PART 2: Extended Lagrange and Hamilton Formalisms for Point Mechanics

The following sections are included:

• Chapter 4 Extended Lagrange Formalism

  • The extended Lagrangian

  • Extended set of Euler-Lagrange equations

  • Correlation between L and L_e

    • Trivial extended Lagrangian

    • Non-trivial extended Lagrangian

  • Noether’s theorem in the extended Lagrangian formalism

  • Excursus: Relativistic path integral with extended Lagrangians

• Chapter 5 The Extended Hamiltonian in Point Mechanics

  • Extended Hamiltonian from the generalized action functional

  • Extended Hamiltonian as Legendre transform of an extended Lagrangian

  • Extended set of canonical equations

  • Canonical quantization in the extended Hamiltonian formalism

• Chapter 6 Theory of Extended Canonical Transformations

  • Extended canonical transformations

  • Infinitesimal extended canonical transformations, generalized Noether theorem

  • Conventional Noether theorem

  • Energy-second-moment map

  • Liouville’s theorem in the extended Hamilton description

  • Galilei and Lorentz transformation as extended canonical transformations

    • The Galilei transformation

    • The Lorentz transformation

• Canonical invariance of extended Poisson brackets

• Extended Hamilton-Jacobi equation