Lagrangian Mechanics
1.10 Constrained Motion
1.10.1 Coordinate Constraints
Consider constraints that do not depend on velocities:
∂2ϕ≡0.
In this case the variation is tangent to the constraint surface if
(∂1ϕ◦Γ)η = 0. (1.180)
Together, equations (1.177) and (1.180) should determine the mo-tion, but how do we eliminate η? The residual of the Lagrange equations is orthogonal90 to any η that is orthogonal to the nor-mal to the constraint surface. A vector that is orthogonal to all vectors orthogonal to a given vector is parallel to the given vec-tor. Thus, the residual of Lagrange’s equations is parallel to the normal to the constraint surface; the two must be proportional:
D(∂2L◦Γ[q])−∂1L◦Γ[q] =λ(∂1ϕ)◦Γ[q]. (1.181) That the two vectors are parallel everywhere along the path does not guarantee that the proportionality factor is the same at each moment along the path, so the proportionality factor λ is some function of time, which may depend on the path under consider-ation. These equations, with the constraint equationϕ◦Γ[q] = 0, are the governing equations. These equations are sufficient to de-termine the pathq and to eliminate the unknown functionλ.
Now watch this
Suppose we form an augmented Lagrangian treating λ as one of the coordinates
L0(t;q, λ; ˙q,λ) =˙ L(t, q,q) +˙ λϕ(t, q,q).˙ (1.182) The Lagrange equations associated with the coordinatesqare just the modified Lagrange equations (1.181), and the Lagrange
equa-90We take two tuple-valued functions of time to be orthogonal if at each instant the dot product of the tuples is zero. Similarly, tuple-valued functions are considered parallel if at each moment one of the tuples is a scalar multiple of the other. The scalar multiplier is in general a function of time.
tion associated with λis just the constraint equation. (Note that λ˙ does not appear in the augmented Lagrangian.) So the La-grange equations for this augmented Lagrangian fully encapsulate the modification to the Lagrange equations that is imposed by the addition of an explicit coordinate constraint, at the expense of in-troducing extra degrees of freedom. Notice that this Lagrangian is of the same form as Lagrangian (1.89) that we used in the deriva-tion ofL=T −V for rigid systems (section 1.6.2).
Alternatively
How do we know that we have enough information to eliminate the unknown function λ from equations (1.181) or that the ex-tra degree of freedom introduced in Lagrangian (1.182) is purely formal?
If λ could be written as a function of the solution state path, then it would be clear that it is determined by the state and can thus be eliminated. Okay, suppose λ can be written as a composition of state-dependent function with the path: λ= Λ◦ Γ[q]. Consider the Lagrangian
L00=L+ Λϕ. (1.183)
This new Lagrangian has no extra degrees of freedom. The La-grange equations for L00 are the Lagrange equations for L with additional terms arising from the product of Λϕ. Applying the Euler-Lagrange operator E (see section 1.9) to this Lagrangian gives91
E[L00] =E[L] +E[Λϕ]
=E[L] + ΛE[ϕ] +E[Λ] ϕ+DtΛ∂2ϕ+∂2ΛDtϕ. (1.184) Composition of E[L00] with Γ[q] gives the Lagrange equations for the path q. Using the fact that the constraint is satisfied on the pathϕ◦Γ[q] = 0 and consequently Dtϕ◦Γ[q] = 0, we have E[L00]◦Γ[q] = (E[L] +λE[ϕ] +Dλ(∂2ϕ))◦Γ[q], (1.185)
91Recall that the Euler-Lagrange operatorEhas the property E[F G] =FE[G] +E[F]G+DtF ∂2G+∂2F DtG.
y
l
m x θ
Figure 1.8 We can formulate the behavior of a pendulum as motion in the plane, constrained to a circle about the pivot.
where we have used λ= Λ◦Γ[q]. If we now use the fact that we are only dealing with coordinate constraints,∂2ϕ= 0 then E[L00]◦Γ[q] = (E[L] +λE[ϕ])◦Γ[q]. (1.186) The Lagrange equations are the same as those derived from the augmented LagrangianL0. The difference is that now we see that λ= Λ◦Γ[q] is determined by the unaugmented state. This is the same as saying that λcan be eliminated.
Considering only the formal validity of the Lagrange equations for the augmented Lagrangian, we could not deduce thatλcould be written as the composition of a state-dependent function Λ with Γ[q]. The explicit Lagrange equations derived from the augmented Lagrangian depend on the accelerations D2q as well as λ so we may not deduce separately that either is the composition of a state-dependent function and Γ[q]. However, now we see that λis such a composition. This allows us to deduce that D2q is also a state-dependent function composed with the path. The evolution of the system is determined from the dynamical state.
The pendulum using constraints
The pendulum can be formulated as the motion of a massive par-ticle in a vertical plane subject to the constraint that the distance to the pivot is constant (see figure 1.8).
In this formulation, the kinetic and potential energies in the Lagrangian are those of an unconstrained particle in a uniform
gravitational acceleration. A Lagrangian for the unconstrained particle is
L(t;x, y;vx, vy) = 12m(vx2+vy2)−mgy. (1.187) The constraint that the pendulum moves in a circle of radius l about the pivot is92
x2+y2−l2 = 0. (1.188)
The augmented Lagrangian is
L0(t;x, y, λ;vx, vy,λ) =˙ 12m(vx2+v2y)−mgy+λ(x2+y2−l2).(1.189) The Lagrange equations for the augmented Lagrangian are
mD2x−2λx= 0 (1.190)
mD2y+mg−2λy= 0 (1.191)
x2+y2−l2 = 0. (1.192)
These equations are sufficient to solve for the motion of the pen-dulum.
It should not be surprising that these equations simplify if we switch to “polar” coordinates
x=rsinθ y =−rcosθ. (1.193)
Substituting this into the constraint equation we determine that r =l, a constant. Forming the derivatives and substituting into the other two equations we find
ml(cosθD2θ−sinθ(Dθ)2)−2λsinθ= 0 (1.194) ml(sinθD2θ+ cosθ(Dθ)2) +mg+ 2λcosθ= 0. (1.195) Multiplying the first by cosθand the second by sinθand adding, we find
mlD2θ+mgsinθ= 0, (1.196)
92This constraint has the same form as the constraints used in the demonstra-tion thatL = T −V can be used for rigid systems. Here it is a particular example of a more general set of constraints.
which we recognize as the correct equation for the pendulum. This is the same as the Lagrange equation for the pendulum using the unconstrained generalized coordinateθ. For completeness, we can find λin terms of the other variables
λ= mD2x 2x =−1
2l(mgcosθ+ml(Dθ)2). (1.197) This confirms thatλis really the composition of a function of the state with the state path. Notice that 2lλ is a force—it is the sum of the outward component of the gravitational force and the centrifugal force. Using this interpretation in the two coordinate equations of motion we see that the terms involving λ are the forces that must be applied to the unconstrained particle to make it move on the circle required by the constraints. Equivalently, we may think of 2lλ as the tension in the pendulum rod that holds the mass.93
Building systems from parts
The method of using augmented Lagrangians to enforce con-straints on dynamical systems provides us with a way of building the analysis of a compound system by combining the results of the analysis of the parts of the system and the coupling between them.
Consider the compound spring-mass system shown at the top of figure 1.9. We could analyze this as a monolithic system with two configuration coordinates x1 and x2, representing the extensions of the springs from their equilibrium lengths X1 and X2.
An alternative procedure is to break the system into several parts. In our spring-mass system we can choose two parts, one is a spring and mass attached to the wall, and the other is a spring and mass with its attachment point at an additional configuration coordinate ξ. We can formulate a Lagrangian for each part sepa-rately. We can then choose a Lagrangian for the composite system as the sum of the two component Lagrangians with a constraint ξ=X1+x1 to accomplish the coupling.
93Indeed, if we had scaled the constraint equations as we did in the discussion of Newtonian constraint forces we could have identifiedλwith the the magni-tude of the constraint forceF. However, thoughλwill in general be related to the constraint forces it will not be one of them. We chose to leave the scaling as it naturally appeared rather than make things turn out artificially pretty.
x1 X1
ξ x1 X1
k1
m1
k1 k2
k2
m1 m2
m2 X2 x2
X2 x2
Figure 1.9 A compound spring-mass system is decomposed into two subsystems. We have two springs and masses that may only move hori-zontally. The equilibrium positions of the springs areX1 and X2. The systems are coupled by the position-coordinate constraintξ=X1+x1.
Let’s see how this works. The Lagrangian for the subsystem attached to the wall is
L1(t, x1,x˙1) = 12m1x˙21−12k1x21 (1.198) and the Lagrangian for the subsystem that attaches to it is L2(t;ξ, x2; ˙ξ,x˙2) = 12m2( ˙ξ+ ˙x2)2−12k2x22. (1.199) We construct a Lagrangian for the system composed from these parts as a sum of the Lagrangians for each of the separate parts,
with a coupling term to enforce the constraint:
L(t;x1, x2, ξ, λ; ˙x1,x˙2,ξ,˙ λ)˙
=L1(t, x1,x˙1) +L2(t;ξ, x2; ˙ξ,x˙2) +λ(ξ−(X1+x1)) (1.200) Thus we can write Lagrange’s equations for the four configuration coordinates, in order, as follows:
m1D2x1=−k1x1−λ (1.201)
m2(D2ξ+D2x2) =−k2x2 (1.202)
m2(D2ξ+D2x2) =λ (1.203)
0 =ξ−(X1+x1) (1.204)
Notice that in this systemλis the force of constraint, holding the system together. We can now eliminate the “glue” coordinates ξ and λto obtain the equations of motion in the coordinates x1 and x2:
m1D2x1+m2(D2x1+D2x2) +k1x1 = 0 (1.205) m2(D2x1+D2x2) +k2x2 = 0 (1.206) This strategy can be generalized. We can make a library of primitive components. Each component may be characterized by a Lagrangian with additional degrees of freedom for theterminals where that component may be attached to others. We then can construct composite Lagrangians by combining components using constraints to glue together the terminals.
Exercise 1.34: Combining Lagrangians
a. Make another primitive component that is compatible with the spring-mass structures described in this section. For example, make a pendu-lum that can attach to the spring-mass system. Build a combination and derive the equations of motion. Be careful, the algebra is horrible if you choose bad coordinates.
b. For a nice little project, construct a family of compatible mechanical parts, characterized by appropriate Lagrangians, that can be combined in a variety of ways to make interesting mechanisms. Remember that in a good language the result of combining pieces should be a piece of the same kind that can be further combined with other pieces.
Exercise 1.35: Bead on a triaxial surface
Consider again the motion of a bead constrained to move on a triaxial surface from exercise 1.18. Reformulate this using rectangular coordi-nates as the generalized coordicoordi-nates with an explicit constraint that the bead stay on the surface. Find a Lagrangian and show that the Lagrange equations are equivalent to those found in exercise 1.18.
Exercise 1.36: Motion of a tiny golf ball
Consider the motion of a golf ball idealized as a point mass constrained to a frictionless smooth surface of varying height h(x, y) in a uniform gravitational field with accelerationg.
a. Find an augmented Lagrangian for this system, and derive the equa-tions governing the motion of the point mass inxandy.
b. Under what conditions is this approximated by a potential function V(x, y) =mgh(x, y)?
c. Assume that we have anh(x, y) that is axisymmetric aboutx=y= 0. Can you find such anhthat yields motions with closed orbits?