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Notes on Lagrangian continuum mechanics

V. M. Jiménez

Abstract

In Continuum Mechanic a simple material body $\mathcal{B}$ is represeted by a three-dimensional differentiable manifold and the configuration space is given by the space of embeddings $Emb \left( \mathcal{B} , \mathbb{R}^{n} \right)$. We use the topology of infinite-dimensional manifold of this space, to present the first variation formula for Lagrangian mechanics.

Notes on Lagrangian continuum mechanics

Abstract

In Continuum Mechanic a simple material body is represeted by a three-dimensional differentiable manifold and the configuration space is given by the space of embeddings . We use the topology of infinite-dimensional manifold of this space, to present the first variation formula for Lagrangian mechanics.
Paper Structure (3 sections, 4 theorems, 50 equations)

This paper contains 3 sections, 4 theorems, 50 equations.

Table of Contents

  1. Manifold of differentiable maps. Embeddings
  2. Lagrangian mechanics
  3. Example

Key Result

Lemma 1.1

Let $M,N$ be (finite dimensional) manifolds. Then $J^{\infty} \left( M , N \right)$ is a second contable metrizable manifold.

Theorems & Definitions (8)

  • Lemma 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: First variation formula
  • Remark 3.1