The Lagrange-D'Alembert Principle from the Viewpoint of ODE

TL;DR

The paper presents a rigorous covariant formulation of the Lagrange-D'Alembert principle within an ODE framework, introducing constrained dynamics on an extended phase space and expressing motion through $ [\mathscr T] = f + N$ with $ [\mathscr T] = \frac{d}{dt}\frac{\partial \mathscr T}{\partial \dot x} - \frac{\partial \mathscr T}{\partial x}$ under $ \varphi(t,x,\dot x)=0 $. It proves that reaction forces $N$ and the space of virtual displacements depend only on the geometry of the constraint manifold $S=\{\varphi=0\}$, independent of the analytic representation of the constraints, and provides explicit multiplier expressions for ideal constraints. A pullback formalism to the reduced configuration space $Y$ yields generalized forces $Q$ and recovers the classical Lagrangian form $L=T-V$, with $T$ the kinetic energy and $V$ the potential, enabling treatment of holonomic and non-holonomic cases within a unified variational framework. Overall, the work strengthens the mathematical foundations of constrained dynamics in ODEs and offers a robust, geometrically grounded approach for ideal constraint systems with generalized potentials.

Abstract

We formulate the Lagrange-D'Alembert principle as a pure mathematical theory that meets modern standards of rigor. While we note several new aspects of the principle, the article is primarily methodological.