Holonomic constraints : an analytical result

TL;DR

This paper derives explicit analytical formulas for Lagrange multipliers in systems with linear holonomic constraints, simplifying the computation of constraint forces in complex mechanical systems.

Contribution

It provides the first explicit analytical solutions for Lagrange multipliers in systems with multiple linear holonomic constraints, enhancing computational efficiency.

Findings

Explicit formulas for 1 to 5 bonds provided

Analytical approach simplifies constraint force calculation

Facilitates simulation of complex constrained systems

Abstract

Systems subjected to holonomic constraints follow quite complicated dynamics that could not be described easily with Hamiltonian or Lagrangian dynamics. The influence of holonomic constraints in equations of motions is taken into account by using Lagrange multipliers. Finding the value of the Lagrange multipliers allows to compute the forces induced by the constraints and therefore, to integrate the equations of motions of the system. Computing analytically the Lagrange multipliers for a constrained system may be a difficult task that is depending on the complexity of systems. For complex systems, it is most of the time impossible to achieve. In computer simulations, some algorithms using iterative procedures estimate numerically Lagrange multipliers or constraint forces by correcting the unconstrained trajectory. In this work, we provide an analytical computation of the Lagrange multipliers for a set of linear holonomic constraints with an arbitrary number of bonds of constant length. In the appendix of the paper, one would find explicit formulas for Lagrange multipliers for systems having 1, 2, 3, 4 and 5 bonds of constant length, linearly connected.