6.3: Lagrange Equations from d’Alembert’s Principle
( \newcommand{\kernel}{\mathrm{null}\,}\)
d’Alembert’s Principle of virtual work
The Principle of Virtual Work provides a basis for a rigorous derivation of Lagrangian mechanics. Bernoulli introduced the concept of virtual infinitessimal displacement of a system mentioned in chapter 5.9.1. This refers to a change in the configuration of the system as a result of any arbitrary infinitessimal instantaneous change of the coordinates δri, that is consistent with the forces and constraints imposed on the system at the instant t. Lagrange’s symbol δ is used to designate a virtual displacement which is called "virtual" to imply that there is no change in time t, i.e. δt=0. This distinguishes it from an actual displacement dri of body i during a time interval dt when the forces and constraints may change.
Suppose that the system of n particles is in equilibrium, that is, the total force on each particle i is zero. The virtual work done by the force Fi moving a distance δri is given by the dot product Fi⋅δri. For equilibrium, the sum of all these products for the N bodies also must be zero
N∑iFi⋅δri=0
Decomposing the force Fi on particle i into applied forces FAi and constraint forces fCi gives
N∑iFAi⋅δri+N∑ifCi⋅δri=0 The second term in Equation ??? can be ignored if the virtual work due to the constraint forces is zero. This is rigorously true for rigid bodies and is valid for any forces of constraint where the constraint forces are perpendicular to the constraint surface and the virtual displacement is tangent to this surface. Thus if the constraint forces do no work, then ??? reduces to
N∑iFAi⋅δri=0
This relation is the Bernoulli’s Principle of Static Virtual Work and is used to solve problems in statics.
Bernoulli introduced dynamics by using Newton’s Law to related force and momentum.
Fi=˙pi
Equation ??? can be rewritten as Fi−˙pi=0
In 1742, d’Alembert developed the Principle of Dynamic Virtual Work in the form
N∑i(Fi−˙pi)⋅δri=0
Using equations ??? plus ??? gives
N∑i(FAi−˙pi)⋅δri+N∑i(fCi⋅δri=0
For the special case where the forces of constraint are zero, then Equation ??? reduces to d’Alembert’s Principle
N∑i(FAi−˙pi)⋅δri=0
d’Alembert’s Principle, by a stroke of genius, cleverly transforms the principle of virtual work from the realm of statics to dynamics. Application of virtual work to statics primarily leads to algebraic equations between the forces, whereas d’Alembert’s principle applied to dynamics leads to differential equations.
Transformation to generalized coordinates
In classical mechanical systems the coordinates δri usually are not independent due to the forces of constraint and the constraint-force energy contributes to Equation ???. These problems can be eliminated by expressing d’Alembert’s Principle in terms of virtual displacements of n independent generalized coordinates qi of the system for which the constraint force term ∑nifCi⋅δqi=0. Then the individual variational coefficients δqi are independent and (FAi−˙pi)⋅δqi=0 can be equated to zero for each value of i.
The transformation of the N-body system to n independent generalized coordinates qk can be expressed as
ri=ri(q1,q2,q3…,qn,t)
Assuming n independent coordinates, then the velocity vi can be written in terms of general coordinates qk using the chain rule for partial differentiation.
vi≡dridt=n∑j∂ri∂qj˙qj+∂ri∂t
The arbitrary virtual displacement δri can be related to the virtual displacement of the generalized coordinate δqj by
δri=n∑j∂ri∂qjδqj
Note that by definition, a virtual displacement considers only displacements of the coordinates, and no time variation δt is involved.
The above transformations can be used to express d’Alembert’s dynamical principle of virtual work in generalized coordinates. Thus the first term in d’Alembert’s Dynamical Principle, ??? becomes
n∑iFAi⋅δri=n∑i,jFAi⋅∂ri∂qjδqj=n∑jQjδqj
where Qj are called components of the generalized force,1 defined as
Qj≡n∑iFAi⋅∂ri∂qj
Note that just as the generalized coordinates qj need not have the dimensions of length, so the Qj do not necessarily have the dimensions of force, but the product Qjδqj must have the dimensions of work. For example, Qj could be torque and δqj could be the corresponding infinitessimal rotation angle.
The second term in d’Alembert’s Principle ??? can be transformed using Equation ???
n∑i˙pi⋅δri=n∑imi¨ri⋅δri=(n∑imi¨ri⋅∂ri∂qj)δqj
The right-hand side of ??? can be rewritten as
(n∑imi¨ri⋅∂ri∂qj)δqj=n∑i{ddt(mi˙ri⋅∂ri∂qj)−mi˙ri⋅ddt(∂ri∂qj)}δqj Note that Equation ??? gives that
∂vi∂˙qj=∂ri∂qj
therefore the first right-hand term in ??? can be written as
ddt(mi˙ri⋅∂ri∂qj)=ddt(mivi⋅∂vi∂˙qj)
The second right-hand term in ??? can be rewritten by interchanging the order of the differentiation with respect to t and qj
ddt(∂ri∂qj)=∂vi∂qj
Substituting ??? and ??? into ??? gives
n∑i˙pi⋅δri=(n∑imi¨ri⋅∂ri∂qj)δqj=N∑i{ddt(mivi⋅∂vi∂˙qj)−mivi⋅∂vi∂qj}δqj Inserting ??? and ??? into d’Alembert’s Principle ??? leads to the relation
n∑i(FAi−˙pi)⋅δri=−N∑j{ddt(∂∂˙qj(∑i12miv2i))−∂∂qj(N∑i12miv2i)−Qj}δqj=0
The ∑ni12miv2i term can be identified with the system kinetic energy T. Thus d’Alembert Principle reduces to the relation
N∑j[{ddt(∂T∂˙qj)−∂T∂qj}−Qj]δqj=0
For cartesian coordinates T is a function only of velocities (˙x,˙y,˙z) and thus the term ∂T∂qj=0. However, as discussed in appendix 19.3, for curvilinear coordinates ∂T∂qj≠0 due to the curvature of the coordinates as is illustrated for polar coordinates where v=˙rˆr+r˙θˆθ.
{ddt(∂T∂˙qj)−∂T∂qj}=Qj
where n≥j≥1. That is, this leads to n Euler-Lagrange equations of motion for the generalized forces Qj. As discussed in chapter 5.8, when m holonomic constraint forces apply, it is possible to reduce the system to s=n−m independent generalized coordinates for which Equation ??? applies.
In 1687 Leibniz proposed minimizing the time integral of his “vis viva", which equals 2T. That is,
δ∫t2t1Tdt=0
The variational Equation ??? accomplishes the minimization of Equation ???. It is remarkable that Leibniz anticipated the basic variational concept prior to the birth of the developers of Lagrangian mechanics, i.e., d’Alembert, Euler, Lagrange, and Hamilton.
Lagrangian
The handling of both conservative and non-conservative generalized forces Qj is best achieved by assuming that the generalized force Qj=∑niFAi⋅∂ˉri∂qj can be partitioned into a conservative velocity-independent term, that can be expressed in terms of the gradient of a scalar potential, −∇Ui, plus an excluded generalized force QEXj which contains the non-conservative, velocity-dependent, and all the constraint forces not explicitly included in the potential Uj. That is,
Qj=−∇Uj+QEXj
Inserting ??? into ???, and assuming that the potential U is velocity independent, allows ??? to be rewritten as
∑j[{ddt(∂(T−U)∂˙qj)−∂(T−U)∂qj}−QEXj]δqj=0
The standard definition of the Lagrangian is
L≡T−U
then ??? can be written as N∑j[{ddt(∂L∂˙qj)−∂L∂qj}−QEXj]δqj=0
Note that if all the generalized coordinates are independent, then the square bracket terms are zero for each value of j, which leads to the general Euler-Lagrange equations of motion.
{ddt(∂L∂˙qj)−∂L∂qj}=QEXj
where n≥j≥1.
Chapter 6.5.3 will show that the holonomic constraint forces can be factored out of the generalized force term QEXj which simplifies derivation of the equations of motion using Lagrangian mechanics. The general Euler-Lagrange equations of motion are used extensively in classical mechanics because conservative forces play a ubiquitous role in classical mechanics.
1This proof, plus the notation, conform with that used by Goldstein [Go50] and by other texts on classical mechanics.