Lagrange’s Equations

A.1 Introduction

When we wish to use Newton’s laws to write the equations of motion of a particle or a system of particles, we must be careful to include all the forces of the system. The Lagrangean form of the equations of motion that we derive herein has the advantage that we can ignore all forces that do no work (e. g., forces at frictionless pins, forces at a point of rolling contact, forces at frictionless guides, and forces in inextensible connections). In the case of conservative systems (i. e., systems for which the total energy remains constant), the Lagrangean method gives us an automatic procedure for obtaining the equations of motion provided only that we can write the kinetic and potential energies of the system.

A.2 Degrees of Freedom

Before proceeding to develop the Lagrange equations, we must characterize our dynamical systems in a systematic way. The most important property of this sort for our present purpose is the number of independent coordinates that we must know to completely specify the position or configuration of our system. We say that a system has n degrees of freedom if exactly n coordinates serve to completely define its configuration.

example 1 A free particle in space has three degrees of freedom because we must know three coordinates—x, y, z, for example – to locate it.

example 2 A wheel that rolls without slipping on a straight track has one degree of freedom because either the distance from some base point or the total angle of rotation will enable us to locate it completely.

A.3 Generalized Coordinates

We usually think of coordinates as lengths or angles. However, any set of parameters that enables us to uniquely specify the configuration of the system can serve as coordinates. When we generalize the meaning of the term in this manner, we call these new quantities “generalized coordinates.”

example 3 Consider a bar rotating in a plane about a point O. The angle of rotation with respect to some base line is suggested as an obvious coordinate for specifying the position of the bar. However, the area swept over by the bar would do equally well and therefore could be used as a generalized co­ordinate.

If a system has n degrees of freedom, then n generalized coordinates are neces­sary and sufficient to determine the configuration.

A.4 Lagrange’s Equations

In deriving these equations, we consider systems having two degrees of freedom and hence are completely defined by two generalized coordinates q1 and q2. However, the results are easily extended to systems with any number of degrees of freedom.

Suppose our system consists of n particles. For each particle, we can write by Newton’s second law

MiXi = Xi

Miji = Yi (A.1)

MiZi = Zi

where xi, yi, and Zi are the rectangular Cartesian coordinates of the ith particle; Mi is the mass; and Xi, Yi, and Z are the resultants of all forces acting on it in the x, y, and Z directions, respectively.

If we multiply both sides of Eqs. (A.1) by Sxi, Syi, and Szi, respectively, and add the equations, we have

Mi (Xi SXi + yi Syi + Zi Szi) = Xi SXi + Yi Syi + Zi Szi (A.2)

The right-hand side of this equation represents the work done by all of the forces acting on the ith particle during the virtual displacements Sxi, Syi, and Szi. Hence, forces that do no work do not contribute to the right-hand side of Eq. (A.2) and may be omitted from the equation. To obtain the corresponding equation for the entire system, we sum both sides of Eq. (A.2) for all particles. Thus

n n

Mi (xi Sxi + yi Syi + zi Szi) = (Xi Sxi + Yi Syi + Zi Szi) (A.3)

i=1 i=1

Now, because our system is completely located in space if we know the two generalized coordinates q1 and q2, we must be able to write xi, yi, and zi as well as

(A.4)

(A.5)

д Xi д Xi д qi 4 д q2 42 д yi д yi 8yi =— 8qi + — 8q2 д qi дq2 д Zi д Zi 8 Zi =— 8qi + — 8q2 д qi д q2 If we substitute these into Eq. (A.3) and rearrange the terms, we obtain

(A.6)

(A.7)

[1] Presently, Dr. Patil is Associate Professor in the Department of Aerospace and Ocean Engineering at Virginia Polytechnic and State University.

[2] Portions of section 2.5 including figures 2.8 and 2.9 are excerpted from Simitses and Hodges (2006) and (2010), used with permission.

[3] Each function must satisfy at least all boundary conditions on displacement and rotation (often called the “geometric” boundary conditions). It is not necessary that they satisfy the force and moment boundary conditions, but satisfaction of them may improve accuracy. However, it is not easy, in general, to find functions that satisfy all boundary conditions.

[4] Each function must be continuous and p times differentiable, where p is the order of the highest spatial derivative in the Lagrangean. The pth derivative of at least one function must be nonzero. Here, from Eq. (3.297), p = 2.

[5] If more than one function is used, they must be chosen from a set of functions that is complete. This means that any function on the interval 0 < x < I with the same boundary conditions as the problem under consideration can be expressed to any degree of accuracy as a linear combination of the functions in the set.

a computer code based on a panel method for design and analysis of subsonic isolated airfoils (see Drela, 1992).

[7] Specify an altitude, which fixes the parameter p.

[8] Specify an initial guess for MTO of, say, zero.

[9] If one does not have ready access to a root-finding application, this step may be replaced by the

[10] Rusak (2011), private comm.