Hamiltonian Dynamics. Cylindrical Constraint Problem  A particle of mass m is attracted to the origin by a force proportional to the distance.  The.

Slides:

Advertisements

Similar presentations

Angular Quantities Correspondence between linear and rotational quantities:

Advertisements

Sect. 8.2: Cyclic Coordinates & Conservation Theorems

Advanced Molecular Dynamics Velocity scaling Andersen Thermostat Hamiltonian & Lagrangian Appendix A Nose-Hoover thermostat.

Physics 430: Lecture 17 Examples of Lagrange’s Equations

Lecture 9 Hamilton’s Equations conjugate momentum cyclic coordinates Informal derivation Applications/examples 1.

Dynamics of Rotational Motion

Shell Model with residual interactions – mostly 2-particle systems Simple forces, simple physical interpretation.

Lagrangian and Hamiltonian Dynamics

Symmetry. Phase Continuity Phase density is conserved by Liouville’s theorem.  Distribution function D  Points as ensemble members Consider as a fluid.

Central Forces. Two-Body System  Center of mass R  Equal external force on both bodies.  Add to get the CM motion  Subtract for relative motion m2m2.

R F For a central force the position and the force are anti- parallel, so r  F=0. So, angular momentum, L, is constant N is torque Newton II, angular.

Classical Model of Rigid Rotor

Motion near an equilibrium position can be approximated by SHM

Lagrangian. Using the Lagrangian  Identify the degrees of freedom. One generalized coordinate for eachOne generalized coordinate for each Velocities.

Hamiltonian. Generalized Momentum  The momentum can be expressed in terms of the kinetic energy.  A generalized momentum can be defined similarly. Kinetic.

Mechanical Energy and Simple Harmonic Oscillator 8.01 Week 09D

Motion of a mass at the end of a spring Differential equation for simple harmonic oscillation Amplitude, period, frequency and angular frequency Energetics.

Lecture 17 Hydrogenic atom (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made.

Work Let us examine the work done by a torque applied to a system. This is a small amount of the total work done by a torque to move an object a small.

Stanford University Department of Aeronautics and Astronautics Introduction to Symmetry Analysis Brian Cantwell Department of Aeronautics and Astronautics.

Section I: (Chapter 1) Review of Classical Mechanics Newtonian mechanics Coordinate transformations Lagrangian approach Hamiltonian with generalized momenta.

Forces and Newton’s Laws. Forces Forces are ________ (magnitude and direction) Contact forces result from ________ ________ Field forces act ___ __ __________.

Focusing Properties of a Solenoid Magnet

Copyright Kaplan AEC Education, 2005 Dynamics Outline Overview DYNAMICS, p. 193 KINEMATICS OF A PARTICLE, p. 194 Relating Distance, Velocity and the Tangential.

Concluding Remarks about Phys 410 In this course, we have … The physics of small oscillations about stable equilibrium points Driven damped oscillations,

Two-Body Systems.

Concluding Remarks about Phys 410 In this course, we have … The physics of small oscillations about stable equilibrium points Re-visited Newtonian mechanics.

Sect. 3.4: The Virial Theorem Skim discussion. Read details on your own! Many particle system. Positions r i, momenta p i. Bounded. Define G  ∑ i r i.

Gravitation. Gravitational Force and Field Newton proposed that a force of attraction exists between any two masses. This force law applies to point masses.

AP Physics C I.F Oscillations and Gravitation. Kepler’s Three Laws for Planetary Motion.

The Two-Body Problem. The two-body problem The two-body problem: two point objects in 3D interacting with each other (closed system) Interaction between.

Two-Body Central-Force Problems Hasbun Ch 8 Taylor Ch 8 Marion & Thornton Ch 8.

Final review Help sessions scheduled for Dec. 8 and 9, 6:30 pm in MPHY 213 Your hand-written notes allowed No numbers, unless you want a problem with numbers.

In the Hamiltonian Formulation, the generalized coordinate q k & the generalized momentum p k are called Canonically Conjugate quantities. Hamilton’s.

Physics 3210 Week 3 clicker questions. A particle of mass m slides on the outside of a cylinder of radius a. What condition must be satisfied by the force.

Ch. 8: Hamilton Equations of Motion Sect. 8.1: Legendre Transformations Lagrange Eqtns of motion: n degrees of freedom (d/dt)[(∂L/∂q i )] - (∂L/∂q i )

Advanced Molecular Dynamics Velocity scaling Andersen Thermostat Hamiltonian & Lagrangian Appendix A Nose-Hoover thermostat.

D’Alembert’s Principle the sum of the work done by

Physics 3210 Week 2 clicker questions. What is the Lagrangian of a pendulum (mass m, length l)? Assume the potential energy is zero when  is zero. A.

Four-potential of a field Section 16. For a given field, the action is the sum of two terms S = S m + S mf – Free-particle term – Particle-field interaction.

Chapter 15 Oscillations. Periodic motion Periodic (harmonic) motion – self-repeating motion Oscillation – periodic motion in certain direction Period.

Moment Of Inertia.

1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.

Phy 303: Classical Mechanics (2) Chapter 3 Lagrangian and Hamiltonian Mechanics.

Vibrations and Waves Hooke’s Law Elastic Potential Energy Simple Harmonic Motion.

Hamiltonian Mechanics (For Most Cases of Interest) We just saw that, for large classes of problems, the Lagrangian terms can be written (sum on i): L.

The Hamiltonian method

Simple Harmonic Motion Physics is phun!. a) 2.65 rad/s b) m/s 1. a) What is the angular velocity of a Simple Harmonic Oscillator with a period of.

Canonical Equations of Motion -- Hamiltonian Dynamics

Advanced Computer Graphics Spring 2014 K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology.

Central Force Umiatin,M.Si. The aim : to evaluate characteristic of motion under central force field.

Introductory Video: Simple Harmonic Motion Simple Harmonic Motion.

Syllabus Note : Attendance is important because the theory and questions will be explained in the class. II ntroduction. LL agrange’s Equation. SS.

Particle Kinematics Direction of velocity vector is parallel to path Magnitude of velocity vector is distance traveled / time Inertial frame – non accelerating,

A Simple Example Another simple example: An illustration of my point that the Hamiltonian formalism doesn’t help much in solving mechanics problems. In.

Introduction to Lagrangian and Hamiltonian Mechanics

Waves and Oscillations_LP_3_Spring-2017

Focusing Properties of a Solenoid Magnet

Test 2 review Test: 7 pm in 203 MPHY

Classical Mechanics Lagrangian Mechanics.

Hamiltonian Mechanics

Advanced Computer Graphics Spring 2008

Advanced Molecular Dynamics

Physics 319 Classical Mechanics

Oscillations Energies of S.H.M.

Chapter 15 Oscillations 1.

Presentation transcript:

Hamiltonian Dynamics

Cylindrical Constraint Problem  A particle of mass m is attracted to the origin by a force proportional to the distance.  The particle is constrained to move on the surface of a cylinder of radius a.  Use Hamilton’s equations to find the equations of motion.

Lagrangian Form  The generalized coordinates are cylindrical coordinates. Radial term constant for the constraintRadial term constant for the constraint  Hamilton’s method starts with a Lagrangian.

Conjugate Replacement  The conjugate momenta are found and used in place of generalized velocities. Replace in the LagrangianReplace in the Lagrangian Replace in the HamiltonianReplace in the Hamiltonian

Hamilton’s Equations  Hamilton’s equations give the equations of motion. Constant angular momentum Simple harmonic oscillations in z

Electromagnetic Hamiltonian  The Lagrangian can be written in terms of a generalized potential. Both E and B derive from potentials , A. Cartesian coordinates for momentum and velocity  The Hamiltonian is a function of momentum and potentials. , A depend on position and time

Electromagnetic Energy  The Jacobean energy integral equals the kinetic plus electrostatic potential. Kinetic energyKinetic energy Kinetic “momentum”Kinetic “momentum”  The kinetic energy and momentum are related the usual way. But they are not conjugateBut they are not conjugate next