This module requires you to read a textbook such as Fowles and Cassiday on material relevant to the following topics.

Module M2 Lagrangian Mechanics and Oscillations Prerequisite: Module C1 This module requires you to read a textbook such as Fowles and Cassiday on material relevant to the following topics. Topics: Hamilton s Variational Principle Lagrange s Equations and Generalized Coordinates Generalized Momenta Lagrange Multipliers and Forces of Constraint Generalized Forces Hamilton s Equations Equilibrium and Stability Oscillators With One Degree of Freedom Coupled Oscillators Vibrating Systems Continuum Limit References: Fowles and Cassiday Analytical Mechanics 7 th ed (Thomson/Brooks/Cole 2005) Chapter 10, Chapter 11 Thornton and Marion Classical Dynamics of Particles and Systems 5 th ed (Thomson/Brooks/Cole 2004) Chapter 7 (sections 1-11), Chapter 12, Chapter 13 (sections 1-4) Read through A Graphing Checklist for Physics Graduate Students before submitting any plots in this module. See the course website for the link. Copyright 2016 Marshall Thomsen 1

Assignment: Note that many of these problems are greatly influenced by (and in some cases virtually identical to) Fowles and Cassiday 0. With your first submission, you must supply me with the title, author and edition of the textbook you are using for this module. I will not grade your submission without this information. By submitting the information, you are indicating that you have ready access to this book, that is, that you have either borrowed or bought the book. It is not the intent of this course to require you to purchase several expensive textbooks. For many modules, you should be able to find older editions at greatly reduced prices if you are unable to borrow a copy from a friend or from the library. Please see me if you are having difficulty locating an appropriate book. In the problems 1-6, you must use the Lagrangian approach. 1. An object of mass m is suspended from an ideal massless vertical spring of spring constant k. Assuming there is no friction in the system, derive the differential equation of motion (i.e., an expression for the acceleration), assuming the object moves only in the vertical direction. 2. A solid cylinder rolls without slipping down a ramp inclined at an angle θ with respect to the horizontal. Determine its acceleration. 3. In a situation similar to problem 2, now assume the ramp sits on a frictionless horizontal surface, so that it is free to move also. Take M to be the mass of the ramp and m to be the mass of the cylinder. Find the acceleration of the ramp. 4. In the figure below, the table is frictionless, block 1 has mass m 1, block 2 has mass m 2, and the rope is uniform with total mass m 3 and length l. The mass of the pulley is negligible. Find an expression for the acceleration of the blocks. 1 2 Copyright 2016 Marshall Thomsen 2

5. Consider a simple pendulum confined to move in a plane, but with the support rod replaced by an elastic string of stiffness k (equivalent to a spring constant) and unstretched length l o. Ignore the mass of the string and take the mass of the bob to be m. Find the differential equations of motion for this system using polar coordinates. Assume that, due to the weight of the pendulum bob, the string is always stretched past its unstretched length. El astic support stri ng 6. Expressing the square of the speed of a particle in spherical coordinates as v 2 = r 2 + r 2 θ 2 + ( rsinθ) 2 φ 2, find the differential equations of motion for a particle of mass m in a central potential V(r). Re-express these equations in terms of the radial force. Note that a central potential has no θ or φ dependence. Copyright 2016 Marshall Thomsen 3

7. Before beginning this problem, you may wish to read the supplemental material at the end of the problems. A track of mass M has a surface representing one quarter of a circle of radius b. The track sits on a frictionless, horizontal surface. A small block of mass m slides without friction down the track, after having been released from rest when θ=0. Using the Lagrange multiplier method and the coordinate system shown, a. show that the motion is governed by the three equations, M x T + m x T + mbθ sinθ = const ( 1' ) gcosθ b θ x T sinθ = 0 2' ( ) bθ 2 + gsinθ + x T cosθ + λ m = 0 ( 3' ) b. Letting γ=m/(m+m), show that 2 and 3 can be rewritten as gcosθ b θ + γb d dt b θ 2 + gsinθ γb d dt ( θ sinθ)sinθ = 0 ( 2" ) ( θ sinθ)cosθ + λ m = 0 ( 3" ) c. From these equations, show that the constraint force on the block is given by: % mg sinθ + 2sinθ + 2γ ( ' sin3 θ + 3γ cos 2 θ sinθ γ 2 cos 2 θ sin 3 θ * ' ( 1 γ sin 2 θ) 2 * & ) x T θ r M Copyright 2016 Marshall Thomsen 4

8. For each of the following potentials, determine (i) the equilibrium position(s) (ii) the conditions under which the equilibrium is stable, and (iii) the oscillation frequency, ω, if the equilibrium is stable. Take the mass of the particle in the potential to be m. a. V ( x) = 1 2 kx 2 + k' x 4 where k > 0 b. V r c. V x )" $ % *# r & a ( ) = V o + 12 ' ( ) = Bxe x / a where a > 0 " $ a% 6, # r '. where V & o > 0, a > 0, r > 0-9. A double pendulum is constructed by attaching a string of length L to a pivot point. At the base of the string is an object of mass m 1 (you can treat it as a point mass). Tied to the bottom of this object is a second string of length L, and at the base of that is a second point-like mass, m 2. Let α=m 2 /m 1. a. Derive a general expression for the normal mode frequencies for this system, expressing your final answer in terms of L and α. You may assume small amplitude oscillations. b. Simplify this expression in the special case that α=1. c. Simplify this expression in the special case that α<<1 (but not α=0!). d. Simplify this expression in the special case that α>>1 (but not infinitely large!). e. Create a computer generated plot of the normal mode frequencies as a function of α. Copyright 2016 Marshall Thomsen 5

10. An array of 4 identical particles of mass m are joined by identical springs of stiffness k. Both ends of this array are attached to fixed bracket with an additional spring. Determine the normal mode frequencies for longitudinal oscillations. Is there any mode in which all of the particles are always moving in the same direction, that is, where they are all moving to the left at one time and all moving to the right at another, but never some moving to the left and some moving to the right? 11. Find a derivation of the equation for waves on a uniform string, and write it out in detail, carefully justifying each step. The point of this exercise to make sure you understand the origin of this equation. The equation you derive should have the form 2 q = v 2 2 q x 2 t 2 where q(x,t) represents the displacement of part of the string (at position x) from its equilibrium position. 12. Conceptual Question: Suppose a student who has never learned about Lagrangians before is about to take the Physics GRE. You offer to give him or her a quick introduction to solving problems using this approach in the hopes that it will be enough to get by on that test. Describe the general approach to applying the Lagrangian technique to generate (but not solve) the differential equations of motion. Make it clear in your discussion how to handle situations where more than one object in the system is in motion (such as blocks connected by a rope) and how the presence of constraints in the system (such as a rope connecting the blocks) affects your analysis. Your essay should be about 250 words long (one double spaced, typed page). 13. Conceptual Question: Suppose you are given that a particle of mass m moves in one dimension under the influence of a known potential energy function, V(x). Describe in words the process you would go through to determine (i) if there are any equilibrium points, (ii) if any of those points are stable (include here a definition of what it means for the equilibrium point to be stable), and (iii) how you would determine the oscillation frequency for small oscillations about any stable equilibrium point (discuss here why this usually works only for small oscillations). Your response should be descriptive, including few if any equations. It should be about 250 words long (one double spaced, typed page). Copyright 2016 Marshall Thomsen 6

Further Insight Into Lagrange Multipliers This material is designed to be read in conjunction with Fowles & Cassiday section 10.7. In the basic Lagrangian approach to analyzing physics problems, the focus is on energy. Forces of constraint those forces that constrain a particle to follow a certain trajectory typically do not change the energy of the particle (just its direction of motion) and thus do not enter into the analysis. For instance, if a block slides down a frictionless ramp, there is a normal force exerted by the ramp on the block. That normal force is a force of constraint in that its value is whatever is necessary to maintain contact between the ramp and the block. Viewed from another perspective, the function of the normal force is to take a two dimensional problem (unconstrained motion in the xy plane) and constrain it to one dimensional motion along the surface of the ramp. As noted in earlier sections of Chapter 10, this normal force is always perpendicular to the displacement of the block and hence it does no work on the block. Since it does no work on the block, it does not influence the energy analysis and so can be ignored in the simple Lagrangian formalism. Now suppose we do not want to ignore the force of constraint. That is, suppose as part of our analysis, we need to calculate that force. Conceptually, the strategy is as follows: we imagine a world where the block can move in the xy plane completely unconstrained. Then we ask ourselves what additional force will be necessary in order to constrain the block to the desired path? The mathematical procedure for this is as follows: 1. Set up expressions for the kinetic and potential energies under the assumption that the motion is unconstrained. In the ramp case, this means assuming that the ramp is not present. 2. Construct the Lagrangian in the usual way, L=T-V. 3. Write out an equation f( )=0, where f is a function of some or all of the coordinates in the system and describes how those coordinates are related to each other to produce the desired constraint. Note in this example, I am assuming there is just one constraint and hence one constraining force to calculate. 4. Modify the Lagrange equations to read d + λ f = 0, where λ is q i dt!q i q i known as the Lagrange multiplier. At this stage, it is an unknown function. As discussed in the textbook, λ determines the constraining force or toque. That is, if our goal is to find the constraining force, we need to solve for λ. Calculate the derivatives in the above equation for each qi, as you would for any regular Lagrange problem, but do not yet apply the constraint. We are still imagining a world where the block is free to move in the entire xy plane. Copyright 2016 Marshall Thomsen 8

5. After you have calculated the derivatives, then apply the constraint between the variables (in the ramp case, you would set y equal to a constant times x). The Lagrange equations then supply you with the information about what it takes (i.e., what size force is needed) to constraint the block to move along the surface of the ramp. The ordering of operations in steps 4 and 5 is essential. We first need to work through Lagrange s equations as if the particle were unconstrained in order to get enough information to determine what it will take to constrain the motion. A concrete example may clarify these steps. A block of mass m slides down a frictionless ramp that makes an angle of θ with respect to the horizontal. We will use the Lagrange multiplier approach to determine both the acceleration of the block and the normal force exerted by the ramp on the block. y x First we assume the block is free to move in two dimensions: T = 1 2 m!x2 + 1 2 m!y2 V = mgy L = T V = 1 2 m!x2 + 1 2 m!y2 mgy Next we describe the constraint. We know that x and y are related by tanθ. Getting the signs right gives y = x tanθ where I have taken the origin of my coordinate system to be at the top of the ramp, so that as x increases, y becomes more negative. We need to rewrite this constraint in the form required in step 3: f x, y ( ) = y + x tanθ = 0 Now I will work out the Lagrange equations for x and for y: Copyright 2016 Marshall Thomsen 9

x = 0 x d dt y = mg!x = m!x d dt!x = m!!x f x = tanθ!x + λ f = 0 0 m!!x + λ tanθ = 0 ( Eq.1) x!y = m!y d dt!y = m!!y f y =1 y d dt!y + λ f = 0 mg m!!y + λ = 0 ( Eq. 2) y Just to emphasize, the final term in each equation gives us the flexibility to apply the constraint. If we set λ=0, then this problem reduces to a particle free to move in the xy plane. We will of course not set λ=0 and instead constrain the particle to move along the ramp: y = x tanθ!!y =!!x tanθ ( Eq. 3) From Eq. 2 : ( ) mg m!!y + λ = 0 mg + m!!x tanθ + λ = 0 λ = mg m!!x tanθ Eq. 4 From Eqs.1& 4 : m!!x + λ tanθ = 0 m!!x + ( mg m!!x tanθ)tanθ = 0 ( ) + mgtanθ = 0 m!!x 1+ tan 2 θ # sinθ &!!x = g tanθ % ( 1+ tan 2 θ = g tanθ tanθ $ cosθ ' = g = g 1+ sin2 θ cos 2 θ + sin 2 θ # 1 & % ( cos 2 θ cos 2 θ $ cos 2 θ ' From Eq. 3 & 5!!y =!!x tanθ = ( gsinθ cosθ)tanθ = gsin 2 θ From Eq. 4 & 5 λ = mg m!!x tanθ = mg m( gsinθ cosθ)tanθ = mg 1 sin 2 θ = gsinθ cosθ ( Eq. 5) ( ) = mgcos 2 θ Now to connect these to the much simpler introductory physics approach to this problem:! a =!!xî +!!y ĵ = gsinθ cosθ î gsin2 θ ĵ = gsinθ cosθ î sinθ ĵ ( ) That is, the box slides down the ramp (that is the direction of the above unit vector) with an acceleration whose magnitude is g sinθ. Copyright 2016 Marshall Thomsen 10

Finally, to determine the normal force (which is what this problem is all about!), the textbook argues that the generalized forces (which are typically the actual force or a torque) are found through the calculation, Q i = λ f q i F x = λ f x = mgcos2 θ tanθ ( ) = mgcosθ sinθ F y = λ f y = mgcos2 θ ( 1) = mgcos 2 θ! F = F x î + F y ĵ = mgcosθ sinθî + mgcos2 θ ĵ = mgcosθ sinθî + cosθ ĵ ( ) This correctly describes the constraint force (in this case the normal force) as pointing perpendicular to the incline with a magnitude of mg cosθ. Copyright 2016 Marshall Thomsen 11