidly detects chaos as deviation vectors become aligned, and identifies quasiperiodic motion by vectors remaining independent. It distinguishes weak and strong chaos based on exponential decay rates.
1) The document discusses exact solutions of nonequilibrium steady states in many-body quantum systems using the matrix product ansatz.
2) It presents an exactly solvable model of a boundary driven XXZ spin chain, where the nonequilibrium steady state can be represented as a matrix product operator with a quadratic algebra.
3) Transport properties like the spin conductivity can be computed exactly using conservation laws that emerge from the integrability of the model.
Fourier Series for Continuous Time & Discrete Time SignalsJayanshu Gundaniya
- Fourier introduced Fourier series in 1807 to solve the heat equation in a metal plate. The heat equation is a partial differential equation describing the distribution of heat in a body over time.
- Prior to Fourier's work, there was no known solution to the heat equation in the general case. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves.
- This superposition or linear combination of sine and cosine waves is called the Fourier series. It allows any periodic function to be decomposed into the sum of simple oscillating functions. Although originally introduced for heat problems, Fourier series have wide applications in mathematics and physics.
Geometric and viscosity solutions for the Cauchy problem of first orderJuliho Castillo
This document summarizes a doctoral dissertation on geometric and viscosity solutions to first order Cauchy problems. It introduces two types of solutions - viscosity solutions and minimax solutions - which are generally different. The aim is to show that iterating the minimax procedure over shorter time intervals approaches the viscosity solution. This extends previous work relating geometric and viscosity solutions in the symplectic case. The document outlines characteristics methods, generating families, Clarke calculus tools, and a proof constructing generating families to relate iterated minimax solutions to viscosity solutions.
The document discusses the Schrodinger equation and its applications in quantum mechanics. It covers:
1. The postulates of quantum mechanics including that systems are described by wavefunctions and observables are represented by Hermitian operators.
2. Examples of operators for observables like position, momentum, energy.
3. The time-independent Schrodinger equation for a time-independent potential and its solution for an infinite square well potential.
4. Other examples like an infinite square well potential trapping an electron and calculating its energy levels and wavefunctions.
This document discusses nonlocal cosmology and modifications to Einstein's theory of gravity. It presents three cases of nonlocal modified gravity models:
1. When P(R)=R and Q(R)=R, nonsingular bounce cosmological solutions were found with scale factor a(t)=a0(σeλt+τe-λt).
2. When P(R)=R-1 and Q(R)=R, several power-law cosmological solutions were obtained, including a(t)=a0|t-t0|α.
3. For the case P(R)=Rp and Q(R)=Rq, the trace and 00 equations of motion were transformed into an equivalent
Serway, raymond a physics for scientists and engineers (6e) solutionsTatiani Andressa
This document contains a chapter outline and sample questions and solutions for a physics and measurement chapter. The chapter outline lists topics like standards of length, mass and time, density and atomic mass, and dimensional analysis. The questions and solutions provide examples of calculations involving converting between units, determining densities, and applying dimensional analysis.
1. This document contains solutions to problems from Chapter 1 of the textbook "The Physics of Vibrations and Waves".
2. It provides detailed calculations and explanations for problems related to simple harmonic motion, including determining restoring forces, stiffness, frequencies, and solving differential equations of motion.
3. Examples include a simple pendulum, a mass on a spring, and vibrations of strings, membranes, and gas columns.
This document discusses dynamical systems. It defines a dynamical system as a system that changes over time according to fixed rules determining how its state changes from one time to the next. It then covers:
- The two parts of a dynamical system: state space and function determining next state.
- Classification of systems as deterministic/stochastic, discrete/continuous, linear/nonlinear, and autonomous/nonautonomous.
- Examples of discrete and continuous models, differential equations, and linear vs nonlinear systems.
- Terminology including phase space, phase curve, phase portrait, and attractors.
- Analysis methods including fixed points, stability, and perturbation analysis.
- Examples of harmonic oscillator,
This document provides an introduction and overview of quantum Monte Carlo methods. It begins by reviewing the Metropolis algorithm and how it can be used to evaluate integrals and quantum mechanical operators. It then outlines the key topics which will be covered, including the path integral formulation of quantum mechanics, diffusion Monte Carlo, and calculating the one-body density matrix and excitation energies. The document proceeds to explain how the path integral formulation leads to the Schrodinger equation in the limit of small time steps, and how imaginary time evolution can be used to project out the ground state wavefunction. It concludes by providing examples of applying these methods to calculate properties of hydrogen, molecular hydrogen, and the one-body density matrix of silicon.
This document provides an overview of classical and quantum mechanics concepts relevant to statistical thermodynamics. It begins with an introduction to classical mechanics using Lagrangian and Hamiltonian formulations. It then discusses limitations of classical mechanics and introduces key concepts in quantum mechanics, including the Schrodinger equation. Examples are provided of a particle in a box and harmonic oscillator to illustrate differences between classical and quantum descriptions of particle behavior.
1) Markov models and hidden Markov models describe systems that transition between states based on probabilities, where the next state depends only on the current state.
2) Markov models assume each state corresponds to a directly observable event, while hidden Markov models allow states to be hidden and observations to depend probabilistically on the current state.
3) Transition and initial state probabilities can be described using a transition matrix in Markov models to calculate the probability of state sequences.
This document provides an introduction to Hidden Markov Models (HMMs). It begins by explaining the key differences between Markov Models and HMMs, noting that in HMMs the states are hidden and can only be indirectly observed through observations. It then outlines the main elements of an HMM - the number of states, observations, state transition probabilities, observation probabilities, and initial state distribution. An example HMM is provided. Finally, it briefly introduces three common problems in HMMs - determining the most likely model given observations, determining the most likely state sequence, and determining the model parameters that are most likely to have generated the observations.
The document provides an overview of quantum computation and algorithms. It introduces concepts from quantum physics like quantum states, observables, and measurement. It discusses how classical computation can be done using reversible gates. It also covers quantum gates, universal quantum gate sets, and the quantum complexity class BQP. A key example covered is Grover's search algorithm for searching an unstructured database on a quantum computer. The document aims to give a quick introduction to the foundations and applications of quantum computation.
This document discusses quantum chaos in clean many-body systems. It begins by outlining the topic and noting that quantum chaos fits into many-body physics and statistical mechanics. It then discusses how the quantum chaos conjecture relates semiclassical physics to many-body systems. Specifically, it discusses how quantum ergodicity, decay of correlations, and Loschmidt echo relate to the integrability-breaking phase transition in spin chains. It also briefly mentions how quantum chaos appears in non-equilibrium steady states of open many-body systems.
This document provides an introduction to quantum Monte Carlo methods. It discusses using Monte Carlo integration to evaluate multi-dimensional integrals that arise in quantum mechanical problems. Variational Monte Carlo is introduced as using a trial wavefunction to sample configuration space and estimate observables, like the energy. The Metropolis algorithm is described as a way to generate Markov chains that sample a given probability distribution. This allows using Monte Carlo methods to solve the electronic structure problem by approximating many-body wavefunctions and integrals over configuration space.
The document introduces second order differential equations and their solutions. It defines an initial value problem for a second order equation as consisting of the equation and two initial conditions. Linear equations are introduced, which can be written in standard form or with constant or variable coefficients. The dynamical system formulation converts a second order equation to a system of first order equations. Undamped and damped free vibrations are discussed. Examples are provided, including finding the solution to an initial value problem, and determining the quasi-frequency, quasi-period, and equilibrium crossing time.
This document contains sample problems and questions related to thermodynamic processes and the first law of thermodynamics. It defines key terms like work (w), heat (q), internal energy change (ΔU), and enthalpy change (ΔH) for various thermodynamic processes including isobaric, isochoric, isothermal, reversible adiabatic, and irreversible processes. It then provides examples of calculating w, q, ΔU, and ΔH for gas expansion/compression processes under different conditions. Finally, it includes some multiple choice questions testing understanding of concepts like signs of w and q and properties of closed, open, and isolated systems.
This document presents two theorems about repunit Lehmer numbers. Theorem 1 states that for any fixed base g > 1, there are only finitely many positive integers n such that the repunit number un = (gn - 1)/(g - 1) is a Lehmer number, and these can all be effectively computed. Theorem 2 states that there are no Lehmer numbers of the form un when 2 ≤ g ≤ 1000. The document provides background on Lehmer numbers and repunit numbers, establishes some preliminary results, and gives the proof of Theorem 1 by considering primitive divisors of the repunit numbers.
This chapter introduces discrete and continuous dynamical systems through examples. Discrete examples include rotations and expanding maps of the circle, as well as endomorphisms and automorphisms of the torus. Continuous examples include flows generated by autonomous differential equations. Periodic points are also defined and analyzed for specific examples. Basic constructions for building new dynamical systems from existing ones are described.
Poster Gauge systems and functions, hermitian operators and clocks as conjuga...vcuesta
This document discusses gauge systems with constraints and complete observables at the quantum level. It examines two cases: a non-relativistic free particle with one constraint and a model with two constraints. For the particle, choosing a clock such that its Poisson bracket with the constraint is one solves problems of the clock not being defined at all times and the operator for a complete observable not being self-adjoint. For the two constraint model, choosing clocks conjugate to the constraints results in self-adjoint operators. The self-adjointness of operators for complete observables in more complex systems like field theory and general relativity requires further study.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Quantum Transitions And Its Evolution For Systems With Canonical And Noncanon...vcuesta
1. The document studies the quantum transitions and time evolution of the phase space coordinates for the one-dimensional harmonic oscillator with both canonical and noncanonical symplectic structures.
2. For the canonical case, the solutions to the classical equations of motion and the quantum transitions between energy levels are obtained. The time evolution of the transition amplitudes between states is also determined.
3. An analogous analysis is performed for the noncanonical case, where modified commutation relations and modified expressions for the creation/annihilation operators are obtained. The quantum transitions and their time evolution are determined.
1) The Gram-Schmidt process is used to transform a basis into an orthogonal basis. It works by making each new basis vector orthogonal to the preceding ones.
2) QR factorization states that any matrix A can be decomposed into A = QR, where Q is orthogonal and R is upper triangular.
3) The QR factorization can be used to solve the least squares problem. The solution is obtained by back substituting into Rx=Q^Tb.
Poster Partial And Complete Observablesguest9fa195
This document discusses a method for finding complete observables, which are gauge invariant quantities, in systems with and without constraints.
In systems without constraints, the method begins with a "clock" (partial observable) and a constant of motion to obtain a complete observable. For systems with constraints, it begins with a set of clocks equal to the number of constraints and another partial observable, obtaining a gauge invariant complete observable.
Examples are provided for a particle in a gravitational field, a model with two constraints, and a field theory model. Different choices of clocks and partial observables are shown to result in different complete observables. The method provides complete observables in terms of initial conditions and constants of motion.
This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
The time scale Fibonacci sequences satisfy the Friedmann-Lema\^itre-Robertson-Walker (FLRW) dynamic equation on time scale, which are an exact solution of Einstein's field equations of general relativity for an expanding homogeneous and isotropic universe. We show that the equations of motion correspond to the one-dimensional motion of a particle of position $F(t)$ in an inverted harmonic potential. For the dynamic equations on time scale describing the Fibonacci numbers $F(t)$, we present the Lagrangian and Hamiltonian formalism. Identifying these with the equations that describe factor scales, we conclude that for a certain granulation, for both the continuous and the discrete universe, we have the same dynamics.
Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshop
Reachability Analysis Control of Non-Linear Dynamical SystemsM Reza Rahmati
The document summarizes an approach for algorithmically synthesizing control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using ellipsoidal calculus. The method uses a first-order Taylor approximation of the nonlinear dynamics combined with a conservative approximation of the Lagrange remainder to transform the system into an affine form. The reachable sets are then over-approximated using ellipsoidal operations, allowing the application of ellipsoidal calculus techniques. An iterative algorithm is proposed to compute stabilizing controllers by solving constrained optimization problems at each step to drive the system to a terminal ellipsoidal set within a finite number of steps while satisfying input constraints.
Reachability Analysis "Control Of Dynamical Non-Linear Systems" M Reza Rahmati
The document summarizes an approach for algorithmically synthesizing control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using ellipsoidal calculus. The method uses a first-order Taylor approximation of the nonlinear dynamics combined with a conservative approximation of the Lagrange remainder to transform the system into an affine form. The reachable sets are then over-approximated using ellipsoidal operations. An iterative algorithm is proposed to compute stabilizing controllers by solving constrained optimization problems to drive the system state into a target ellipsoidal set within a finite number of steps while satisfying input constraints.
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
This document summarizes a research paper about nonexistence results for certain Griesmer codes of dimension 4 over finite fields. It begins by providing background on optimal linear codes and the Griesmer bound. It then presents two theorems: Theorem 2 improves valid ranges for the parameter r in an earlier theorem, and Theorem 3 proves that the Griesmer bound is attained for specific code parameters when q is greater than or equal to 7. The document provides proofs for Theorems 2 and 3 using a geometric method involving partitions of projective spaces and properties of minihypers and arcs.
Similar to Proceedings A Method For Finding Complete Observables In Classical Mechanics (20)
Gauge systems and functions, hermitian operators and clocks as conjugate func...vcuesta
This document summarizes a research article about gauge systems and constraints in physics. It discusses two key problems that can arise: 1) Clocks may not be well-defined over the entire phase space. 2) Quantum operators associated with complete observables may not be self-adjoint. The summary proposes selecting clocks such that their Poisson brackets with constraints are equal to 1. This is shown to solve the two problems for several example systems, including a free particle and a system with two constraints. Clocks and complete observables are constructed for the examples, and it is verified that the operators are self-adjoint.
Generalizations For Cartans Equations And Bianchi Identities For Arbitrary Di...vcuesta
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Topological Field Theories In N Dimensional Spacetimes And Cartans Equations
Proceedings A Method For Finding Complete Observables In Classical Mechanics
1. A method for finding complete observables in classical mechanics
Vladimir Cuesta † and Jos´ David Vergara † †
e
Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de M´ xico, M´ xico
o e e e
Abstract. In the present work a new method for finding complete observables is discussed. In first place is presented the algorithm
for systems without constraints, and in second place the method is exemplified for gauge systems. In the case of systems with first class
constraints we begin with a set of clocks (non gauge invariant quantities) that are equal to the number of constraints and another non gauge
invariant quantity, being all partial observables, and we finish with a gauge invariant quantity or complete observable. The starting point
is to consider a partial observable and a clock or clocks being both functions of the phase space variables, that is a function of the phase
space variables P (q, p) and a clock T (q, p) or clocks T1 (q, p), . . . , Tn (q, p), where n is the number of first class constraint. Later, we
take the equations of motion for the system and we found constants of motion and with the help of these at different times, we can find a
gauge invariant phase space function associated with the partial observable P (q, p) and the set of clock or clocks.
Introduction
Loop Quantum Gravity is one of the most known intents to build a quantum theory of gravity, for making this theory successful,
one must solve the set of constraints that appear in the theory, or equivalently one should find functions that can commute with
all gravity constraints and in this way quantizing the theory using the methods developed in Loop Quantum Gravity.
One outstanding result in Loop Quantum Gravity is that the length, area and volume operator have discrete spectra. However
these mathematical entities are not fully gauge invariant. Following with this line of reasoning we can find inside the scientific
literature three crucial concepts: a partial observable (a non fully gauge invariant quantity), a system of clock or clocks (that in
the case of first class systems this number is equal to the number of first class constraints) and using different procedures we can
obtain a complete observable (a gauge invariant quantity) [1-3]. Different studies have shown that these gauge invariant quantities
could have two kinds of spectra, we mean discrete and continuous spectra [4-7]. However, the main reason to study this problem
is the fact that in Loop Quantum Gravity we must find gauge invariant quantities with respect to all Gravity constraints and then
the following question remains open: Do the spectra of these gauge invariant quantities are discrete or continuous?
In the present section we will show the algorithm (the interested reader can compare this method with the presented in
[5-6]), we need two phase space functions that do not commute with the set of constraints or a hamiltonian (a clock and a partial
observable) for finishing with a phase space function that commute with all the constraints or with the hamiltonian.
Now, the idea is to find for the system constants of motion and define these at two different values of the evolution
parameter, we will take one given at the initial value of the parameter and another to arbitrary value. In this expressions the
complete observable will correspond to write the partial observable in terms of initial values. And the parameter of evolution
will correspond to the clock in terms of initial values. In this way the complete observable will commute with all the constraints
or will be a constant of motion for a system without constraints.
We begin with the set of equations of motion q i = E i (q1 , . . . , qn , p1 , . . . , pn , N1 , . . . , Na ) and pi =
˙ ˙
i
F (q1 , . . . , qn , p1 , . . . , pn , N1 , . . . , Na ) where 2n is the dimension of the phase space and a is the number of first class constraints
and Na are Lagrange multipliers, the second step is to combine the previous equations and to eliminate the dependence on time,
we obtain a set of differential relations ρs (dq1 , . . . , dqn , dp1 , . . . , dpn ) = 0 where s = 2n − a, solving these differential
equations it is possible to obtain 2n − a constants of motion (in the case of systems without constraints the number of constants
of motion are 2n − 1 and probably these are not independent). To finish we choose a set of clocks Ta (q1 , . . . , qn , p1 , . . . , pn )
and a partial observable f (q1 , . . . , qn , p1 , . . . , pn ), now if we take τa = Ta (q1 (t = 0), . . . , qn (t = 0), p1 (t = 0), . . . , pn (t = 0))
and as a complete observable the initial condition F (q1 , . . . , qn , p1 , . . . , pn , τ1 , . . . , τa ) = f (q1 (t = 0), . . . , qn (t = 0), p1 (t =
0), . . . , pn (t = 0)) the dependence lies on the parameters τ1 , . . . , τa and the constants of motion C1 , . . . , C2n−a .
† vladimir.cuesta@nucleares.unam.mx
†† vergara@nucleares.unam.mx
2. A method for finding complete observables in classical mechanics 2
Particle on a gravitational field
We begin with one of the simplest models, a massive particle immerse in a constant gravitational field, the importance of this
system lies into the fact that it has a finite number of degrees of freedom and not constraints, the phase space coordinates are
1 p
(xµ ) = (q, p), with hamiltonian H = 2m p2 − mgq, the equations of motion are q = m and p = mg. In this case a constant of
˙ ˙
motion is the hamiltonian, making the selection of the clock as T = p and as a partial observable f = q one can find the complete
observable in the following way: we take the parameter τ = p0 = p(t = 0) and the complete observable as F = q0 = q(t = 0),
p2 p2 τ2
and using the constant of motion at t = 0 and t arbitrary, we write 2m − mgq = 2m − mgq0 = 2m − mgq0 and solving for q0
0
τ 2 −p2
the complete observable is F1 = q + 2m2 g .
Model with two constraints
The present model has the characteristic of being a system with a finite number of degrees of freedom more precisely has 2
degrees of freedom per point of the phase space. In this case the phase is labeled by the coordinates (xµ ) = (q1 , q2 , q3 , p1 , p2 , p3 )
and the set of constraints is
1 1
D1 = [−(p1 )2 + (p2 )2 + (p3 )2 ], D2 = − [q1 p1 + q2 p2 + q3 p3 ], (1)
2 2
with equations of motion
1 1 1 1 1 1
q1 = −N p1 − M q1 , q2 = N p2 − M q2 , q3 = N p3 − M q3 , p1 = M p1 , p2 = M p2 , p3 = M p3 , (2)
˙ ˙ ˙ ˙ ˙ ˙
2 2 2 2 2 2
and in this case we obtain the constants of motion C1 , C2 , C3 and C4
p2 p3 p1 p2 p2
C1 = , C2 = , C3 = q 3 − p 1 q 2 , C 4 = q 1 p1 + 1 q 3 , (3)
p1 p1 p3 p3
if we take as a partial observable f = q1 and as a pair of clocks T1 = q3 and T2 = p1 , the final complete observable is
p 1 q 1 p3 + q 3 p1
F = − τ1 . (4)
p3 τ2
However, taking T1 = q1 − q2 and T2 = p2 + p2 as partial observable we obtain the complete observable
2 2
1 2
2
−C1 C3 p1 + C1 C4 p1 C3 p2 − 2C1 C3 C4 p2 + C1 C4 p2 + p4 τ1 − C1 p4 τ1
2
1 1
2 2
1 1
2
1
F = ± ,
C1 (p2 − 1)
2
1 C1 (p2 − 1)
2
1
this shows that the set of complete observables depend of the clock in the same way as [5].
sl(2, ) model
The model sl(2, ) is an outstanding model [7] quadratic in the phase space coordinates, it resembles the structure of the
constraints for General Relativity, the model has 2 degrees of freedom per point of the space. In this case the phase space
coordinates are (xµ ) = (u1 , u2 , v 1 , v 2 , p1 , p2 , π 1 , π 2 ) with constraints
1 1
H1 = (p 2 − v 2 ), H2 = (π 2 − u 2 ), D = u· p − v· π, (5)
2 2
whose Poisson bracket between them are {H1 , H2 } = D, {H1 , D} = −2H1 , {H2 , D} = 2H2 , with this set of constraints one
can find the following equations of motion
˙ ˙ ˙
u = N p + λu, v = M π − λv, p = M u − λp, π = N v + λπ, ˙ (6)
after a straiightforward calculation we find the constants of motion
O1 = u1 p2 − p1 u2 , O 2 = u 1 v 1 − p1 π 1 , O 3 = u 1 v 2 − p1 π 2 ,
O4 = u2 v 1 − p2 π 1 , O5 = u 2 v 2 − p2 π 2 , O6 = π 1 v 2 − v 1 π 2 . (7)
1 2 1 1
if we take τ1 = u i , τ2 = u i , τ3 = v i as clocks and f1 = π as a partial observable and following the algorithm presented in
the present work we obtain the complete observable
τ2 (u1 v 1 − p1 π 1 ) − τ1 (u2 v 1 − p2 π 1 )
F1 = , (8)
u 1 p2 − p 1 u 2
3. A method for finding complete observables in classical mechanics 3
For finishing this section we can take the 3 clocks T1 = u1 , T2 = u2 and T3 = v 1 , with partial observable
f2 = A sin(u2 + βπ 2 ), and we can deduce that the complete observable is
O1 + βO2 O4
F2 = A sin τ2 − β τ1 . (9)
O1 O1
An example in field theory
As a final example we show a model of field theory, in this case the original model can be thought as a model with 2
degrees of freedom per point. In this model the coordinates (xµ ) = (e, φ, πe , πφ ) label the phase space that we are working, the
hamiltonian for the theory is (see [8] for instance)
2
eπe πe πφ
H=− + + eV = 0, (10)
2D 2 D
dD(φ)
where D = dφ and V = V (φ). Using D(φ) = Qφ, V (φ) = λ we obtain the equations of motion
2
eπe πφ ˙ πe πe
e=
˙ − + N, φ= N, πe =
˙ − λ N, πφ = 0,
˙ (11)
q2 Q Q 2q2
and we find the following constants of motion
2 2
C1 = πφ , C2 = φ − Q ln(πe − 2λq2 ), C3 = (πe − 2q2 λ)e − 2πφ Qπe , (12)
if we take as clock πe and as partial observable f = e the complete observable that we obtain is
2
(πe − 2q2 λ)e − 2πφ Qπe + 2πφ Qτ
F = . (13)
τ 2 − 2q2 λ
And for finishing the present work we can suppose that T = aφ2 + bπe as a a different clock and if we choose as partial
τ1 −a2 C1
observable f2 = e the result is F = a1 .
Conclusions and perspectives
In the present work we have presented a method for finding complete observables we mean functions of the phase space that
commute with the hamiltonian for systems without constraints or that commute with all the constraints for gauge systems. In the
first case we need a clock and a partial observable for finishing with a complete one or a constant of motion, in the second case we
begin with a set of clocks with the number equal to the number of first class constraints and we finish with a complete observable
or a gauge invariant function. In all the cases we put the complete observables in terms of initial conditions. This method can
illustrate the difficulties found in Loop Quantum Gravity to solve the constraints or to obtain gauge invariant quantities with
respect to the full set of General Relativity constraints.
The procedure presented in this work differs of [6], this algorithm lies in to solve the flow (or a formal serie) of the set
of clocks under the set of first class constraints and inserting the solution into the flow of the partial observable and the final
result is the complete observable. However, the physical interpretation in that article is not clear. In fact, in the present work we
have presented a different procedure and we have interpreted the final result or the complete observable as initial conditions or
equivalently as a function of full gauge invariant quantities and a set of parameters τ1 , . . . , τa .
[1] V. Cuesta, M. Montesinos and J. D. Vergara, Phys. Rev. D 76, 025025 (2008),
[2] C. Rovelli, Class. Quant. Grav. 8 (1991) 297-316,
[3] C. Rovelli, Phys. Rev. D, 65, 124013 (2002),
[4] M. Montesinos, Gen. Rel. Grav. 33, 1 (2001),
[5] B. Dittrich, Gen. Rel. Grav. 39 (2007) 1891-1927,
[6] B. Dittrich and T. Thiemann, J. Math. Phys. 50, 012503 (2009),
[7] M. Montesinos, C. Rovelli and T. Thiemann, Phys. Rev. D, 60 044009 (1999),
[8] T. Banks and M. O’loughlin, Nucl. Phys. B, 362, 3 (1991) 649-664.