Lagrangian Mechanics I Monday, ÕÉ September Óþõõ Physics ÕÕÕ

Lagrangian Mechanics I Monday, ÕÉ September óþÕÕ Physics ÕÕÕ

How Lagrange revolutionized Newtonian mechanics

Newton based his dynamical theory on the notion of force, for which he gave a somewhat incomplete denition, but which he made plausible by appealing to the common experience of pushes and pulls to which the human body may be subject. In the century following New- ton’s publication of the Philosophiæ Naturalis Principia Mathematica (mathematical prin- ciples of natural philosophy), mathematicians sought to clarify and generalize the foundation he laid. e most important developments before Einstein were due to Leonhard Euler (Õßþß–Õßì), Joseph-Louis Lagrange (Õßìä–ÕÕì), and William Rowan Hamilton (Õþ¢-Õä¢). Euler and Lagrange developed the general formulation of the calculus of variations, Lagrange applied the method of virtual velocities developed initially by Johann Bernoulli (Õääß–Õß¦), along with generalized coordinates, to produce a powerful new approach to solving dynamics problems. is method today is called Lagrangian mechanics and remains the method of choice for solving a variety of mechanics problems involving conservative forces. Hamil- ton provided a third approach, derived from Lagrange’s, which proved fundamental to the foundation of statistical mechanics and quantum mechanics. We will investigate Hamilton’s approach aer making a careful study of Lagrange’s. One way to develop Lagrangian mechanics is from Hamilton’s principle, which states

Of all the possible paths along which a dynamical system may move from one point to another in conguration space within a specied time interval, the actual path followed is that which minimizes the time integral of the dierence between the kinetic and potential energy of the system.

is approach has the advantage of simplicity and power, and is roughly the approach used by Helliwell and Sahakian, who are inspired by the motion of a (relativistic) particle in a uni- form gravitational eld at the conclusion of Chapter ì. e disadvantage of this approach is that it seems quite disconnected from the Newtonian roots from which it sprang. Fur- thermore, the Irish mathematician and physicist William Rowan Hamilton wasn’t born until thirteen years aer Lagrange published his great work, Mecanique´ analytique, in Õß. To supplement the text, we will develop here Lagrange’s approach following a path that more closely approximates history. Our concession to modernity will be to use the terms kinetic energy, potential energy, and work, rather than those Lagrange and his contemporaries actually employed. We will develop Lagrange’s approach by studying a relatively simple dynamical system, aer which we will generalize.

Physics ÕÕÕÕ Peter N. Saeta Õ. EXAMPLE SYSTEM

Õ. Example System A small bead of mass m is threaded on a wire bent in the shape of a parabola of equation z = αxó, where z is along the vertical and α is a constant. e wire is mounted to a motor and made to rotate at the constant angular speed ω about the z axis. We seek the equation(s) of motion of the bead.

In the Newtonian approach, we prepare an isolation diagram showing all the forces applied to the bead (gravity and the normal force supplied by the wire), and set F = ma. Our rst step in developing the Lagrangian approach is to make the very mundane observation that Q Fext − mr¨ = þ (Õ)

at is, the sum of all forces, including the inertial force −mr¨, is zero, just as in static equilib- rium. is is not particularly revolutionary, as far as it goes, but is sometimes employed as a way of avoiding sign errors by including “inertial forces” on free-body diagrams. e insight that goes under the name d’Alembert’s principle, aer Jean le Rond d’Alembert (ÕßÕß–Õßì), but is based as well on the work of Johann Bernoulli and Leonhard Euler, is to imagine the work done in an innitesimal, instantaneous “virtual” displacement of the bead δr in a direction consistent with the constraints on the bead’s motion. In this case, the bead’s motion through three-dimensional space is really conned to the one-dimensional parabolic curve of the rotating wire. Such a constraint, which reduces the number of coordinates (also called degrees of freedom) necessary to specify the position of the particle, is called holonomic.Õ Lagrange’s approach signicantly simplies dynamics problems that involve holonomic constraints. When we include the inertial forces, the sum of all forces vanishes by Eq. (Õ), so no work is done in the virtual displacement,

Q Fcon + Q Fapp − mr¨ ⋅ δr = þ (ó)

where we have separated the external forces on m into constraint forces and (all the other) applied forces. At this point we note that the virtual displacement is along the wire and therefore perpendicular to the forces of constraint. Hence, ∑ Fcon ⋅ δr = þ, which implies d’Alembert’s principle, Q Fapp − mr¨ ⋅ δr = þ (ì) a nontrivial simplication of the problem. In our example, the only applied force is gravity, which in our coordinate system takes the form F = −mgzˆ. e acceleration in cylindrical coordinates is r¨ = ρ¨ − ρϕ˙ó ρˆ + (ρϕ¨ + óρ˙ϕ˙)ϕˆ + z¨zˆ (¦)

ÕA nonholonomic constraint limits the range of motion of a particle or system without reducing the number of degrees of freedom. For example, a particle that moves inside a spherical shell, such that r < a. In such a case, the approach we develop here has no advantage over the more familiar Newtonian approach.

Peter N. Saetaó Physics ÕÕÕ ó. GENERALIZED COORDINATES where ϕ˙ = ω = constant.ó e virtual displacement δr must be along the wire (z = αρó), and therefore have slope dz = óαρ Ô⇒ δz = óαρδρ dρ Substituting into Eq. (ì), we have

[−mgzˆ − m (ρ¨ − ρωó) ρˆ − mz¨zˆ] ⋅ (ρˆ + óαρzˆ) δρ = þ (¢)

or −(g + z¨)óαρ − ρ¨ + ρωó = þ (ä) At this point, we need again to invoke the constraint equation z = αρó to obtain

z˙ = óαρρ˙ z¨ = óαρ˙ó + óαρρ¨

from which we nd the equation of motion,

óαρ(g + óαρ˙ó + óαρρ¨) + ρ¨ − ρωó = þ ρ¨(Õ + ¦αóρó) + ¦αóρρ˙ó + ρ(óαg − ωó) = þ (ß)

You may derive this same equation through more involved means using the Newtonian approach. e virtual-displacement trick quickly eliminated the constraint forces, allowing us to proceed more directly to the equations of motion. On the other hand, we still needed the somewhat complicated expression for the acceleration in cylindrical coordinates, Eq. (¦), as well as vector components in taking the dot product.

ó. Generalized Coordinates Lagrange’s key insight was to generalize the method of virtual work to use an arbitrary set of generalized coordinates, which are a set of real numbers that suce to specify the position of the particle uniquely. at is, we assume that we have a set of coordinates qi such that we may express the particle position as r = r(qi , t). e coordinates qi may be any convenient values, such as angles, distances, or combinations thereof. In this case, we will be using cylindrical coordinates, (ρ, ϕ, z), which lack explicit time dependence, but to keep the discussion general, we will seek to redo the derivation of the previous section using the coordinates qi. e particle’s velocity may be expressed by the chain rule of vector calculus:

dr ∂r ∂r v = r˙ = = Q q˙i + () dt i ∂qi ∂t

óe course text uses r for the distance from the z axis in cylindrical coordinates. To avoid confusion between r = ρρˆ + zzˆ and a unit vector in the radial direction, I will use ρ for the radial distance in the following.

Physics ÕÕÕì Peter N. Saeta ó.Õ Virtual work in generalized coordinates ó. GENERALIZED COORDINATES

dq where the q˙ ≡ i are called the generalized velocities of the particle. Note that although i dt (by assumption) r is a function of qi and t, but not of the generalized velocities q˙i, the velocity v is a function of both the generalized coordinates and the generalized velocities, which we treat independently. erefore, from Eq. () we have

∂r˙ ∂r = (É) ∂q˙i ∂qi as though we can just cancel the dots. is only works because the equations of transforma- tion, r(q j, t), have no explicit dependence on the coordinate velocities q˙j. We will need this relation shortly.

Question Õ Suppose, instead, that the particle position is a function of both the generalized coordinates and the generalized velocities, r = r(qi , q˙i , t). Explain why Eq. (É) no longer holds.

ó.Õ Virtual work in generalized coordinates Wenow seek to express Eq. (ì) using generalized coordinates. In particular, we will attack the term mr¨ ⋅ δr. Again using the chain rule, we have ∂r δr = Q δqi (Õþ) i ∂qi

∂r where there is no term ∂t because the virtual displacement happens instantaneously. erefore, ∂r ∂r mr¨ ⋅ δr = Q mr¨ ⋅ δqi = mr¨ ⋅ δqi (summation convention) (ÕÕ) i ∂qi ∂qi With malice of forethought, consider now the derivative

d ∂r ∂r d ∂r r˙ ⋅ = r¨ ⋅ + r˙ ⋅ (Õó) dt ∂qi ∂qi dt ∂qi e second term on the right-hand side is a second derivative: the position vector is dier- entiated with respect to both the generalized coordinate qi and the time. Interchanging the order of derivatives, this term becomes v⋅∂v~∂qi. Meanwhile, the rst time on the right-hand side of Eq. (Õó) is just the term we need. Isolating this term gives

∂r d ∂r ∂r˙ r¨ ⋅ = r˙ ⋅ − r˙ ⋅ (Õì) ∂qi dt ∂qi ∂qi e nal term may be simplied by noting

∂r˙ ∂v ∂ Õ r˙ ⋅ = v ⋅ = v ⋅ v ∂qi ∂qi ∂qi ó

Peter N. Saeta¦ Physics ÕÕÕ ó. GENERALIZED COORDINATES ó.Õ Virtual work in generalized coordinates and we can pull a similar trick with the rst term on the right-hand side of Eq. (Õì) by invok- ing Eq. (É) to replace coordinates with velocities. erefore,

∂r d ∂ Õ ∂ Õ r¨ ⋅ = υó − υó (Õ¦) ∂qi dt ∂q˙i ó ∂qi ó

Multiplying through by mass m and calling T = mυó~ó the kinetic energy of the particle, we have ∂r d ∂T ∂T mr¨ ⋅ = − (Õ¢) ∂qi dt ∂q˙i ∂qi or d ∂T ∂T mr¨ ⋅ δr = − δqi (Õä) dt ∂q˙i ∂qi where we are implicitly summing over the i. We now need to express the other term in Eq. (ì) in terms of the generalized coordinates,

app app ∂r F ⋅ δr = Q F ⋅ δqi = Q Fi δqi (Õß) i ∂qi i

where the Fi are the components of the generalized force, dened by

app ∂r Fi = F ⋅ (Õ) ∂qi

Hence, Eq. (ì) expressed in generalized coordinates becomes

∂T d ∂T Q Fi + − ¡ δqi = þ (ÕÉ) i ∂qi dt ∂q˙i

ì Since each of the virtual displacements δqi is independent of the others, this implies that the term in braces must vanish for each generalized coordinate:

∂T d ∂T Fi + − = þ (óþ) ∂qi dt ∂q˙i

Question ó What are the physical dimensions of the generalized force Fi dened by Eq. (Õ)?

ìSometimes it will be more convenient to use redundant coordinates, such as ρ and z in the example, in which case we will have to ensure that their variations are coordinated. is has the additional benet of yielding the constraint forces, should they be of interest. For the moment, however, we will assume that the qi constitute a minimal set necessary to specify the position of the particle, which in this case is the single coordinate ρ.

Physics ÕÕÕ¢ Peter N. Saeta ó.ó Conservative forces ì. EXAMPLE

ó.ó Conservative forces A further simplication is possible if the applied forces are conservative, and can thus be derived from a scalar potential. Recall that the curl of a gradient vanishes, since

R R R ˆx ˆy zˆ R ó ó ó ó ó ó R ∂ ∂ ∂ R ∂ U ∂ U ∂ U ∂ U ∂ U ∂ U ∇×(∇U) = R R = ˆx − +ˆy − +zˆ − = þ R ∂x ∂y ∂z R ∂y∂z ∂z∂y ∂z∂x ∂x∂z ∂x∂y ∂y∂x R ∂U ∂U ∂U R R ∂x ∂y ∂z R by the equality of mixed partial derivatives. If a force satises ∇ × F = þ, then we can nd a scalar function U such that F = −∇U. en

app ∂r ∂r ∂U ∂x ∂U ∂y ∂U ∂z ∂U Fi = F ⋅ = −∇U ⋅ = − + + = − (óÕ) ∂qi ∂qi ∂x ∂qi ∂y ∂qi ∂z ∂qi ∂qi Hence, ∂(T − U) d ∂T − = þ (óó) ∂qi dt ∂q˙i Furthermore, since we are assuming that the applied force depends only on positions, not generalized velocities, ∂U = þ. We thus dene the Lagrangian, ∂q˙i

L(qi , q˙i , t) = T − U (óì)

to yield (nally) Lagrange’s equations:

∂L d ∂L − = þ (ó¦) ∂qi dt ∂q˙i

If some of the applied forces may be derived from scalar potentials, but others cannot, then we can simply return to Eq. (óþ) and separate the applied forces into conservative and nonconservative. e conservative forces are treated as above and are included in the Lagrangian, while the nonconservative forces remain as in Eq. (óþ):

∂L d ∂L noncons − + Fi = þ (ó¢) ∂qi dt ∂q˙i In practice, however, the Lagrangian formulation is most benecial when any nonconservative forces may be neglected.

ì. Example We have loaded the elephant gun; let’s shoot the bead-on-the-parabolic-wire problem. e potential energy is simple: U = mgz. To nd the kinetic energy in cylindrical coordinates, start with the velocity, r˙ = v = ρ˙ρˆ + ρϕ˙ϕˆ + z˙zˆ (óä)

Peter N. Saetaä Physics ÕÕÕ ¦. GENERAL PROCEDURE so that Õ m T = mv ⋅ v = ρ˙ó + ρóϕ˙ó + z˙ó (óß) ó ó and the Lagrangian is m L = (ρ˙ó + ρóωó + z˙ó) − mgz (ó) ó Using the equation of the parabola, we can eliminate z to get m L = (ρ˙ó + ρóωó + ¦αóρóρ˙ó) − mgαρó (óÉ) ó Note that we must do this because the particle has only a single degree of freedom. at is, only one generalized coordinate is required to uniquely specify the position of the particle. Later we will see how to manage constraints without reducing the number of coordinates. Now we use Lagrange’s equation,

d ∂L ∂L d − = mωóρ + ¦mαóρ˙óρ − ómgαρ − (mρ˙ + ¦mαóρóρ˙) = þ (ìþ) dt ∂ρ˙ ∂ρ dt

Carrying out the dierentiation, and dividing by m, we obtain the equation of motion for the coordinate ρ:

ρ(ωó + ¦αóρ˙ó) − ógαρ − ρ¨ − ¦αóρóρ¨ − αóρρ˙ó = þ ρ¨(Õ + ¦αóρó) + ¦αóρρ˙ó + ρ(óαg − ωó) = þ (ìÕ) which is the same equation we had before. It may seem like this was an awful lot of work to get just what we had before, but most of that work was spent deriving Eq. (ó¦). In curvilin- ear coordinates it is invariably simpler to compute the velocity than the acceleration, so the Lagrangian is more straightforward to obtain than the Newtonian equations. Furthermore, once L has been computed, Eq. (ó¦) provides a very straightforward path to the equation(s) of motion.

¦. General Procedure Let us take a moment to summarize what we have seen so far. e general approach to solving a dynamics problem using the Lagrangian formalism is as follows:

Õ. Select a minimal set of generalized coordinates qi, one for each degree of freedom.

ó ó. Express the square of the velocity in terms of the generalized coordinates, υ (qi , q˙i , t), Õ ó and then the kinetic energy T = ó mυ . ì. Express the potential energy, U, of the particle in terms of the generalized coordinates.

¦. Form the Lagrangian, L(qi , q˙i , t) = T − U.

Physics ÕÕÕß Peter N. Saeta ¦. GENERAL PROCEDURE

¢. Obtain the equation of motion for each generalized coordinate from the Lagrangian ∂L d ∂L using the Lagrange equation − = þ. Remember that the time derivative ∂qi dt ∂q˙i in this equation dierentiates every variable (qi, q˙i, and t) that appears in ∂L~∂q˙i.

Although it may seem complicated at rst blush, in many respects Lagrange’s method is much simpler than the Newtonian approach. Energy is a scalar, which makes it easier to work with than forces and accelerations. It eliminates forces of constraint entirely. Once you have settled on a set of generalized coordinates, your path to obtain the equations of motion lies clear before you, although it may require a page or two of careful algebra.

Peter N. Saeta Physics ÕÕÕ