Preface
There are two distinct yet equivalent approaches to solving a problem in rigid body mechanics: the Newtonian approach, based on Newton's laws, and Lagrange's approach, based on the postulate called the principle of virtual powers, and which lead to Lagrangian or analytical mechanics.
Although both approaches yield equivalent results, they differ on a number of points both in terms of conception as well as formulation. In addition to the usual ingredients - velocity, acceleration, mass and forces - analytical mechanics involves a new concept that does not exist in Newtonian mechanics, which is given the enigmatic name "virtual velocity". It is based on this concept that virtual powers are defined. While Newton's laws state relationships between vector quantities (force and acceleration), the principle of virtual powers, is written in terms of virtual powers, which are scalar quantities. Analytical mechanics is also distinguished by the fact that parameterization plays a primordial role here: given the same mechanical problem, it is possible to choose different parameterizations and the resulting equations - and thus, the information they yield - differ based on the chosen parameterization. Another salient feature demonstrated in analytical mechanics is that once the parameterization is chosen, the kinematical behavior of the system, a vector description in essence, is condensed into a scalar function, called the parameterized kinetic energy.
While Newtonian mechanics brings into play physical concepts that are easy to apprehend, Lagrange theory appears more complicated because of the virtual velocity and the very statement of the principle of virtual powers. However, two technical advantages compensate for this conceptual difficulty:
- (i) It is seen that the physicist's task is practically reduced to choosing the appropriate parameterization for the system under study. Once the parameterization is chosen, the Lagrange equations systematically lead to as many equations as there are unknowns (if the problem is well-posed). Of course, it is also possible to obtain a sufficient number of equations using the Newtonian approach, but there is no systematic way of doing so. One must first carry out an analysis of the applied forces and must often subdivide the mechanical system being studied in an adequate manner and then write the equations for the subsystems.
- (ii) The operations carried out in analytical mechanics - especially the calculation of the parameterized kinetic energy and its derivatives - are purely algebraic and, therefore, programmable. This explains the success of analytical mechanics in the study of complex systems containing a large number of kinematic parameters, where it is more difficult to obtain the equations of motion using the Newtonian approach.
This book strives to explain the subtleties of analytical mechanics and to help the reader master the techniques to obtain Lagrange's equations in order to fully use the potential of this elegant and efficient formulation. It is meant for students doing their bachelors or masters degree in Engineering, who are interested in a comprehensive study of analytical mechanics and its applications. It is also meant for those who teach mechanics, engineers and anyone else who wishes to review the fundamentals of this field. Although the content does not require any prior knowledge of mechanics, it is preferable for the reader to be familiar with the Newtonian approach.
The format adopted in this book
When writing this book, the authors laid out the objective of revisiting analytical mechanics and presenting it from a different angle both in form and style. This was done:
- - by adopting a more axiomatic and formal framework than a conventional course,
- - and by taking special efforts concerning notations to arrive at mathematical expressions that are both precise and concise.
By "axiomatic framework", we mean that all through this book, the chapters are constructed in a manner that is similar to a mathematical discussion, where the ingredients are presented in the following order:
- the definitions to establish the vocabulary used,
- the theorems, where results are proven and where we specify the hypotheses, clearly indicating the conditions of applicability for this or that result,
- and finally, examples to illustrate the nuances of the theory and the mechanisms of the calculation.
While the theory is constructed in a deductive manner and forms a monolithic block, each theorem is written in a self-contained and condensed form - that is, hypotheses followed by results - in order to make it "ready to use".
Synopsis
This book contains 11 chapters and two appendices:
Chapter 1 reviews the basic ingredients of kinematics: time, space and the observer (or reference frame). We present here the key concept of the derivative of a vector with respect a reference frame and introduce the concept of a "common reference frame", which is used to connect or relate two different reference frames.
Chapter 2 focuses on an important operation in analytical mechanics, namely the parameterization of the mechanical system being studied. This operation consists of choosing a certain number of primitive parameters of the system, expressing all existing constraints in terms of these parameters and, finally, classifying the constraint equations into two categories, called "primitive" and "complementary" equations. This task, incumbent on the physicist, is specific to analytical mechanics and has no equivalent in Newtonian mechanics. It is important because, as we will see, the Lagrange equations that are obtained (and, consequently, the information that may be extracted from them) are essentially dependent on the choice of parameterization.
The parameterization of the system leads to the definition of the parameterized velocities and the parameterized kinetic energy, the concept on which the Lagrange kinematic formula is based.
Chapter 3 reviews the conventional concept of efforts that includes forces and torques. These can be classified as either internal efforts and external efforts, or given efforts and constraint efforts. The virtual powers of efforts are calculated in Chapter 5 depending on how the efforts are categorized.
Chapter 4, dedicated to virtual kinematics, introduces new kinematic quantities that are the counterparts of those introduced in Chapter 1 and are labeled "virtual": the virtual derivative of a vector with respect to a reference frame, the virtual velocity of a particle, the virtual velocity fields in a rigid body or a system of rigid bodies, and, finally, the virtual angular velocity of a rigid body. This chapter provides formulae to calculate these quantities and, notably, the formulae for the composition of virtual velocities.
Chapter 5 deals with virtual power, which is, grosso modo, the product of an effort, seen in Chapter 3, and a virtual velocity, seen in Chapter 4. The presentation closely follows the conventional presentation of real powers in Newtonian mechanics and we obtain several results that are analogous to those obtained for real powers. Two results are, nevertheless, specific to analytical mechanics: the virtual power expressed as a linear form and the power of the quantities of acceleration.
Chapter 6 shows how to exploit the principle of virtual powers using the results obtained in the previous chapters in order to arrive at the final product of analytical mechanics, namely the famous Lagrange equations. In this chapter, we also see several examples which illustrate how important the choice of parameterization is and what consequences it has on the obtained results. This chapter concludes with the statement of Lagrange equations in a non-Galilean reference frame.
Chapters 7 and 8 are concerned with perfect joints. The chief advantage of these joints is that the generalized forces present in the right-hand side of the Lagrange equations are then zero or may be easily calculated using Lagrange multipliers. The concept of the perfect joint also exists in Newtonian mechanics, but they are defined there in a simpler manner, with more obvious consequences. In analytical mechanics, the definition of a perfect joint is less natural inasmuch as it involves the parameterization and the virtual velocities that are compatible with the complementary constraint equations. It is, therefore, important to verify that the perfect character of a joint is intrinsic, i.e. it does not depend on the chosen parameterization. For this reason, a large section is dedicated to the invariance of virtual velocity fields with respect to the parameterization.
Chapter 9 is dedicated to the first integrals, which offer the advantage of yielding first-order differential equations that are easier to solve. The first integral called "Painlevé's first integral" has no equivalent in Newtonian mechanics and presents the unique feature of being able to exist for systems that receive energy from the exterior. The energy integral resembles that of Newtonian mechanics. However, the conditions for application are slightly different.
Chapter 10 shows how the Lagrange equations are simplified in the particular case of equilibrium. The chapter also contains a brief discussion on the question of the stability of an equilibrium...
