5. Lagrangian Mechanics

Classical mechanics can be re-formulated starting principles of calculus of variations, usually referred as analytical mechanics. Under some assumptions, that will be discussed during the derivation, analytical mechanics is equivalent to Newton mechanics.

Lagrange equations - II kind. Lagrange equations of the II kind provides pure equations of motion, in which constraint forces do not appear. Given a system with \(N\) degrees of freedom, that can be described with \(N\) independent generalized coordinates \(\{ q^k \}_{k=1:N}\), Lagrange equations of the II kind are a set of \(N\) \(2^{nd}\) order ODEs in the generalized coordinates.

The equivalence with Newton’s approach to classical mechanics is discussed in detail for different kind of systems (points, rigid bodies,…): starting from Newton’s dynamical equations of motion (strong form), D’Alembert approach (weak form) is derived, and Lagrange equations are derived from that with a proper choice of test functions. Lagrange equations are then recast as the stationariety condition of a functional, providing a variational approach to classical mechanics.

Tip

Lagrange equations of the II kind could be the best approach for small-dimensional problems, when there’s no interest in evaluating constraint reactions.

Lagrange equations - I kind. Lagrange equations of the I kind provides a set of DAEs, explicitly including constraints as independent equations and adding their effects - their constraint reactions - in the dynamical equations of the degrees of freedom as Lagrange multipliers.

Tip

Lagrange equations of the I kind could be the best approach for numerical approach to generic mechanical systems, as it is a less problem-dependent approach, without any case-dependent manipulation.

Lagrangian mechanics, with expliicitly time-dependent Lagrangian function. Energy conservation is related to explicit time independent of its Lagrangian function, and can be related - beside the absence of non-conservative actions - to the absence of any time-dependent external input, or the choice of an inertial reference frame. Consequence of explicit time dependence in the Lagrangian functions are discussed, with examples.

Constraints in Lagrangian mechanics. Constraints in Lagrangian mechanics are discussed: holonomic and non-holonomic are discussed, and an approach to ideal non-holonomic (semi-linear?) constraints in Lagrange mechanics is outlined.

Properties. Some properties of Lagrangian approach to mechanics are discussed. As an example, Lagrange mechanics provides a symmetric form of the (linearised?) governing equations, without any additional effort. This could be quite useful, especially for exploiting numerical methods for symmetric (and definite positive, sometimes) matrices.