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- Thornton S., Marion J. Classical Dynamics of Particles and Systems
- Thornton S., Marion J. Classical Dynamics of Particles and Systems
- Classical Dynamics of Particles and Systems 5th Ed - s. Thornton, j. Marion backmocadiwus.ml
- Classical Dynamics of Particles and Systems

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pdf book: Classical Dynamics of Particles and Systems by Marion and Thornton. appendices) from Classical Dynamics of Particles and Systems, Fifth Edition, by Stephen T. Thornton and Jerry B. Marion. It is intended for use only by. chapter contents preface problems solved in student solutions manual vii matrices, vectors, and vector calculus newtonian mechanics—single particle

Thornton and Jerry B. It is intended for use only by instructors using Classical Dynamics as a textbook, and it is not available to students in any form. The problem numbers of those solutions in the Student Solutions Manual are listed on the next page. As a result of surveys received from users, I continue to add more worked out examples in the text and add additional problems. There are now problems, a significant number over the 4th edition. A few of the problems are quite challenging. Many of them require numerical methods.

There are now problems, a significant number over the 4th edition. A few of the problems are quite challenging. Many of them require numerical methods. Having this solutions manual should provide a greater appreciation of what the authors intended to accomplish by the statement of the problem in those cases where the problem statement is not completely clear.

Please inform me when either the problem statement or solutions can be improved.

Specific help is encouraged. The instructor will also be able to pick and choose different levels of difficulty when assigning homework problems. We have solutions for your book! Show by division and by direct expansion in a Taylor series that For what range of x is the series valid?

Step-by-step solution:. JavaScript Not Detected. The Taylor series expansion of the function about the origin is expressed as, …… 1. Comment 0.

Expand At the value of And Thus, Hence,. For Hence, is discontinuous at Thus, is valid for. View a full sample. Classical Dynamics of Particles and Systems 5th Edition. Jerry B. Marion , Stephen T. Thornton Authors: We see that if the torque t vanishes at all times the angular momentum is conserved.

This can happen not only if the force is zero, but also if the force always points to the reference point. This is the case in a central force problem such as motion of a planet about the sun.

So far we have talked about a system consisting of only a single particle, possibly inuenced by external forces. The conguration.

Let Fi be the total force acting on particle i. It is the sum of the forces produced by each of the other particles and that due to any external force.

Let Fji be the force particle j exerts on particle i and let FiE be the external force on particle i. Here we are assuming forces have identiable causes, which is the real meaning of Newtons second law, and that the causes are either individual particles or external forces. Thus we are assuming there are no three-body forces which are not simply the sum of two-body forces that one object exerts on another.

Thus the internal forces cancel in pairs in their eect on the total momentum, which changes only in response to the total external force. As an obvious but very important consequence3 the total momentum of an isolated system is conserved. The total angular momentum is also just a sum over the individual particles, in this case of the individual angular momenta: There are situations and ways of describing them in which the law of action and reaction seems not to hold.

One should not despair for the validity of momentum conservation. The Law of Biot and Savart only holds for time-independent current distributions. Unless the currents form closed loops, there will be a charge buildup and Coulomb forces need to be considered. More generally, even the sum of the momenta of the current elements is not the whole story, because there is momentum in the electromagnetic eld, which will be changing in the time-dependent situation.

This is not automatically zero, but vanishes if one assumes a stronger form of the Third Law, namely that the action and reaction forces between two particles acts along the line of separation of the particles. If the force law is independent of velocity and rotationally and translationally symmetric, there is no other direction for it to point.

For spinning particles and magnetic forces the argument is not so simple in fact electromagnetic forces between moving charged particles are really only correctly viewed in a context in which the system includes not only the particles but also the elds themselves. For such a system, in general the total energy, momentum, and angular momentum of the particles alone will not be conserved, because the elds can carry all of these quantities.

But properly dening the energy, momentum, and angular momentum of the electromagnetic elds, and including them in the totals, will result in quantities conserved as a result of symmetries of the underlying physics.

This is further discussed in section 8. The conservation laws are very useful because they permit algebraic solution for part of the velocity. Similarly if L is conserved, the components of v which are perpendicular to r are determined in terms of the xed constant L. With both conserved, v. Examples of the usefulness of conserved quantities are everywhere, and will be particularly clear when we consider the two body central force problem later.

But rst we continue our discussion of general systems of particles. As we mentioned earlier, the total angular momentum depends on the point of evaluation, that is, the origin of the coordinate system used. We now show that it consists of two contributions, the angular momentum about the center of mass and the angular momentum of a ctitious point object located at the center of mass.

The rst term of the nal form is the sum of the angular momenta of the particles about their center of mass, while the second term is the angular momentum the system would have if it were collapsed to a point at the center of mass. What about the total energy? Thus the kinetic energy of the system can also be viewed as the sum of the kinetic energies of the constituents about the center of mass, plus the.

If the forces on the system are due to potentials, the total energy will be conserved, but this includes not only the potential due to the external forces but also that due to interparticle forces, Uij ri , rj. In general this contribution will not be zero or even constant with time, and the internal potential energy will need to be considered.

One exception to this is the case of a rigid body. As these distances do not vary, neither does the internal potential energy. These interparticle forces cannot do work, and the internal potential energy may be ignored. The rigid body is an example of a constrained system, in which the general 3n degrees of freedom are restricted by some forces of constraint which place conditions on the coordinates ri , perhaps in conjunction with their momenta.

In such descriptions we do not wish to consider or specify the forces themselves, but only their approximate eect. The forces are assumed to be whatever is necessary to have that effect. It is generally assumed, as in the case with the rigid body, that the constraint forces do no work under displacements allowed by the constraints.

We will consider this point in more detail later. If the constraints can be phrased so that they are on the coordinates and time only, as i r1 , These constraints determine hypersurfaces in conguration space to which all motion of the system is conned. In general this hypersurface forms a 3n k dimensional manifold.

Consider, for example, a mass on one end of a rigid light rod. We might choose as geny eralized coordinates the standard spherical angles and. Thus the constrained subspace is two di- x mensional but not at rather it is the surface of a sphere, which Generalized coordinates , for mathematicians call S 2. It is nata particle constrained to lie on a ural to reexpress the dynamics in sphere. The use of generalized non-cartesian coordinates is not just for constrained systems.

Before we pursue a discussion of generalized coordinates, it must be pointed out that not all constraints are holonomic. The standard example is a disk of radius R, which rolls on a xed horizontal plane. It is constrained to always remain vertical, and also to roll without slipping on the plane.

As coordinates we can choose the x and y of the center of the disk, which are also the x and y of the contact point, together with the angle a xed line on the disk makes with the downward direction, , and the angle the axis of the disk makes with the x axis,.

The fact that these involve velocities does not automatically make them nonholonomic. In the simpler one-dimensional problem in which the disk is conned to the yz plane, rolling along. A vertical disk free to roll on a plane.

A xed line on the disk makes an angle of with respect to the vertical, and the axis of the disk makes an angle with the x-axis. The long curved path is the trajectory of the contact point. The three small paths are alternate trajectories illustrating that x, y, and can each be changed without any net change in the other coordinates. This cannot be done with the two-dimensional problem. We can see that there is no constraint among the four coordinates themselves because each of them can be changed by a motion which leaves the others unchanged.

Rotating without moving the other coordinates is straightforward. By rolling the disk along each of the three small paths shown to the right of the disk, we can change one of the variables x, y, or , respectively, with no net change in the other coordinates.

Thus all values of the coordinates4 can be achieved in this fashion. There are other, less interesting, nonholonomic constraints given by inequalities rather than constraint equations.

A bug sliding down a bowling ball obeys the constraint r R. But this is heading far aeld. Before we get further into constrained systems and DAlemberts Principle, we will discuss the formulation of a conservative unconstrained system in generalized coordinates. Thus we wish to use 3n generalized coordinates qj , which, together with time, determine all of the 3n cartesian coordinates ri: The t dependence permits there to be an explicit dependence of this relation on time, as we would have, for example, in relating a rotating coordinate system to an inertial cartesian one.

A small change in the coordinates of a particle in conguration space, whether an actual change over a small time interval dt or a virtual change between where a particle is and where it might have been under slightly altered circumstances, can be described by a set of xk or by a set of qj.

We may think of U q, t: Note that if the coordinate transformation is time-dependent, it is possible that a time-independent potential U x will lead to a time-dependent po tential U q, t , and a system with forces described by a time-dependent potential is not conservative. The denition in 1. The qk do not necessarily have units of distance.

For example, one qk might be an angle, as in polar or spherical coordinates. The corresponding component of the generalized force will have the units of energy and we might consider it a torque rather than a force.

We have seen that, under the right circumstances, the potential energy can be thought of as a function of the generalized coordinates qk , and the generalized forces Qk are given by the potential just as for ordinary cartesian coordinates and their forces. Now k. The last term is due to the possibility that the coordinates xi q1 , Only the rst term arises if the relation between x and q is time independent.

The second and third terms are the sources of the r r and r 2 terms in the kinetic energy when we consider rotating coordinate systems6. But in an anisotropic crystal, the eective mass of a particle might in fact be dierent in dierent directions. Lets work a simple example: In generalized coordinates, it is quadratic but not homogeneous7 in the velocities, and with an arbitrary dependence on the coordinates.

In general, even if the coordinate transformation is time independent, the form of the kinetic energy is still coordinate dependent and, while a purely quadratic form in the velocities, it is not necessarily diagonal. It involves quadratic and lower order terms in the velocities, not just quadratic ones. The coecients of these unit vectors can be understood graphically with geometric arguments. Similar geometric arguments 2 are usually used to nd the form of the kinetic energy in spherical coordinates, but the formal approach of 1.

It is important to keep in mind that when we view T as a function of coordinates and velocities, these are independent arguments evaluated at a particular moment of time. Thus we can ask independently how T varies as we change xi or as we change xi , each time holding the other variable xed. If the trajectory of the system in conguration space, r t , is known, the velocity as a function of time, v t is also determined. Viewed as functions of time, this gives nothing beyond the information in the trajectory.

But at any given time, r and p provide a complete set of initial conditions, while r alone does not. We dene phase space as the set of possible positions This space is called the tangent bundle to conguration space. For cartesian coordinates it is almost identical to phase space, which is in general the cotangent bundle to conguration space. Equivalently, it is the set of possible initial conditions, or the set of possible motions obeying the equations of motion.

For a single particle in cartesian coordinates, the six coordinates of phase space are the three components of r and the three components of p.

At any instant of time, the system is represented by a point in this space, called the phase point, and that point moves with time according to the physical laws of the system. These laws are embodied in the force function, which we now consider as a function of p rather than v, in addition to r and t. This is to be distinguished from the trajectory in conguration space, where in order to know the trajectory you must have not only an initial point position but also an initial velocity.

We have spoken of the coordinates of phase space for a single particle as r and p, but from a mathematical point of view these together give the coordinates of the phase point in phase space.

Only half of this velocity is the ordiWe will assume throughout that the force function is a well dened continuous function of its arguments. The path traced by the phase point as it travels through phase space is called the phase curve.

For a system of n particles in three dimensions, the complete set of initial conditions requires 3n spatial coordinates and 3n momenta, so phase space is 6n dimensional. While this certainly makes visualization dicult, the large dimensionality is no hindrance for formal developments. Also, it is sometimes possible to focus on particular dimensions, or to make generalizations of ideas familiar in two and three dimensions.

For example, in discussing integrable systems 7. Thus for a system composed of a nite number of particles, the dynamics is determined by the rst order ordinary dierential equation 1.

All of the complication of the physical situation is hidden in the large dimensionality of the dependent variable and in the functional dependence of the velocity function V , t on it.

There are other systems besides Newtonian mechanics which are controlled by equation 1. Collectively these are known as dynamical systems. The populations of three competing species could be described by eq.

The dimensionality d of in 1. A dth order dierential equation in one independent variable may always be recast as a rst order dierential equation in d variables, so it is one example of a dth order dynamical system.

The space of these dependent variables is called the phase space of the dynamical system. Newtonian systems always give rise to an even-order This is not to be confused with the simpler logistic map, which is a recursion relation with the same form but with solutions displaying a very dierent behavior.

Even for constrained Newtonian systems, there is always a pairing of coordinates and momenta, which gives a restricting structure, called the symplectic structure11 , on phase space. If the force function does not depend explicitly on time, we say the system is autonomous. This gives a visual indication of the motion of the systems point.

Undamped Damped Figure 1. Velocity eld for undamped and damped harmonic oscillators, and one possible phase curve for each system through phase space. The velocity eld is everywhere tangent to any possible path, one of which is shown for each case. Note that qualitative features of the motion can be seen from the velocity eld without any solving of the dierential equations; it is clear that in the damped case the path of the system must spiral in toward the origin.

The paths taken by possible physical motions through the phase space of an autonomous system have an important property. Because This almost implies that the phase curve the object takes through phase space must be nonintersecting Then the motion of the system is relentlessly upwards in this direction, though still complex in the others. For the undamped one-dimensional harmonic oscillator, the path is a helix in the three dimensional extended phase space.

Most of this book is devoted to nding analytic methods for exploring the motion of a system. In several cases we will be able to nd exact analytic solutions, but it should be noted that these exactly solvable problems, while very important, cover only a small set of real problems. 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.