Presentation is loading. Please wait.

Presentation is loading. Please wait.

Hamilton-Jacobi. Time-Dependent Generator  A generator determines a canonical transformation. The transform generally changes the form of H.The transform.

Published by Modified over 9 years ago

Similar presentations

Presentation on theme: "Hamilton-Jacobi. Time-Dependent Generator  A generator determines a canonical transformation. The transform generally changes the form of H.The transform."— Presentation transcript:

1 Hamilton-Jacobi

2 Time-Dependent Generator  A generator determines a canonical transformation. The transform generally changes the form of H.The transform generally changes the form of H. If time-dependent, the value of H also changes.If time-dependent, the value of H also changes.

3 Independent Hamiltonian  The easiest integration for H is if H is independent of all variables. Select  to give that resultSelect  to give that result  Coordinates and momenta are constants of the motion. 2f constants2f constants

4 Hamilton-Jacobi Equation  Hamilton-Jacobi is a partial differential equation. First order, generally second degreeFirst order, generally second degree f+1 independent variables: q j, tf+1 independent variables: q j, t f+1 constants, one additive, others are  k.f+1 constants, one additive, others are  k.

5 Principal Function  The Lagrangian is directly related to the generator. Generator  is Hamilton’s principal functionGenerator  is Hamilton’s principal function since setting one set at t and the other at t 1 :

6 Principal Function and Action  The action is defined when H does not involve time.  is the principal function  is the principal function Additive constant is possibleAdditive constant is possible  Subsitute to simplify HJ equation. Time-independent HTime-independent H Time variable separatesTime variable separates E is not independentE is not independent

7 Path Equations  Choose energy symmetrically Simplifies action relations Gives f parameterized equations  One can pick one for E. Path described without time One equation for location on the path next

Download ppt "Hamilton-Jacobi. Time-Dependent Generator  A generator determines a canonical transformation. The transform generally changes the form of H.The transform."

Similar presentations

Indefinite Integrals 6.1. Integration - Antidifferentiation - method of solution to a differential equation INdefinite Integral Integration Symbol Variable.

Indefinite Integrals 6.1. Integration - Antidifferentiation - method of solution to a differential equation INdefinite Integral Integration Symbol Variable.
Eric Prebys, FNAL.  We have focused largely on a kinematics based approach to beam dynamics.  Most people find it more intuitive, at least when first.

Eric Prebys, FNAL.  We have focused largely on a kinematics based approach to beam dynamics.  Most people find it more intuitive, at least when first.
Advanced Molecular Dynamics Velocity scaling Andersen Thermostat Hamiltonian & Lagrangian Appendix A Nose-Hoover thermostat.

Advanced Molecular Dynamics Velocity scaling Andersen Thermostat Hamiltonian & Lagrangian Appendix A Nose-Hoover thermostat.
Poisson Brackets. Matrix Form  The dynamic variables can be assigned to a single set. q 1, q 2, …, q n, p 1, p 2, …, p nq 1, q 2, …, q n, p 1, p 2, …,

Poisson Brackets. Matrix Form  The dynamic variables can be assigned to a single set. q 1, q 2, …, q n, p 1, p 2, …, p nq 1, q 2, …, q n, p 1, p 2, …,
Hamiltonian Formalism

Hamiltonian Formalism
9/25/2013PHY 711 Fall 2013 -- Lecture 131 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 13: Finish reading.

9/25/2013PHY 711 Fall Lecture 131 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 13: Finish reading.
The equipartition theorem: a classical but sometimes useful result Photons* and Planck’s black body radiation law c V -> 3R Dulong-Petit limit Example.

The equipartition theorem: a classical but sometimes useful result Photons* and Planck’s black body radiation law c V -> 3R Dulong-Petit limit Example.
Lagrangian and Hamiltonian Dynamics

Lagrangian and Hamiltonian Dynamics
Work. Generalized Coordinates  Transformation rules define alternate sets of coordinates. 3N Cartesian coordinates x i f generalized coordinates q m.

Work. Generalized Coordinates  Transformation rules define alternate sets of coordinates. 3N Cartesian coordinates x i f generalized coordinates q m.
Symmetry. Phase Continuity Phase density is conserved by Liouville’s theorem.  Distribution function D  Points as ensemble members Consider as a fluid.

Symmetry. Phase Continuity Phase density is conserved by Liouville’s theorem.  Distribution function D  Points as ensemble members Consider as a fluid.
Action-Angle. 1-Dimensional CT  Define a 1-D generator S’. Time-independent H.Time-independent H. Require new conjugate variables to be constants of.

Action-Angle. 1-Dimensional CT  Define a 1-D generator S’. Time-independent H.Time-independent H. Require new conjugate variables to be constants of.
Mechanics.

Mechanics.
Generators. Phase Space  A contact transformation acts on a Hamiltonian. Transform on T* Q.Transform on T* Q.  The function  does not always give a.

Generators. Phase Space  A contact transformation acts on a Hamiltonian. Transform on T* Q.Transform on T* Q.  The function  does not always give a.
Transformations.

Transformations.
Canonical Transformations and Liouville’s Theorem

Canonical Transformations and Liouville’s Theorem
Separability. Prinicipal Function  In some cases Hamilton’s principal function can be separated. Each W depends on only one coordinate.Each W depends.

Separability. Prinicipal Function  In some cases Hamilton’s principal function can be separated. Each W depends on only one coordinate.Each W depends.
Hamiltonian Dynamics. Cylindrical Constraint Problem  A particle of mass m is attracted to the origin by a force proportional to the distance.  The.

Hamiltonian Dynamics. Cylindrical Constraint Problem  A particle of mass m is attracted to the origin by a force proportional to the distance.  The.
Advanced Celestial Mechanics. Questions Question 1 Explain in main outline how statistical distributions of binary energy and eccentricity are derived.

Advanced Celestial Mechanics. Questions Question 1 Explain in main outline how statistical distributions of binary energy and eccentricity are derived.
Hamiltonian. Generalized Momentum  The generalized momentum was defined from the Lagrangian.  Euler-Lagrange equations can be written in terms of p.

Hamiltonian. Generalized Momentum  The generalized momentum was defined from the Lagrangian.  Euler-Lagrange equations can be written in terms of p.
Noether. Generalized Momentum  Variables q, q’ are not functionally independent.  The Lagrangian provides canonically conjugate variable. generalized.

Noether. Generalized Momentum  Variables q, q’ are not functionally independent.  The Lagrangian provides canonically conjugate variable. generalized.

Similar presentations

Ads by Google