chapter contents preface problems solved in student solutions manual vii matrices, vectors, and vector calculus newtonian mechanics—single particle Related Posts: pdf book: PHYSICS FOR SCIENTIST AND ENGINEERS by Serway, 8th edition CLICK HERE TO DOWNLOAD pdf book. Classical Dynamics of Particles and Systems,. 5th edition (Brooks/Cole), from S.T. Thornton, and J.B. Marion. Useful references: See the bibliography below.
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Thornton, Jerry B. Marion completely free. Marion PDF This smash hit established mechanics content, composed for the propelled undergrad maybe a couple semester course, gives an entire record of the traditional mechanics of particles, frameworks of particles, and inflexible bodies. Vector analytics is utilized broadly to investigate topics. The Lagrangian detailing of mechanics is acquainted ahead of schedule with demonstrate its intense critical thinking capacity.. Marion — The way to deal with molecule flow at this level denotes the main introduction for a material science understudy to a totally independent physical hypothesis. Also, in this specific book, the introduction is as much a stylish achievement as pedantic.
These are the historically universal and still most widely used terms for the equinoxes, but are potentially confusing because in the southern hemisphere the vernal equinox does not occur in spring and the autumnal equinox does not occur in autumn. The equivalent common language English terms spring equinox and autumn or fall equinox are even more ambiguous. They are still not universal, however, as not all cultures use a solar-based calendar where the equinoxes occur every year in the same month as they do not in the Islamic calendar and Hebrew calendar , for example.
The northward equinox occurs in March when the Sun crosses the equator from south to north, and the southward equinox occurs in September when the Sun crosses the equator from north to south. These terms can be used unambiguously for other planets. They are rarely seen, although were first proposed over years ago.
Due to the precession of the equinoxes , however, the constellations where the equinoxes are currently located are Pisces and Virgo , respectively.
Sunrise and sunset can be defined in several ways, but a widespread definition is the time that the top limb of the Sun is level with the horizon. Sunrise , which begins daytime, occurs when the top of the Sun's disk appears above the eastern horizon.
At that instant, the disk's centre is still below the horizon. The Earth's atmosphere refracts sunlight. As a result, an observer sees daylight before the top of the Sun's disk appears above the horizon. Their combination means that when the upper limb of the Sun is on the visible horizon, its centre is 50 arcminutes below the geometric horizon, which is the intersection with the celestial sphere of a horizontal plane through the eye of the observer.
The real equality of day and night only happens in places far enough from the equator to have a seasonal difference in day length of at least 7 minutes,  actually occurring a few days towards the winter side of each equinox. The times of sunset and sunrise vary with the observer's location longitude and latitude , so the dates when day and night are equal also depend upon the observer's location. A third correction for the visual observation of a sunrise or sunset is the angle between the apparent horizon as seen by an observer and the geometric or sensible horizon.
This is known as the dip of the horizon and varies from 3 arcminutes for a viewer standing on the sea shore to arcminutes for a mountaineer on Everest. The date on which the day and night are exactly the same is known as an equilux; the neologism , believed to have been coined in the s, achieved more widespread recognition in the 21st century. Prior to this, the word "equilux" was more commonly used as a synonym for isophot , and there was no generally accepted term for the phenomenon.
In the mid-latitudes, daylight increases or decreases by about three minutes per day at the equinoxes, and thus adjacent days and nights only reach within one minute of each other. The date of the closest approximation of the equilux varies slightly by latitude; in the mid-latitudes, it occurs a few days before the spring equinox and after the fall equinox in each respective hemisphere.
Geocentric view of the astronomical seasons This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. December Learn how and when to remove this template message In the half-year centered on the June solstice, the Sun rises north of east and sets north of west, which means longer days with shorter nights for the northern hemisphere and shorter days with longer nights for the southern hemisphere.
In the half-year centered on the December solstice, the Sun rises south of east and sets south of west and the durations of day and night are reversed. Also on the day of an equinox, the Sun rises everywhere on Earth except at the poles at about and sets at about local solar time. These times are not exact for several reasons: Most places on Earth use a time zone which differs from the local solar time by minutes or even hours.
Day length is also affected by the variable orbital speed of the Earth around the Sun. This combined effect is described as the equation of time. Thus even locations which lie on their time zone's reference meridian will not see sunrise and sunset at and At the March equinox they are 7—8 minutes later, and at the September equinox they are about 7—8 minutes earlier. Sunrise and sunset are commonly defined for the upper limb of the solar disk, rather than its center.
The upper limb is already up for at least a minute before the center appears, and the upper limb likewise sets later than the center of the solar disk. Also, when the Sun is near the horizon, atmospheric refraction shifts its apparent position above its true position by a little more than its own diameter. This makes sunrise more than two minutes earlier and sunset an equal amount later.
It is therefore important to have methods other than searching for analytic solutions to deal with dynamical systems. Phase space provides one method for nding qualitative information about the solutions. Another approach is numerical. Newtons Law, and more generally the equation 1. Thus it is always subject to numerical solution given an initial conguration, at least up until such point that some singularity in the velocity function is reached. This gives a new approximate value for at the An exception can occur at an unstable equilibrium point, where the velocity function vanishes.
The motion can just end at such a point, and several possible phase curves can terminate at that point. This is not to say that nu- Integrating the motion, for a damped harmonic oscillator.
An analytical solution, if it can be found, is almost always preferable, because It is far more likely to provide insight into the qualitative features of the motion. Numerical solutions must be done separately for each value of the parameters k, m, and each value of the initial conditions x0 and p0.
Numerical solutions have subtle numerical problems in that they are only exact as t 0, and only if the computations are done exactly. Sometimes uncontrolled approximate solutions lead to surprisingly large errors. This is a very unsophisticated method. The errors made in each step for r and p are typically O t 2. In principle therefore we can approach exact results for a nite time evolution by taking smaller and smaller time steps, but in practise there are other considerations, such as computer time and roundo errors, which argue strongly in favor of using more sophisticated numerical techniques, with errors of higher order in t.
These can be found in any text on numerical methods. Nonetheless, numerical solutions are often the only way to handle a real problem, and there has been extensive development of techniques for eciently and accurately handling the problem, which is essentially one of solving a system of rst order ordinary dierential equations. As we just saw, Newtons equations for a system of particles can be cast in the form of a set of rst order ordinary dierential equations in time on phase space, with the motion in phase space described by the velocity eld.
This could be more generally discussed as a dth order dynamical system, with a phase point representing the system in a d-dimensional phase space, moving with time t along the velocity eld, sweeping out a path in phase space called the phase curve.
The phase point t is also called the state of the system at time t. Many qualitative features of the motion can be stated in terms of the phase curve.
At other points, the system does not stay put, but there may be sets of states which ow into each other, such as the elliptical orbit for the undamped harmonic oscillator. These are called invariant sets of states. In a rst order dynamical system14 , the xed points divide the line into intervals which are invariant sets.
Even though a rst-order system is smaller than any Newtonian system, it is worthwhile discussing briey the phase ow there. We have been assuming the velocity function is a smooth function generically its zeros will be rst order, and near the xed point 0 we will have V c 0. Of course there are other possibilities: But this kind of situation is somewhat articial, and such a system is structually unstable. Thus the simple zero in the velocity function is structurally stable. Note that structual stability is quite a dierent notion from stability of the xed point.
In this discussion of stability in rst order dynamical systems, we see that generically the stable xed points occur where the velocity function decreases through zero, while the unstable points are where it increases through zero. Thus generically the xed points will alternate in stability, dividing the phase line into open intervals which are each invariant sets of states, with the points in a given interval owing either to the left or to the right, but never leaving the open interval. This form of solution is called terminating motion.
The stability of the ow will be determined by this d-dimensional square matrix M. Generically the eigenvalue equation, a dth order polynomial in , will have d distinct solutions.
Because M is a real matrix, the eigenvalues must either be real or come in complex conjugate pairs. For the real case, whether the eigenvalue is positive or negative determines the instability or stability of the ow along the direction of the eigenvector. Thus we see that the motion spirals in towards the xed point if u is negative, and spirals away from the xed point if u is positive. Stability in these directions is determined by the sign of the real part of the eigenvalue. In general, then, stability in each subspace around the xed point 0 depends on the sign of the real part of the eigenvalue.
Then 0 is an attractor and is a strongly stable xed point. On the other hand, if some of the eigenvalues have positive real parts, there are unstable directions. Starting from a generic point in any neighborhood of 0 , the motion will eventually ow out along an unstable direction, and the xed point is considered unstable, although there may be subspaces along which the ow may be into 0.
Some examples of two dimensional ows in the neighborhood of a generic xed point are shown in Figure 1. Note that none of these describe the xed point of the undamped harmonic oscillator of Figure 1. We have discussed generic situations as if the velocity eld were chosen arbitrarily from the set of all smooth vector functions, but in fact Newtonian mechanics imposes constraints on the velocity elds in many situations, in particular if there are conserved quantities.
Eect of conserved quantities on the ow If the system has a conserved quantity Q q, p which is a function on phase space only, and not of time, the ow in phase space is considerably changed. Strongly stable Strongly stable Unstable xed spiral point. Figure 1. Four generic xed points for a second order dynamical system. Unless this conserved quantity is a trivial function, i. In the terms of our generic discussion, the gradient of Q gives a direction orthogonal to the image of M, so there is a zero eigenvalue and we are not in the generic situation we discussed.
If this point is a maximum or a saddle of U, the motion along a descending path will be unstable. Such a xed point is called stable15 , but it is not strongly stable, as the ow does not settle down to 0. This is the situation we saw for the undamped harmonic oscillator. The curves 2 of constant E in phase space are ellipses, and each motion orbits the appropriate ellipse, as shown in Fig. This contrasts to the case of the damped oscillator, for which there is no conserved energy, and for which the origin is a strongly stable xed point.
A xed point is stable if it is in arbitrarity small neighborhoods, each with the property that if the system is in that neighborhood at one time, it remains in it at all later times. As an example of a conservative system with both sta0. There is a stable equix Each The velocity eld in phase space and several possible orbits are shown. Near the stax ble equilibrium, the trajectories are approximately ellipses, as they were for the harmonic os-1 cillator, but for larger energies they begin to feel the asymmetry of the potential, and Figure 1.
Motion in a cubic potenthe orbits become egg-shaped. If the system has total energy precisely U xu , the contour line crosses itself. This contour actually consists of three separate orbits. An orbit with this critical value of the energy is called a seperatrix, as it seperates regions in phase space where the orbits have dierent qualitative characteristics.
Quite generally hyperbolic xed points are at the ends of seperatrices. Exercises 1. The Earth has a mass of 6. Newtons gravitational constant is 6.
There are some situations in which we wish to focus our attention on a set of particles which changes with time, such as a rocket ship which is emitting gas continuously. Let M t be the mass of the rocket and remaining fuel at time t, assume that the fuel is emitted with velocity u with respect to the rocket, and call the velocity of the rocket v t in an inertial coordinate system.
If the external force on the rocket is F t and the external force on the innitesimal amount of exhaust is innitesimal, the fact that F t is the rate of change of the total momentum gives the equation of motion for the rocket.
Because this pair of unit vectors dier from point to point, the er and e along the trajectory of a moving particle are themselves changing with time. Assuming F to be dierentiable, show that the error which accumulates in a nite time interval T is of order t 1. Do this for several t, and see whether the error accumulated in one period meets the expectations of problem 1.
Give the xed points, the invariant sets of states, and describe the ow on each of the invariant sets. Ignore one of the horizontal directions, and describe the dynamics in terms of the angle.
Show all xed points, seperatrices, and describe all the invariant sets of states. This can be plotted on a strip, with the understanding that the left and right edges are identied. Chapter 2 Lagranges and Hamiltons Equations In this chapter, we consider two reformulations of Newtonian mechanics, the Lagrangian and the Hamiltonian formalism.
The rst is naturally associated with conguration space, extended by time, while the latter is the natural description for working in phase space. Lagrange developed his approach in in a study of the libration of the moon, but it is best thought of as a general method of treating dynamics in terms of generalized coordinates for conguration space. It so transcends its origin that the Lagrangian is considered the fundamental object which describes a quantum eld theory.
Hamiltons approach arose in in his unication of the language of optics and mechanics. It too had a usefulness far beyond its origin, and the Hamiltonian is now most familiar as the operator in quantum mechanics which determines the evolution in time of the wave function. We begin by deriving Lagranges equation as a simple change of coordinates in an unconstrained system, one which is evolving according to Newtons laws with force laws given by some potential.
Lagrangian mechanics is also and especially useful in the presence of constraints, so we will then extend the formalism to this more general situation. This particular combination of T r with U r to get the more complicated L r, r seems an articial construction for the inertial cartesian coordinates, but it has the advantage of preserving the form of Lagranges equations for any set of generalized coordinates.
As we did in section 1. We are treating the Lagrangian here as a scalar under coordinate transformations, in the sense used in general relativity, that its value at a given physical point is unchanged by changing the coordinate system used to dene that point. The rst term vanishes because qk depends only on the coordinates xk and t, but not on the xk. From the inverse relation to 1. Lagranges equation involves the time derivative of this. It is called the stream derivative, a name which comes from uid mechanics, where it gives the rate at which some property dened throughout the uid, f r, t , changes for a xed element of uid as the uid as a whole ows.
We write it as a total derivative to indicate that we are following the motion rather than evaluating the rate of change at a xed point in space, as the partial derivative does. In fact, from 2.
Lagranges equation in cartesian coordinates says 2. It is primarily for this reason that this particular and peculiar combination of kinetic and potential energy is useful.
Note that we implicity assume the Lagrangian itself transformed like a scalar, in that its value at a given physical point of conguration space is independent of the choice of generalized coordinates that describe the point. The change of coordinates itself 2.
We now wish to generalize our discussion to include contraints. At the same time we will also consider possibly nonconservative forces. As we mentioned in section 1. We will assume the constraints are holonomic, expressible as k real functions r1 , There may also be other forces, which we will call FiD and will treat as having a dynamical eect. These are given by known functions of the conguration and time, possibly but not necessarily in terms of a potential. This distinction will seem articial without examples, so it would be well to keep these two in mind.
In each of these cases the full conguration space is R3 , but the constraints restrict the motion to an allowed subspace of extended conguration space.
In section 1. The rod exerts the constraint force to avoid compression or expansion. The natural assumption to make is that the force is in the radial direction, and therefore has no component in the direction of allowed motions, the tangential directions. Consider a bead free to slide without friction on the spoke of a rotating bicycle wheel3 , rotating about a xed axis at xed angular velocity. That is, for the polar angle of inertial coordinates,: Here the allowed subspace is not time independent, but is a helical sort of structure in extended conguration space.
We expect the force exerted by the spoke on the bead to be in the e Unlike a real bicycle wheel, we are assuming here that the spoke is directly along a radius of the circle, pointing directly to the axle. This is again perpendicular to any virtual displacement, by which we mean an allowed change in conguration at a xed time. It is important to distinguish this virtual displacement from a small segment of the trajectory of the particle.
In this case a virtual displacement is a change in r without a change in , and is perpendicular to e. So again, we have the net virtual work of the constraint forces is zero.
It is important to note that this does not mean that the net real work is zero. In a small time interval, the displacement r includes a component rt in the tangential direction, and the force of constraint does do work! We will assume that the constraint forces in general satisfy this restriction that no net virtual work is done by the forces of constraint for any possible virtual displacement.
We can multiply by an arbitrary virtual displacement i. This gives an equation which determines the motion on the constrained subspace and does not involve the unspecied forces of constraint F C. We drop the superscript D from now on. Suppose we know generalized coordinates q1 ,.
Dierentiating 2. The rst term in the equation 2. The generalized force Qj has the same form as in the unconstrained case, as given by 1. Notice that Qj depends only on the value of U on the constrained surface. This is Lagranges equation, which we have now derived in the more general context of constrained systems. Some examples of the use of Lagrangians Atwoods machine consists of two blocks of mass m1 and m2 attached by an inextensible cord which suspends them from a pulley of moment of inertia I with frictionless bearings.
This one degree of freedom parameterizes the line which is the allowed subspace of the unconstrained conguration space, a three dimensional space which also has directions corresponding to the angle of the pulley and the height of the second mass. The constraints restrict these three variables because the string has a xed length and does not slip on the pulley. Note that this formalism has permitted us to solve the problem without solving for the forces of constraint, which in this case are the tensions in the cord on either side of the pulley.
As a second example, reconsider the bead on the spoke of a rotating bicycle wheel. The velocity-independent term in T acts just like a potential would, and can in fact be considered the potential for the centrifugal force.
This is because the force of constraint, while it does no virtual work, does do real work. Finally, let us consider the mass on the end of the gimballed rod. Notice that this is a dynamical system with two coordinates, similar to ordinary mechanics in two dimensions, except that the mass matrix, while diagonal, is coordinate dependent, and the space on which motion occurs is not an innite at plane, but a curved two dimensional surface, that of a sphere.
These two distinctions are connectedthe coordinates enter the mass matrix because it is impossible to describe a curved space with unconstrained cartesian coordinates.
The conguration of a system at any moment is specied by the value of the generalized coordinates qj t , and the space coordinatized by these q1 ,.
The time evolution of the system is given by the trajectory, or motion of the point in conguration space as a function of time, which can be specied by the functions qi t. One can imagine the system taking many paths, whether they obey Newtons Laws or not. We consider only paths for which the qi t are dierentiable.
The action depends on the starting and ending points q t1 and q t2 , but beyond that, the value of the action depends on the path, unlike the work done by a conservative force on a point moving in ordinary space.
That means that for any small deviation of the path from the actual one, keeping the initial and nal congurations xed, the variation of the action vanishes to rst order in the deviation. To nd out where a dierentiable function of one variable has a stationary point, we dierentiate and solve the equation found by setting the derivative to zero. But our action is a functional, a function of functions, which represent an innite number of variables, even for a path in only one dimension.
Intuitively, at each time q t is a separate variable, though varying q at only one point makes q hard to interpret. It is not really necessary to be so rigorous, however.
We see that if f is the Lagrangian, we get exactly Lagranges equation. The above derivation is essentially unaltered if we have many degrees of freedom qi instead of just one.
In this section we will work through some examples of functional variations both in the context of the action and for other examples not directly related to mechanics.
The action is T. The rst integral is independent of the path, so the minimum action requires the second integral to be as small as possible.
Is the shortest path a straight line? The calculus of variations occurs in other contexts, some of which are more intuitive. The classic example is to nd the shortest path between two points in the plane.
The length of a path y x from x1 , y1 to. We see that length is playing the role of the action, and x is playing the role of t. Ignorable Coordinates If the Lagrangian does not depend on one coordinate, say qk , then we say it is an ignorable coordinate. Of course, we still want to solve for it, as its derivative may still enter the Lagrangian and eect the evolution of other coordinates.
To prove that the straight line is shorter than other paths which might not obey this restriction, do Exercise 2. In that sense we might say that the generalized momentum and the generalized force have not been dened consistently. Angular Momentum As a second example of a system with an ignorable coordinate, consider an axially symmetric system described with inertial polar coordinates r, , z , with z along the symmetry axis.
Extending the form of the kinetic energy we found in sec 1. We see that the conserved momentum P is in fact the z-component of the angular momentum, and is conserved because the axially symmetric potential can exert no torque in the z-direction: In section 3. Note, however, that even though the potential is independent of as well, does appear undierentiated in the Lagrangian, and it is not an ignorable coordinate, nor is P conserved6.
Energy Conservation We may ask what happens to the Lagrangian along the path of the motion. We expect energy conservation when the potential is time invariant and there is not time dependence in the constraints, i. It seems curious that we are nding straightforwardly one of the components of the conserved momentum, but not the other two, Ly and Lx , which are also conserved.
The fact that not all of these emerge as conjugates to ignorable coordinates is related to the fact that the components of the angular momentum do not commute in quantum mechanics. This will be discussed further in section 6. H is essentially the Hamiltonian, although strictly speaking that name is reserved for the function H q, p, t on extended phase space rather than the function with arguments q, q, t. What is H physically? In the case of Newtonian mechanics with a potential function, L is a quadratic function of the velocities qi.
As we shall see later, however, there are constrained systems in which the Hamiltonian is conserved but is not the ordinary energy. This gives rise to local gauge invariance, and will be discussed in Chapter 8, but until then we will assume that the phase space q, p , or cotangent bundle, is equivalent to the tangent bundle, i.
The rst two constitute Hamiltons equations of motion, which are rst order equations for the motion of the point representing the system in phase space. Lets work out a simple example, the one dimensional harmonic oscillator.
Note this is just the sum of the 2 2 kinetic and potential energies, or the total energy. These two equations verify the usual connection of the momentum and velocity and give Newtons second law.
The identication of H with the total energy is more general than our particular example. We have concentrated thus far on Newtonian mechanics with a potential given as a function of coordinates only. As the potential is a piece of the Lagrangian, which may depend on velocities as well, we should also entertain the possibility of velocity-dependent potentials. Only by If M were not invertible, there would be a linear combination of velocities which does not aect the Lagrangian.
The degree of freedom corresponding to this combination would have a Lagrange equation without time derivatives, so it would be a constraint equation rather than an equation of motion. But we are assuming that the qs are a set of independent generalized coordinates that have already been pruned of all constraints.
The rst term is a stream derivative evaluated at the time-dependent position of the particle, so, as in Eq. The last term looks like the last term of 2. This suggests that these two terms combine to form a cross product.
Indeed, noting B. Note, however, that this Lagrangian describes only the motion of the charged particle, and not the dynamics of the eld itself. Arbitrariness in the Lagrangian In this discussion of nding the Lagrangian to describe the Lorentz force, we used the lemma that guaranteed that the divergenceless magnetic eld B can be written in terms This is but one of many consequences of the Poincar lemma, discussed in e section 6. If we do, we have completely unchanged electromagnetic elds, which is where the physics lies.
We have here an example which points out that there is not a unique Lagrangian which describes a given physical problem, and the ambiguity is more that just the arbitrary constant we always knew was involved in the potential energy. This ambiguity is quite general, not depending on the gauge transformations of Maxwell elds. While this can be easily checked by evaluating the Lagrange equations, it is best understood in terms of the variation of the action.
The variation of path that one makes to nd the stationary action does not change the endpoints qjF and qjI , so the dierence S 2 S 1 is a.
This ambiguity is not usually mentioned in elementary mechanics, because if we restict our attention to Lagrangians consisting of canonical kinetic energy and potentials which are velocity-independent, a change 2. Dissipation Another familiar force which is velocity dependent is friction. Even the constant sliding friction met with in elementary courses depends on the direction, if not the magnitude, of the velocity. We saw above that a potential linear in velocities produces a force perpendicular to v, and a term higher order in velocities will contribute to the acceleration.
This situation cannot handled by Lagranges equations. An extension to the Lagrange formalism, involving Rayleighs dissipation function, is discussed in Ref. Exercises 2. This is a dierent statement than Eq. Show it will not generally be true if U S is not restricted to depend only on the dierences in positions.
Which of these quantities are conserved? Then 1. This is of the form of a variational integral with two variables. Show that the variational equations do not determine the functions x and y , but do determine that the path is a straight line. Explain why this equality of the lengths is obvious in terms of alternate parameterizations of the path.
On the hoop there is a bead of mass m, which. The only external force is gravity. Derive the Lagrangian and the Lagrange equation using the polar angle as the unconstrained generalized coordinate. Find the condition on such that there is an equilibrium point away from the axis. The engine rotated a vertical shaft with an angular velocity proportional to its speed. On opposite sides of this shaft, two hinged rods each held a metal weight, which was attached to another such rod hinged to a sliding collar, as shown.
L As the shaft rotates faster, the balls move outm m wards, the collar rises and uncovers a hole, 1 1 releasing some steam. Assume all hinges are L frictionless, the rods massless, and each ball has mass m1 and the collar has mass m2. Tell Governor for a steam enwhether the equilibrium is stable or not.
As the equilibrium at the top is unstable, the top cylinder will begin to roll on the bottom cylinder. Find the angle at which the separation ocR curs, and nd the minimum value of s for which this situation holds.
An inextensible massless string attached to m goes through the hole and is connected to another particle of mass M , which moves vertically only. Give a full set of generalized unconstrained coordinates and write the Lagrangian in terms of these. Assume the string remains taut at all times and that the motions in question never have either particle reaching the hole, and there is no friction of the string sliding at the hole.
Are there ignorable coordinates? Reduce the problem to a single second order dierential equation. This is a spherical pendulum. Find the Lagrangian and the equations of motion. Find the canonical momenta for a charged particle moving in an electromagnetic eld and also under the inuence of a non-electromagnetic force described by a potential U r.
Chapter 3 Two Body Central Forces Consider two particles of masses m1 and m2 , with the only forces those of their mutual interaction, which we assume is given by a potential which is a function only of the distance between them, U r1 r2.
In a mathematical sense this is a very strong restriction, but it applies very nicely to many physical situations. The classical case is the motion of a planet around the Sun, ignoring the eects mentioned at the beginning of the book. But it also applies to electrostatic forces and to many eective representations of nonrelativistic interparticle forces.
Our original problem has six degrees of freedom, but because of the symmetries in the problem, many of these can be simply separated and solved for, reducing the problem to a mathematically equivalent problem of a single particle moving in one dimension. First we reduce it to a one-body problem, and then we reduce the dimensionality.
For the other three, we rst use the cartesian components of the relative coordinate r: Thus the kinetic energy is transformed to the form for two eective particles of mass M and , which is neither simpler nor more complicated than it was in the original variables.
We have not yet made use of the fact that U only depends on the magnitude of r. In fact, the above reduction applies to any two-body system without external forces, as long as Newtons Third Law holds. In the problem under discussion, however, there is the additional restriction that the potential depends only on the magnitude of r, that is, on the distance between the two particles, and not on the direction of r. Thus we now convert from cartesian to spherical coordinates r, , for r.
Note that r sin is the distance of the particle from the z-axis, so P is just the z-component of the angular momentum, Lz. Of course all. Thus L is a constant1 , and the motion must remain in a plane perpendicular to L and passing through the origin, as a consequence of the fact that r L. It simplies things if we choose our coordinates so that L is in the z-direction.
Before we proceed, a comment may be useful in retrospect about the reduction in variables in going from the three dimensional one-body problem to a one dimensional problem. Here we reduced the phase space from six variables to two, in a problem which had four conserved quantities, L and H. But we have not yet used the conservation of H in this reduction, we have only used the three conserved quantities L. Where have these dimensions gone? This is generally true for an ignorable coordinate the corresponding momentum becomes a time-constant parameter, and the coordinate disappears from the remaining problem.
We can simplify the problem even more by using the one conservation law left, that of energy. Qualitative features of the motion are largely determined by the range over which the argument of the square root is positive, as for.
Thus the motion is restricted to this allowed region. Generically the force will not vanish there, so E Ue c r rp for r rp , and the integrals in 3. The radius rp is called a turning point of the motion. If there is also a maximum value of r for which the velocity is real, it is also a turning point, and an outgoing orbit will reach this maximum and then r will start to decrease, conning the orbit to the allowed values of r. If there are both minimum and maximum values, this interpretation of Eq.
But there is no particular reason for this period to be the geometrically natural periodicity 2 of , so that dierent values of r may be expected in successive passes through the same angle in the plane of the motion.
There would need to be something very special about the attractive potential for the period to turn out to be just 2, but indeed that is the case for Newtonian gravity.
We have reduced the problem of the motion to doing integrals. In general that is all we can do explicitly, but in some cases we can do the integral analytically, and two of these special cases are very important physically.
Consider rst the force of Newtonian gravity, or equivalently the Coulomb attraction of unlike charged particles. Thus there is always at least one turning point, umax , corresponding to the minimum distance rmin.
Then the argument of the square root must factor into [ u umax u umin ], although if umin is negative it is not really the minimum u, which can never get past zero. The integral 3. For example, see 2. Pericenter is also used, but not as generally as it ought to be. What is this orbit? Clearly rp just sets the scale of the whole orbit. The nature of the curve depends on the coecient of x2.
All of these are posible motions. The bound orbits are ellipses, which describe planetary motion and also the motion of comets.
But objects which have enough energy to escape from the sun, such as Voyager 2, are in hyperbolic orbit, or in the dividing case where the total energy is exactly zero, a parabolic orbit. Then as time goes to , goes to a nite value, for a parabola, or some constant less than for a hyperbolic orbit. Let us return to the elliptic case.
An ellipse is a circle stretched uniformly in one direction; the diameter in that direction becomes the major axis of the ellipse, while the perpendicular diameter becomes the minor axis.
The Notice that the center of the el- eccentricity is e and a is the semi-major axis lipse is ea away from the Sun. Kepler tells us not only that the orbit is an ellipse, but also that the sun is at one focus.
To verify that, note the other focus of an ellipse is symmetrically located, at 2ea, 0 , and work out the sum of the distances of any point on the ellipse from the two foci.
How are a and e related to the total energy E and the angular momentum L? In other words, as the planet makes its revolutions around the sun, its perihelion is always in the same direction. That didnt have to be the case one could imagine that each time around, the minimum distance occurred at a slightly dierent or very dierent angle.
Such an eect is called the precession of the perihelion. We will discuss this for nearly circular orbits in other potentials in section 3. What about Keplers Third Law? The area of an ellipse made by stretching a circle is stretched by the same amount, so A is times the semimajor axis times the semiminor axis. We may also ask about trajectories which dier only slightly from this orbit, for which r a is small.
Let us dene the apsidal angle as the angle between an apogee and the next perigee. In the treatment of planetary motion, the precession of the perihelion is the angle though which the perihelion slowly moves, so it is 2 2 per orbit. We have seen that it is zero for the pure inverse force law. There is actually some precession of the planets, due mostly to perturbative eects of the other planets, but also in part due to corrections to Newtonian mechanics found from Einsteins theory of general relativity.
In the late nineteenth century descrepancies in the precession of Mercurys orbit remained unexplained, and the resolution by Einstein was one of the important initial successes of general relativity. The remarkable simplicity of the motion for the Kepler and harmonic oscillator central force problems is in each case connected with a hidden symmetry.
We now explore this for the Kepler problem.
The motion is therefore conned to a plane perpendicular to L, and the vector p L is always in the plane of motion, as are r and p.
While we have just found three conserved quantities in addition to the conserved energy and the three conserved components of L, these cannot all be independent. The rst equation, known as the bac-cab equation, is shown in Appendix A.
While others often use only the last two names, Laplace clearly has priority. There could not be more than ve independent conserved functions depending analytically on the six variables of phase space for the relative motion only , for otherwise the point representing the system in phase space would be unable to move.
In fact, the ve independent conserved quantities on the six dimensional dimensional phase space conne a generic invariant set of states, or orbit, to a one dimensional subspace.
So we see the connection between the existence of the conserved A in the Kepler case and the fact that the orbits are closed. This average will also be zero if the region stays in some bounded part of phase space for which G can only take bounded values, and the averaging time is taken to innity. This is appropriate for a system in thermal equilibrium, for example.
The fact that the average value of the kinetic energy in a bound system gives a measure of the potential energy is the basis of the measurements of the missing mass, or dark matter, in galaxies and in clusters of galaxies.
This remains a useful tool despite the fact that a multiparticle gravitationally bound system can generally throw o some particles by bringing others closer together, so that, strictly speaking, G does not return to its original value or remain bounded. Whatever the relative signs, we are going to consider scattering here, and therefore positive energy solutions with the initial state of nite speed v0 and r.
Rutherford scattering. An particle approaches a heavy nucleus with an impact parameter b, scattering through an angle. Here we use S. For Gaussian units drop the 4 0 , or for Heaviside-Lorentz units drop only the 0. We need to evaluate E and L. The scattering angle therefore depends on b, the perpendicular displacement from the axis parallel to the beam through the nucleus. Particles passing through a given area will be scattered through a given angle, with a xed angle corresponding to a circle centered on the axis, having radius b given by 3.
There is a beam of N particles shot at random impact parameters onto a foil with n scattering centers per. Each particle will be signicantly scattered only by the scattering center to which it comes closest, if the foil is thin enough. A d d We have used the cylindrical symmetry of this problem to ignore the dependance of the scattering.
In Rutherford scattering increases monotonically as b decreases, which is possible only because the force is hard, and a particle aimed right at the center will turn around rather than plowing through. This was a surprize to Rutherford, for the concurrent model of the nucleus, Thompsons plum pudding model, had the nuclear charge spread out over some atomic-sized spherical region, and the Coulomb potential would have decreased once the alpha particle entered this region.
So suciently energetic alpha particles aimed at the center should have passed through undeected instead of scattered backwards. In fact, of course, the nucleus does have a nite size, and this is still true, but at a much smaller distance, and therefore a much larger energy.
It also means that the cross section becomes innite as b0 , and vanishes above that value of. This eect is known as rainbow scattering, and is the cause of rainbows, because the scattering for a given color light o a water droplet is very strongly peaked at the maximum angle of scattering.
This eect is called glory scattering, and can be seen around the shadow of a plane on the clouds below. It is then internally reected once and refracted again on the way out. Find the angular radius of the rainbows circle, and the angular width of the rainbow, and tell whether the red or One way light can scatter from a blue is on the outside. Find the Lagrangian, two conserved quantities, and reduce the e problem to a one dimensional problem.
What is the condition for circular motion at constant r? Identify i as many as possible of these with previously known conserved quantities. Chapter 4 Rigid Body Motion In this chapter we develop the dynamics of a rigid body, one in which all interparticle distances are xed by internal forces of constraint. This is, of course, an idealization which ignores elastic and plastic deformations to which any real body is susceptible, but it is an excellent approximation for many situations, and vastly simplies the dynamics of the very large number of constituent particles of which any macroscopic body is made.
In fact, it reduces the problem to one with six degrees of freedom. While the ensuing motion can still be quite complex, it is tractible. In the process we will be dealing with a conguration space which is a group, and is not a Euclidean space. Degrees of freedom which lie on a group manifold rather than Euclidean space arise often in applications in quantum mechanics and quantum eld theory, in addition to the classical problems we will consider such as gyroscopes and tops.
A macroscopic body is made up of a very large number of atoms. Describing the motion of such a system without some simplications is clearly impossible. Many objects of interest, however, are very well approximated by the assumption that the distances between the atoms This constitutes a set of holonomic constraints, but not independent ones, as we have here 1 n n 1 constraints on 3n coordinates.
We will need to discuss how to represent the latter part of the conguration, including what a rotation is , and how to reexpress the kinetic and potential energies in terms of this conguration space and its velocities.
The rst part of the conguration, describing the translation, can be specied by giving the coordinates of the marked point xed in the body, R t. Often, but not always, we will choose this marked point to be the center of mass R t of the body. In order to discuss other points which are part of the body, we will use an orthonormal coordinate system xed in the body, known as the body coordinates, with the origin at the xed point R.
The constraints mean that the position of each particle of the body has xed coordinates in terms of this coordinate system. Thus the dynamical conguration of the body is completely specied by giving the orientation of these coordinate axes in addition to R. This orientation needs to be described relative to a xed inertial coordinate system, or inertial coordinates, with orthonormal basis ei.
In order to know its components in the In this chapter we will use Greek letters as subscripts to represent the dierent particles within the body, reserving Latin subscripts to represent the three spatial directions.
Because they play such an important role in the study of rigid body motion, we need to explore the properties of orthogonal transformations in some detail.
There are two ways of thinking about an orthogonal transformation A and its action on an orthonormal basis, Eq. Thus A is to be viewed as a rule for giving the primed basis vectors in terms of the unprimed ones 4. This picture of the role of A is called the passive interpretation.
One may also use matrices to represent a real physical transformation of an object or quantity. In particular, Eq. For real rota tion of the physical system, all the vectors describing the objects are changed by the rotation into new vectors V V R , physically dierent from the original vector, but having the same coordinates in the primed basis as V has in the unprimed basis. This is called the active interpretation of the transformation.
Both active and passive views of the transformation apply here, and this can easily lead to confusion. The transformation A t is the physical transformation which rotated the body from some standard orientation, in which the body axes ei were parallel to the lab frame axes ei , to the conguration of the body at time t.
But it also gives the relation of the components of the same position vectors at time t expressed in body xed and lab frame coordinates. If we rst consider rotations in two dimensions, it is clear that they are generally described by the counterclockwise angle through which the basis is rotated,. Clearly taking the transpose simply changes the sign of , which is just what is necessary to produce the inverse transformation. Thus each two dimensional rotation is an orthogonal transformation. It is straightforward to show that these four equations on the four elements of A determine A to be of the form 4.
Let us call this transformation P. Thus any two-dimensional orthogonal matrix is a rotation or is P followed by a rotation. The set of all real orthogonal matrices in two dimensions is called O 2 , and the subset consisting of rotations is called SO 2. In three dimensions we need to take some care with what we mean by a rotation. On the one hand, we might mean that the transformation has some xed axis and is a rotation through some angle about that axis. Let us call that a rotation about an axis.
On the other hand, we might mean all transformations we can produce by a sequence of rotations about various axes. Let us dene rotation in this sense. In two dimensions, straightforward evaluation will verify that if R and R are of the form 4. Thus all rotations are rotations about an axis there. Rotations in. Figure 4. The results of applying the two rotations H and V to a book depends on which is done rst.
Thus rotations do not commute. Here we are looking down at a book which is originally lying face up on a table. V is a rotation about the vertical z-axis, and H is a rotation about a xed axis pointing to the right, each through We can still represent the composition of rotations with matrix multiplication, now of 3 3 matrices.
A graphic illustration is worth trying. Let V be the process of rotating an object through 90 about the vertical z-axis, and H be a rotation through 90 about the x-axis, which goes goes o to our right.
If we start with the book lying face up facing us on the table, and rst apply V and then H, we wind up with the binding down and the front of the book facing us.
If, however, we start from the same position but apply rst H and then V , we wind up with the book standing upright on the table with the binding towards us. Clearly the operations H and V do not commute. It is clear that any composition of rotations must be orthogonal, as any set of orthonormal basis vectors will remain orthonormal under each transformation.
So the rotations are a subset of the set O N of orthogonal matrices. This set of orthogonal matrices is a group, which means that the set O N satises the following requirements, which we state for a general set G.
A set G of elements A, B, C, One then says that the set G is closed under. In our case the group multiplication is ordinary matrix multiplication, the group consists of all 3 3 orthogonal real matrices, and we have just shown that it is closed.
This element is called the inverse of A, and in the case of orthogonal matrices is the inverse matrix, which always exists, because for orthogonal matrices the inverse is the transpose, which always exists for any matrix.
While the constraints 4. But the set of matrices O 3 is not connected in this fashion: To I see it is true, we look at the determinant of A. But the determinant varies continuously as the matrix does, so no continuous variation of the matrix can lead to a jump in its determinant.
The set of all unimodular orthogonal matrices in N dimensions is called SO N. It is a subset of O N , the set of all orthogonal matrices in N dimensions. Clearly all rotations are in this subset. The subset is closed under multiplication, and the identity and the inverses of elements in SO N are also in SO N , for their determinants are clearly 1.
Simultaneously we will show that every rotation in three dimensions is a rotation about an axis. We now show that every A SO 3 has one vector it leaves unchanged or invariant, so that it is eectively a rotation in the plane perpendicular to this direction, or in other words a rotation about the axis it leaves invariant.
The fact that every unimodular orthogonal matrix in three dimensions is a rotation about an axis is known as Eulers Theorem. Of course this determines only the direction of , and only up to sign. If we choose a new coordinate system in which.
Thus A is of the form. Thus we see e that the set of orthogonal unimodular matrices is the set of rotations, and elements of this set may be specied by a vector2 of length.
Thus we see that the rotation which determines the orientation of a rigid body can be described by the three degrees of freedom. Together with the translational coordinates R, this parameterizes the conguration space of the rigid body, which is six dimensional. It is important to recognize that this is not motion in a at six dimensional conguration space, however.
The composition of rotations is by multiplication of the matrices, not by addition of the s. There are other ways of describing the conguration space, two of which are known as Euler angles and Cayley-Klein parameters, but none of these make describing the space very intuitive. We will discuss these applications rst.
Later, when we do need to discuss the conguration in section 4. More precisely, we choose along one of the two opposite directions left invariant by A, so that the the angle of rotation is non-negative and.
This species a point in or on the surface of a three dimensional ball of radius , but in the case when the angle is exactly the two diametrically opposed points both describe the same rotation. Mathematicians say that the space of SO 3 is three-dimensional real projective space P3 R. We have seen that the rotations form a group. Let us describe the conguration of the body coordinate system by the position R t of a given point and the rotation matrix A t: A given particle of the body is xed in the body coordinates, but this, of course, is not an inertial coordinate system, but a rotating and possibly accelerating one.
We need to discuss the transformation of kinematics between these two frames. While our current interest is in rigid bodies, we will rst derive a general formula for rotating and accelerating coordinate systems. We are not assuming at the moment that the particle is part of the rigid body, in which case the bi t would be independent of time. We might dene a body time derivative b: This expression has coordinates in the body frame with basis vectors from the inertial frame. The interpretation of the rst term is suggested by its matrix form: This transformation must be close to an identity, as t 0.
Let us expand it: Here is a matrix which has xed nite elements as t 0, and is called the generator of the rotation. Subtracting 1 from both sides of 4. Then the k also determine the ij: We have still not answered the question, what is V?
When dierentiating a true vector, which is independent of the origin of the coordinate system, rather than a position, the rst term in 4. The velocity v is a vector, as are R and b, the latter because it is the dierence of two positions.
This shows that even the peculiar object b b obeys 4. Applying this to the velocity itself 4. This is a general relation between any orthonormal coordinate system and an inertial one, and in general can be used to describe physics in noninertial coordinates, regardless of whether that coordinate system is imbedded in a rigid body.