Figure 7.12 shows a crude model of a yoyo. A massless

Write down the Lagrangian for a projectile (subject to no air resistance) in terms of its Cartesian coordinates (x, y, z), with z measured vertically upward. Find the three Lagrange equations and show that they are exactly what you would expect for the equations of motion.

Write down the Lagrangian for a one-dimensional particle moving along the x axis and subject to a force F = -kx (with k positive). Find the Lagrange equation of motion and solve it.

Consider a mass in moving in two dimensions with potential energy U (x, y) = Zkr2, where r2 = x2 + y2. Write down the Lagrangian, using coordinates x and y, and find the two Lagrange equations of motion. Describe their solutions. [This is the potential energy of an ion in an "ion trap," which can be used to study the properties of individual atomic ions.]

Consider a mass m moving in a frictionless plane that slopes at an angle a with the horizontal. Write down the Lagrangian in terms of coordinates x, measured horizontally across the slope, and y, measured down the slope. (Treat the system as two-dimensional, but include the gravitational potential energy.) Find the two Lagrange equations and show that they are what you should have expected.

Find the components of V f (r, 0) in two-dimensional polar coordinates. [Hint: Remember that the change in the scalar f as a result of an infinitesimal displacement dr is df =V f dr.]

Consider two particles moving unconstrained in three dimensions, with potential energy U(r1, r2). (a) Write down the six equations of motion obtained by applying Newton's second law to each particle. (b) Write down the Lagrangian L (ri, r2, ri,t2) = T U and show that the six La-grange equations are the same as the six Newtonian equations of part (a). This establishes the validity of Lagrange's equations in rectangular coordinates, which in turn establishes Hamilton's principle. Since the latter is independent of coordinates, this proves Lagrange's equations in any coordinate system.

Do Problem 7.6, but for N particles moving unconstrained in three dimensions (in which case there are 3N equations of motion).

(a) Write down the Lagrangian L (x1, x2, xl, i2) for two particles of equal masses, m1 = m2 = m, confined to the x axis and connected by a spring with potential energy U = -1kx2. [Here x is the extension of the spring, x = (x1 x2 1), where 1 is the spring's unstretched length, and I assume that mass 1 remains to the right of mass 2 at all times.] (b) Rewrite L in terms of the new variables X = i (x1 + x2) (the CM position) and x (the extension), and write down the two Lagrange equations for X and x. (c) Solve for X (t) and x (t) and describe the motion.

Consider a bead that is threaded on a rigid circular hoop of radius R lying in the xy plane with its center at 0, and use the angle 4) of two-dimensional polar coordinates as the one generalized coordinate to describe the bead's position. Write down the equations that give the Cartesian coordinates (x, y) in terms of and the equation that gives the generalized coordinate in terms of (x, y).

A particle is confined to move on the surface of a circular cone with its axis on the z axis, vertex at the origin (pointing down), and half-angle a. The particle's position can be specified by two generalized coordinates, which you can choose to be the coordinates (p, 0) of cylindrical polar coordinates. Write down the equations that give the three Cartesian coordinates of the particle in terms of the generalized coordinates (p, 0) and vice versa.

Consider the pendulum of Figure 7.4, suspended inside a railroad car, but suppose that the car is oscillating back and forth, so that the point of suspension has position xs = A cos wt, ys = 0. Use the angle as the generalized coordinate and write down the equations that give the Cartesian coordinates of the bob in terms of 0 and vice versa.

Lagrange's equations in the form discussed in this chapter hold only if the forces (at least the non constraint forces) are derivable from a potential energy. To get an idea how they can be modified to include forces like friction, consider the following: A single particle in one dimension is subject to various conservative forces (net conservative force \(=F=-\partial U / \partial x\)) and a nonconservative force (let's call it \(F_{\text {fric }}\)) Define the Lagrange as \(\mathcal{L}=T-U\) and show that the appropriate modification is \(\frac{\partial \mathcal{L}}{\partial x}+F_{\mathrm{fric}}=\frac{d}{d t} \frac{\partial \mathcal{L}}{\partial \dot{x}}\).

In Section 7.4 [Equations (7.41) through (7.51)], I proved Lagrange's equations for a single particle constrained to move on a two-dimensional surface. Go through the same steps to prove Lagrange's equations for a system consisting of two particles subject to various unspecified constraints. [Hint: The net force on particle 1 is the sum of the total constraint force VI' and the total nonconstraint force F1, and likewise for particle 2. The constraint forces come in many guises (the normal force of a surface, the tension force of a string tied between the particles, etc.), but it is always true that the net work done by all constraint forces in any displacement consistent with the constraints is zero this is the defining property of constraint forces. Meanwhile, we take for granted that the nonconstraint forces are derivable from a potential energy U(ri, r2, t); that is, F1 = V1U and likewise for particle 2. Write down the difference SS between the action integral for the right path given by r1(t) and r2(t) and any nearby wrong path given by r1(t) E1(t) and r2(t) 62(0. Paralleling the steps of Section 7.4, you can show that SS is given by an integral analogous to (7.49), and this is zero by the defining property of constraint forces.]

Figure 7.12 shows a crude model of a yoyo. A massless string is suspended vertically from a fixed point and the other end is wrapped several times around a uniform cylinder of mass m and radius R. When the cylinder is released it moves vertically down, rotating as the string unwinds. Write down the Lagrangian, using the distance x as your generalized coordinate. Find the Lagrange equation of motion and show that the cylinder accelerates downward withi = 2g/3. [Hints: You need to remember from your introductory physics course that the total kinetic energy of a body like the yoyo is T = lmv2 11(02, where v is the velocity of the center of mass, I is the moment of inertia (for a unform cylinder, I = im R2) and co is the angular velocity about the CM. You can express w in terms of I.]

A mass mi rests on a frictionless horizontal table and is attached to a massless string. The string runs horizontally to the edge of the table, where it passes over a massless, frictionless pulley and then hangs vertically down. A second mass m2 is now attached to the bottom end of the string. Write down the Lagrangian for the system. Find the Lagrange equation of motion, and solve it for the acceleration of the blocks. For your generalized coordinate, use the distance x of the second mass below the tabletop.

Write down the Lagrangian for a cylinder (mass m, radius R, and moment of inertia I) that rolls without slipping straight down an inclined plane which is at an angle a from the horizontal. Use as your generalized coordinate the cylinder's distance x measured down the plane from its starting point. Write down the Lagrange equation and solve it for the cylinder's acceleration x. Remember that T = 4mv2 + z L02, where v is the velocity of the center of mass and co is the angular velocity.

Use the Lagrangian method to find the acceleration of the Atwood machine of Example 7.3 (page 255) including the effect of the pulley's having moment of inertia I. (The kinetic energy of the pulley is 41(.02, where co is its angular velocity.)

A mass m is suspended from a massless string, the other end of which is wrapped several times around a horizontal cylinder of radius R and moment of inertia I, which is free to rotate about a fixed horizontal axle. Using a suitable coordinate, set up the Lagrangian and the Lagrange equation of motion, and find the acceleration of the mass m. [The kinetic energy of the rotating cylinder is i /6)21

In Example 7.5 (page 258) the two accelerations are given by Equations (7.66) and (7.67). Check that the acceleration of the block is given correctly in the limit M 0. [You need to find the components of this acceleration relative to the table.]

A smooth wire is bent into the shape of a helix, with cylindrical polar coordinates p = R and z = X0, where R and X are constants and the z axis is vertically up (and gravity vertically down). Using z as your generalized coordinate, write down the Lagrangian for a bead of mass m threaded on the wire. Find the Lagrange equation and hence the bead's vertical acceleration E. In the limit that R 0, what is i? Does this make sense?

The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a horizontal plane with constant angular velocity w. Write down the Lagrangian for a bead threaded on the rod, using r as your generalized coordinate, where r, are the polar coordinates of the bead. (Notice that 0 is not an independent variable since it is fixed by the rotation of the rod to be 0 = wt.) Solve Lagrange's equation for r(t). What happens if the bead is initially at rest at the origin? If it is released from any point ro > 0, show that r(t) eventually grows exponentially. Explain your results in terms of the centrifugal force mw2r.

Using the usual angle 0 as generalized coordinate, write down the Lagrangian for a simple pendulum of length I suspended from the ceiling of an elevator that is accelerating upward with constant acceleration a. (Be careful when writing T; it is probably safest to write the bob's velocity in component form.) Find the Lagrange equation of motion and show that it is the same as that for a normal, nonaccelerating pendulum, except that g has been replaced by g + a. In particular, the angular frequency of small oscillations is ./(g + a)/l.

A small cart (mass m) is mounted on rails inside a large cart. The two are attached by a spring (force constant k) in such a way that the small cart is in equilibrium at the midpoint of the large. The distance of the small cart from its equilibrium is denoted x and that of the large one from a fixed point on the ground is X, as shown in Figure 7.13. The large cart is now forced to oscillate such that X = A cos wt, with both A and w fixed. Set up the Lagrangian for the motion of the small cart and show that the Lagrange equation has the form + wax = B cos wt where coo is the natural frequency coo = ,/k/m and B is a constant. This is the form assumed in Section 5.5, Equation (5.57), for driven oscillations (except that we are here ignoring damping). Thus the system Figure 7.13 Problem 7.23 described here would be one way to realize the motion discussed there. (We could fill the large cart with molasses to provide some damping.)

We saw in Example 7.3 (page 255) that the acceleration of the Atwood machine is x = (m m2)g/(mi+ m2). It is sometimes claimed that this result is "obvious" because, it is said, the effective force on the system is (m1 m2)g and the effective mass is (m1 + m2). This is not, perhaps, all that obvious, but it does emerge very naturally in the Lagrangian approach. Recall that Lagrange's equation can be thought of as [Equation (7.17)] (generalized force) = (rate of change of generalized momentum). Show that for the Atwood machine the generalized force is (m1 m2)g and the generalized momentum (mi m2).i. Comment.

Prove that the potential energy of a central force F = krni (with n 1) is U = krn+11(n 1). In particular, if n = 1, then F = kr and U = tkr2

In Example 7.7 (page 264), we saw that the bead on a spinning hoop can make small oscillations about any of its stable equilibrium points. Verify that the oscillation frequency S2' defined in (7.79) is equal to /w2 (g 1 coR)2 as claimed in (7.80).

Consider a double Atwood machine constructed as follows: A mass 4m is suspended from a string that passes over a massless pulley on frictionless bearings. The other end of this string supports a second similar pulley, over which passes a second string supporting a mass of 3m at one end and m at the other. Using two suitable generalized coordinates, set up the Lagrangian and use the Lagrange equations to find the acceleration of the mass 4m when the system is released. Explain why the top pulley rotates even though it carries equal weights on each side.

A couple of points need checking from Example 7.6 (page 260). (a) From the point of view of a noninertial frame rotating with the hoop, the bead is subject to the force of gravity and a centrifugal force mw2p (in addition to the constraint force, which is the normal force of the wire). Verify that at the equilibrium points given by (7.71), the tangential components of these two forces balance one another. (A free-body diagram will help.) (b) Verify that the equilibrium point at the top (0 = tr) is unstable. (c) Verify that the equilibrium at the second point given by (7.71) (the one on the left, with 0 negative) is stable.

Figure 7.14 shows a simple pendulum (mass m, length l) whose point of support P is attached to the edge of a wheel (center O, radius R) that is forced to rotate at a fixed angular velocity \(\omega\) . At t = 0, the point P is level with O on the right. Write down the Lagrange and find the equation of motion for the angle . [Hint: Be careful writing down the kinetic energy T. A safe way to get the velocity right is to write down the position of the bob at time t, and then differentiate.] Check that your answer makes sense in the special case that .

Consider the pendulum of Figure 7.4, suspended inside a railroad car that is being forced to accelerate with a constant acceleration a. (a) Write down the Lagrangian for the system and the equation of motion for the angle 0. Use a trick similar to the one used in Equation (5.11) to write the combination of sin 0 and cos 0 as a multiple of sin( 0 + /3). (b) Find the equilibrium angle 0 at which the pendulum can remain fixed (relative to the car) as the car accelerates. Use the equation of motion to show that this equilbrium is stable. What is the frequency of small oscillations about this equilibrium position? (We shall find a much slicker way to solve this problem in Chapter 9, but the Lagrangian method does give a straightforward route to the answer.)

A simple pendulum (mass M and length L) is suspended from a cart (mass m) that can oscillate on the end of a spring of force constant k, as shown in Figure 7.15. (a) Write the Lagrangian in terms of the two generalized coordinates x and 0, where x is the extension of the spring from its equilibrium length. (Read the hint in Problem 7.29.) Find the two Lagrange equations. (Warning: They're pretty ugly!) (b) Simplify the equations to the case that both x and 0 are small. (They're still pretty ugly, and note, in particular, that they are still coupled; that is, each equation involves both variables. Nonetheless, we shall see how to solve these equations in Chapter 11 see particularly Problem 11.19.) Figure 7.15 Problem 7.31

Consider the cube balanced on a cylinder as described in Example 4.7 (page 130). Assuming that b < r, use the Lagrangian approach to find the angular frequency of small oscillations about the top. The simplest procedure is to make the small-angle approximations to before you differentiate to get Lagrange's equation. As usual, be careful in writing down the kinetic energy; this is Z (1111.12 + 162), where v is the speed of the CM and / is the moment of inertia about the CM (2mb2/3). The safe way to find v is to write down the coordinates of the CM and then differentiate.

A bar of soap (mass m) is at rest on a frictionless rectangular plate that rests on a horizontal table. At time t = 0, I start raising one edge of the plate so that the plate pivots about the opposite edge with constant angular velocity co, and the soap starts to slide toward the downhill edge. Show that the equation of motion for the soap has the form x co2x = g sin cot, where x is the soap's distance from the downhill edge. Solve this for x (t), given that x (0) = xo. [You'll need to use the method used to solve Equation (5.48). You can easily solve the homogeneous equation; for a particular solution try x = A sin cot and solve for A.]

Consider the well-known problem of a cart of mass m moving along the x axis attached to a spring (force constant k), whose other end is held fixed (Figure 5.2). If we ignore the mass of the spring (as we almost always do) then we know that the cart executes simple harmonic motion with angular frequency co = .\/k/m. Using the Lagrangian approach, you can find the effect of the spring's mass M, as follows: (a) Assuming that the spring is uniform and stretches uniformly, show that its kinetic energy is 6Mx2. (As usual x is the extension of the spring from its equilibrium length.) Write down the Lagrangian for the system of cart plus spring. (Note: The potential energy is still ikx2.) (b) Write down the Lagrange equation and show that the cart still executes SHM but with angular frequency co = Vkl(m M/3); that is, the effect of the spring's mass M is just to add M/3 to the mass of the cart. Figure 7.16 Problem 7.35

Figure 7.16 is a bird's-eye view of a smooth horizontal wire hoop that is forced to rotate at a fixed angular velocity co about a vertical axis through the point A. A bead of mass m is threaded on the hoop and is free to move around it, with its position specified by the angle (/) that it makes at the center with the diameter AB. Find the Lagrangian for this system using as your generalized coordinate. (Read the hint in Problem 7.29.) Use the Lagrange equation of motion to show that the bead oscillates about the point B exactly like a simple pendulum. What is the frequency of these oscillations if their amplitude is small?

A pendulum is made from a massless spring (force constant k and unstretched length /0) that is suspended at one end from a fixed pivot 0 and has a mass m attached to its other end. The spring can stretch and compress but cannot bend, and the whole system is confined to a single vertical plane. (a) Write down the Lagrangian for the pendulum, using as generalized coordinates the usual angle (/) and the length r of the spring. (b) Find the two Lagrange equations of the system and interpret them in terms of Newton's second law, as given in Equation (1.48). (c) The equations of part (b) cannot be solved analytically in general. However, they can be solved for small oscillations. Do this and describe the motion. [Hint: Let 1 denote the equilibrium length of the spring with the mass hanging from it and write r = 1 + E. "Small oscillations" involve only small values of c and 0, so you can use the small-angle approximations and drop from your equations all terms that involve powers of E or 0 (or their derivatives) higher than the first power (also products of E and 0 or their derivatives). This dramatically simplifies and uncouples the equations.]

Two equal masses, ml = m2 = m, are joined by a massless string of length L that passes through a hole in a frictionless horizontal table. The first mass slides on the table while the second hangs below the table and moves up and down in a vertical line. (a) Assuming the string remains taut, write down the Lagrangian for the system in terms of the polar coordinates (r, 0) of the mass on the table. (b) Find the two Lagrange equations and interpret the 0 equation in terms of the angular momentum of the first mass. (c) Express in terms of t and eliminate 4 from the r equation. Now use the r equation to find the value r = ro at which the first mass can move in a circular path. Interpret your answer in Newtonian terms. (d) Suppose the first mass is moving in this circular path and is given a small radial nudge. Write r (t) = ro E (t) and rewrite the r equation in terms of E (t) dropping all powers of E (t) higher than linear. Show that the circular path is stable and that r(t) oscillates sinusoidally about r0. What is the frequency of its oscillations?

A particle is confined to move on the surface of a circular cone with its axis on the vertical z axis, vertex at the origin (pointing down), and half-angle a. (a) Write down the Lagrangian L in terms of the spherical polar coordinates r and 0. (b) Find the two equations of motion. Interpret the 0 equation in terms of the angular momentum tz, and use it to eliminate 4 from the r equation in favor of the constant fz. Does your r equation make sense in the case that = 0? Find the value ro of r at which the particle can remain in a horizontal circular path. (c) Suppose that the particle is given a small radial kick, so that r(t) = ro E(t), where E(t) is small. Use the r equation to decide whether the circular path is stable. If so, with what frequency does r oscillate about r0?

a) Write down the Lagrangian for a particle moving in three dimensions under the influence of a conservative central force with potential energy U (r), using spherical polar coordinates (r, 0, 4'). (b) Write down the three Lagrange equations and explain their significance in terms of radial accelera- tion, angular momentum, and so forth. (The 0 equation is the tricky one, since you will find it implies that the 0 component of varies with time, which seems to contradict conservation of angular mo- mentum. Remember, however, that to is the component of in a variable direction.) (c) Suppose that initially the motion is in the equatorial plane (that is, 00 = 7r/2 and 60 = 0). Describe the subsequent motion. (d) Suppose instead that the initial motion is along a line of longitude (that is, 4o = 0). Describe the subsequent motion.

The "spherical pendulum" is just a simple pendulum that is free to move in any sideways direction. (By contrast a "simple pendulum" unqualified is confined to a single vertical plane.) The bob of a spherical pendulum moves on a sphere, centered on the point of support with radius r = R, the length of the pendulum. A convenient choice of coordinates is spherical polars, r, 0, 0, with the origin at the point of support and the polar axis pointing straight down. The two variables 0 and 0 make a good choice of generalized coordinates. (a) Find the Lagrangian and the two Lagrange equations. (b) Explain what the 0 equation tells us about the z component of angular momentum tz. (c) For the special case that 0 = const, describe what the 0 equation tells us. (d) Use the 0 equation to replace cb by tz in the 0 equation and discuss the existence of an angle 00 at which 0 can remain constant. Why is this motion called a conical pendulum? (e) Show that if 0 = 00 E, with c small, then 0 oscillates about 00 in harmonic motion. Describe the motion of the pendulum's bob.

Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity w about its vertical axis, as shown in Figure 7.17. Use cylindrical polar coordinates and let the equation of the parabola be z = kp2 . Write down the Lagrangian in terms of p as the generalized coordinate. Find the equation of motion of the bead and determine whether there are positions of equilibrium, that is, values of p at which the bead can remain fixed, without sliding up or down the spinning wire. Discuss the stability of any equilibrium positions you find. Figure 7.17 Problem 7.41

[Computer] In Example 7.7 (page 264), we saw that the bead on a spinning hoop can make small oscillations about its nonzero stable equilibrium points that are approximately sinusoidal, with frequency Q' = ,Ico2 (g I co R)2 as in (7.80). Investigate how good this approximation is by solving the equation of motion (7.73) numerically and then plotting both your numerical solution and the approximate solution (t) = 00 + A cos(Q't 8) on the same graph. Use the following numbers: g = R = 1 and (1)2 = 2, and initial conditions 6(0) = 0 and 0 (0) = 00 + E0, where E0 = 1. Repeat with co = 10. Comment on your results.

Computer] Consider a massless wheel of radius R mounted on a frictionless horizontal axis. A point mass M is glued to the edge, and a massless string is wrapped several times around the perimeter and hangs vertically down with a mass m suspended from its bottom end. (See Figure 4.28.) Initially I am holding the wheel with M vertically below the axle. At t = 0, I release the wheel, and m starts to fall vertically down. (a) Write down the Lagrangian = T U as a function of the angle 0 through which the wheel has turned. Find the equation of motion and show that, provided m < M, there is one position of stable equilibrium. (b) Assuming m < M, sketch the potential energy U(0) for -7 < < 47 and use your graph to explain the equilibrium position you found. (c) Because the equation of motion cannot be solved in terms of elementary functions, you are going to solve it numerically. This requires that you choose numerical values for the various parameters. Take M = g = R = 1 (this amounts to a convenient choice of units) and m = 0.7. Before solving the equation make a careful plot of U (0) against 0 and predict the kind of motion expected when M is released from rest at 0 = 0. Now solve the equation of motion for 0 < t < 20 and verify your prediction. (d) Repeat part (c), but with m = 0.8.

[Computer] If you haven't already done so, do Problem 7.29. One might expect that the rotation of the wheel would have little effect on the pendulum, provided the wheel is small and rotates slowly. (a) Verify this expectation by solving the equation of motion numerically, with the following numbers: Take g and 1 to be 1. (This means that the natural frequency N/g1 1 of the pendulum is also 1.) Take co = 0.1, so that the wheel's rotational frequency is small compared to the natural frequency of the pendulum; and take the radius R = 0.2, significantly less than the length of the pendulum. As initial conditions take 0 = 0.2 and 0 = 0 at t = 0, and make a plot of your solution 0 (t) for 0 < t < 20. Your graph should look very like the sinusoidal oscillations of an ordinary simple pendulum. Does the period look correct? (b) Now plot (t) for 0 < t < 100 and notice that the rotating support does make a small difference, causing the amplitude of the oscillations to grow and shrink periodically. Comment on the period of these small fluctuations.

(a) Verify that the coefficients Ai j in the important expression (7.94) for the kinetic energy of any "natural" system are symmetric; that is, Ai j = A11. (b) Prove that for any n variables v1, , v dv. . j,k jkViVk [Hint: Start with the case that n = 2, for which you can write out the sums in full. Notice that you need the result of part (a).] This identity is useful in many areas of physics; we needed it to prove the expression (7.96) for the generalized momentum pi.

Noether's theorem asserts a connection between invariance principles and conservation laws. In Section 7.8 we saw that translational invariance of the Lagrangian implies conservation of total linear momentum. Here you will prove that rotational invariance of ,C implies conservation of total angular momentum. Suppose that the Lagrangian of an N-particle system is unchanged by rotations about a certain symmetry axis. (a) Without loss of generality, take this axis to be the z axis, and show that the Lagrangian is unchanged when all of the particles are simultaneously moved from (ra, Oa, 0e) to (re, Oa, Oa E) (same c for all particles). Hence show that E .0. (x=i aoc, (b) Use Lagrange's equations to show that this implies that the total angular momentum L, about the symmetry axis is constant. In particular, if the Lagrangian is invariant under rotations about all axes, then all components of L are conserved.

In Chapter 4 (at the end of Section 4.7) I claimed that, for a system with one degree of freedom, positions of stable equilibrium "normally" correspond to minima of the potential energy U (q). Using Lagrangian mechanics, you can now prove this claim. (a) Consider a one-degree system of N particles with positions ra = ra(q), where q is the one generalized coordinate and the transformation between r and q does not depend on time; that is, q is what we have now agreed to call "natural." (This is the meaning of the qualification "normally" in the statement of the claim. If the transformation depends on time, then the claim is not necessarily true.) Prove that the KE has the form T = 4./442, where A = A(q) > 0 may depend on q but not on 4. [This corresponds exactly to the result (7.94) for n degrees of freedom. If you have trouble with the proof here, review the proof there.] Show that the Lagrange equation of motion has the form dU 1 d A .2 A(q).4 = d 2 d q . q q (b) A point go is an equilibrium point if, when the system is placed at go with 4 = 0, it remains there. Show that qo is an equilibrium point if and only if dU I dq = 0. (c) Show that the equilibrium is stable if and only if U is minimum at go. (d) If you did Problem 7.30, show that the pendulum of that problem does not satisfy the conditions of this problem and that the result proved here is false for that system.

Let F = F (q 1, , qn) be any function of the generalized coordinates (q1, , qn) of a system with Lagrangian (qi, , qn, 41, , 4, t). Prove that the two Lagrangians 4, and LI = dF I dt give exactly the same equations of motion.

Consider a particle of mass in and charge q moving in a uniform constant magnetic field B in the z direction. (a) Prove that B can be written as B= V x A with A = 413 x r. Prove equivalently that in cylindrical polar coordinates, A = 1Bp b. (b) Write the Lagrangian (7.103) in cylindrical polar coordinates and find the three corresponding Lagrange equations. (c) Describe in detail those solutions of the Lagrange equations in which p is a constant.

A mass m1 rests on a frictionless horizontal table. Attached to it is a string which runs hori-zontally to the edge of the table, where it passes over a frictionless, small pulley and down to where it supports a mass m2. Use as coordinates x and y the distances of mi and m2 from the pulley. These satisfy the constraint equation f (x, y)=--x-Fy=- const. Write down the two modified Lagrange equa-tions and solve them (together with the constraint equation) for x, 53, and the Lagrange multiplier X. Use (7.122) (and the corresponding equation in y) to find the tension forces on the two masses. Verify your answers by solving the problem by the elementary Newtonian approach.

Write down the Lagrangian for the simple pendulum of Figure 7.2 in terms of the rectangular coordinates x and y. These coordinates are constrained to satisfy the constraint equation f y) Vx2 y2 = 1. (a) Write down the two modified Lagrange equations (7.118) and (7.119). Comparing these with the two components of Newton's second law, show that the Lagrange multiplier is (minus) the tension in the rod. Verify Equation (7.122) and the corresponding equation in y. (b) The constraint equation can be written in many different ways. For example we could have written 1(x, y) x2 + y2 = *2. t Check that using this function would have given the same physical results.

The method of Lagrange multipliers works perfectly well with non-Cartesian coordinates. Consider a mass m that hangs from a string, the other end of which is wound several times around a wheel (radius R, moment of inertia I) mounted on a frictionless horizontal axle. Use as coordinates for the mass and the wheel x, the distance fallen by the mass, and 4, the angle through which the wheel has turned (both measured from some convenient reference position). Write down the modified Lagrange equations for these two variables and solve them (together with the constraint equation) for .3j and and the Lagrange multiplier. Write down Newton's second law for the mass and wheel, and use them to check your answers for x and 4i. Show that X of/ax is indeed the tension force on the mass. Comment on the quantity X afoo.