QM Lecture 4 (not match)

LECTURE 4 From the HAMILTON VARIATIONAL PRINCIPLE to the HAMILTON JACOBI EQUATION (File updated on 10-07-2015)

4.1. The Lagrange formulation and the Hamilton’s variational principle

4.1A Specification of the state of motion

4.1B Time evolution of a classical state: Hamilton’s variational principle

Definition of the classical action

The variational principle leads to the Newton’s Law

The Lagrange equation of motion obtained from the variational principle

Example: The Lagrangian for a particle in an electromagnetic field

4.1C Constants of motion

Cyclic coordinates and the conservation of the generalized momentum

Lagrangian independent of time and the conservation of the Hamiltonian

Case of a potential independent of the velocities: Hamiltonian is the mechanical energy

4.2 The Hamilton formulation of mechanics

4.2A Legendre transformation

4.2B The Hamilton Equations of Motion

i) Recipe for solving problems in mechanics

ii)Properties of the Hamiltonian

4.2C Finding constant of motion before calculating the motion itself

Looking for functions whose Poisson bracket with the Hamiltonian vanishes

Cyclic coordinates

4.2D The modified Hamilton’s principle: Derivation of the Hamilton’s equations from a variational principle

4.3 The Poisson bracket

4.3A Hamiltonian equations in terms of the Poisson brackets

4.3B Fundamental brackets

4.3C The Poisson bracket theorem: Preserving the description of the classical motion in terms of a Hamiltonian

a) Example of motion described by no Hamiltonian

b) Change of coordinates to attain a Hamiltonian description

4.4 Canonical Transformations

4.4A Canonoid transformations (i.e. not quite canonical)

Preservation of the canonical equations with respect to a particular Hamiltonian)

4.4B Canonical transformations

Definition

Canonical transformation theorem

Canonical transformation and the invariance of the Poisson bracket

4.4C Restricted canonical transformation

4.5 How to generate (restricted) canonical transformations

4.5A Generating function of transformations

4.5B Classification of (restricted) canonical transformations

4.5C Time evolution of a mechanics state viewed as series of canonical transformations

i) The generator of the identity transformation

ii) Infinitesimal transformations

iii) The Hamiltonian as a generating function of canonical transformations

iv) Time evolution of a mechanical state viewed as a canonical transformation

4.6 Universality of the Lagrangian

4.6A Invariant of the Lagrangian equation with respect to the configuration space coordinates

4.6B The Lagrangian equation as an invariant operator

4.7 The Hamilton Jacobi equation

4.7A The Hamilton principal function

4.7B Further physical significance of the Hamilton principal function