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Hamiltonian Reduction in Affine Principal Bundles

Miguel Ángel Berbel, Marco Castrillón López

Abstract

This paper presents a Hamiltonian reduction procedure for field theories over affine principal bundles introducing a canonical identification to describe the reduced multisymplectic space without the introduction of a connection. The main goal is to provide a Hamiltonian analogue of the Lagrangian reduction theory developed in M. Castrillón López, P. M. Chacón, and P. L. García. J. Geom. Mech., 5(4):399-414, 2013. The core of this work lies in the derivation of this canonical identification, the reduced Hamilton-Cartan equations, and a reduced covariant bracket that describes the dynamics. Finally, this theoretical framework is illustrated with a fundamental example: molecular strands.

Hamiltonian Reduction in Affine Principal Bundles

Abstract

This paper presents a Hamiltonian reduction procedure for field theories over affine principal bundles introducing a canonical identification to describe the reduced multisymplectic space without the introduction of a connection. The main goal is to provide a Hamiltonian analogue of the Lagrangian reduction theory developed in M. Castrillón López, P. M. Chacón, and P. L. García. J. Geom. Mech., 5(4):399-414, 2013. The core of this work lies in the derivation of this canonical identification, the reduced Hamilton-Cartan equations, and a reduced covariant bracket that describes the dynamics. Finally, this theoretical framework is illustrated with a fundamental example: molecular strands.
Paper Structure (6 sections, 8 theorems, 64 equations)

This paper contains 6 sections, 8 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Lagrangian Reduction in Affine Principal Bundles
  3. Multisymplectic and Polysymplectic bundles
  4. Covariant Bracket on Affine Principal Bundles
  5. Reduction
  6. Application to molecular strands

Key Result

Theorem 1

Let $L: J^1(Q \times_M E) \to \mathbb{R}$ be an $K$-invariant Lagrangian and let $l: C(Q) \times_M J^1E \to \mathbb{R}$ be the reduced Lagrangian. For an open set $U \subset M$, with $\bar{U}$ compact, and a section $(s, e): \bar{U} \to Q \times_M E$, define $\sigma: \bar{U} \to C(Q)$ as in the iden

Theorems & Definitions (18)

  • Theorem 1: Lagrangian Reduction
  • Definition 2
  • Definition 3
  • Theorem 4
  • Proposition 5
  • Remark 6
  • Proposition 7
  • proof
  • Definition 8
  • Lemma 9
  • ...and 8 more