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D'Alamebrt and Hamiltonian principles and classical relativity in Lagrangian mechanics

Bozidar Jovanovic

Abstract

In this note we present invariant formulation of the d'Alambert principle and classical time-dependent Lagrangian mechanics with holonomic constraints from the perspective of moving frames.

D'Alamebrt and Hamiltonian principles and classical relativity in Lagrangian mechanics

Abstract

In this note we present invariant formulation of the d'Alambert principle and classical time-dependent Lagrangian mechanics with holonomic constraints from the perspective of moving frames.
Paper Structure (11 sections, 8 theorems, 60 equations)

This paper contains 11 sections, 8 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. D'Alambert principle and dynamics of relative motions
  3. D'Alambert principle and Hamiltonian principle of least action
  4. Dynamics in the moving frames
  5. D'Alambert principle for systems with holonomic constraints
  6. Time-dependent holonomic constraints
  7. External description of the dynamics: d'Alembert principle
  8. Intrinsical formulation of the dynamics
  9. Space-time formulation of Lagrangian mechanics
  10. Space-time and reference frames
  11. Dynamics

Key Result

Proposition 2.1

(i) The angular velocity vector fields are related as follows (ii) The addition of velocities Let $\Gamma(t)$ be a smooth curve on $M$ and $\gamma(t)=g_t(\Gamma(t))$ be the associated curve on $Q$. Then Conversely, for a given curve $\gamma(t)$ and the associated curve $\Gamma(t)=g_t^{-1}(\gamma(t))$, we have

Theorems & Definitions (15)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • ...and 5 more