Motion of Nonholonomous Rheonomous Systems in the Lagrangian Formalism

Abstract

The main purpose of the paper consists in illustrating a procedure for expressing the equations of motion for a general time-dependent constrained system. Constraints are both of geometrical and differential type. The use of quasi-velocities as variables of the mathematical problem opens the possibility of incorporating some remarkable and classic cases of equations of motion. Afterwards, the scheme of equations is implemented for a pair of substantial examples, which are presented in a double version, acting either as a scleronomic system and as a rheonomic system.

An extended Lagrangian formalism

“…An interesting further development will be the investigation of the expressions (2), for h = 1, especially with regard to their bearing on real physical systems. On the other hand, the context is well suited for letting more the constraint equations more general, embracing linear and nonlinear holonomic restrictions, in order to develop and complete the starting results formulated in [8].…”

An algebraic procedure for reducing the Boltzmann-Hamel equations in nonholonomic systems

“…If (B) is the case, (11) cannot be solved separately from (6), since all the q generally appear in it.…”

“…In this example = 6 and the choice of the Lagrangian coordinates is q = (x C , y C , φ, ψ, θ, θ 1 ), where θ 1 is the angle that P r −O 1 forms with the downward vertical direction. As discussed in [6], the Lagrangian function includes U = −κ 1 g cos θ + m d gρ cos θ 1 , with κ 1 = m d R + m f ρ 1 + m r (ρ 1 + ρ 2 ), and T with matrix A, whose main diagonal and upper triangular part are…”

“…Equations (11) are joined to (6), in order to form a system of σ + equations for the σ + unknown quantities η and q. System (11) + (6) can be written in normal form, since A is a positive-definite square matrix and rank Γ = σ, so that even A Γ = Γ T AΓ is a positive-definite σ × σ symmetrical matrix.…”