Advances in the study of dynamical systems have revolutionized the way that classical mechanics is taught and understood. Classical dynamics of particles and systems solutions Other editions — View all Classical Dynamics: Classical dynamics of particles and systems solutions pdf dynamics of particles and systems solutions pdf The authors cover all the material that one would expect to find in a standard graduate course: Account Options Sign in.
Published on Feb View 2. Cambridge University PressAug 13, — Science. List of Worked Examples. Recent advances in the study of dynamical systems have revolutionized the way that classical mechanics is taught and understood. Saletan No preview available — This site contains solutions Lagrangian and Hamiltonian dynamics, canonical transformations, the Hamilton-Jacobi equation, perturbation methods, and rigid bodies.
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Chapter 3. Sections 3. Chapter 4. Sections 4. Chapter 5. Sections 5. Chapter 6. Sections 6. Chapter 7. Sections 7. Chapter 8. Sections 8. Path 2. For students who have had a good undergraduate course, but one without Hamiltonian dynamics. Comment: A lot depends on the students's background. Therefore some sections are labeled IN, for "If New. At the end of this path we indicate some sections that might be added for optional enrichment or substituted for skipped material.
Chapter 9. Section 9. Suggested material for optional enrichment: Chapter 2. Section 2. Section 3. Section 4. It is assumed that the reader has worked out many problems using the basic techniques of Newtonian mechanics. The brief review presented here thus emphasizes the underlying ideas and introduces the notation and the geometrical approach to mechanics that will be used throughout the book. Because the position of a moving object is specified by the location of every point composing it, we must start by considering how to specify the location of an arbitrary point.
This is done by giving the coordinates of the point in a coordinate system, called the reference system or reference frame. Each point in space is associated with a set of three real numbers, the coordinates of the point, and this association is unique, which means that any given set of coordinates is associated with only one point or that two different points have different sets of coordinates.
Probably the most familiar example of this geometric construction is a Cartesian coordinate system; other examples are spherical polar and cylindrical polar coordinates. Given a reference frame, the position of a point can be specified by giving the radius vector x which goes from the origin of the frame to the point. In a Cartesian frame, the components of x with respect to the three axes X 1, X 2 , X 3 of the frame, are the coordinates x 1 , x 2 , x 3 of the point.
We are assuming here that space is three dimensional and Euclidean and that it has the usual properties one associates with such a space Shilov, ; Doubrovine eta!. This assumption is necessary for much of what we will say, becoming more explicit in the next section, on dynamics. To simplify many of the equations, we will now start using the summation convention, according to which the summation sign I: is omitted together with its limits in equations such as 1. An index that appears twice in a mathematical term is summed over the entire range of that index, which should be specified the first time it appears.
The occasional exceptions to this convention will always be explained. If an index appears singly, the equation applies to its entire range, even in isolated mathematical expressions. Thus, "The a, 1 ; 1 are real" means that the expression a, 1 ; 1 , summed over the range of j, is real for each value of i in the range of i. If an index appears more than twice, its use will be explained.
Accordingly, Eq. When such a particle moves, its position vector changes, and thus a parameter t is needed to label the different space points that the particle occupies.
This concept of successively later positions is an intuitive one depending on the ability to distinguish the order in which events take place, that is, between before and after. For any two positions of the particle we assume that there is no question as to which is the earlier one: given two values t 1 and t2 oft such that t 1 1. The definition of distance is known as the metric of the space for which it is being defined.
The Euclidean metric i. The reason for bringing in the velocity at this point is that trajectories are usually found from other properties of the motion, of which velocity is an example. Another reason is that one is often interested not only in the trajectory itself, but also in other properties of the motion such as velocity. It is convenient to write v in terms of distance l along the trajectory.
Lets be any parameter that increases smoothly and monotonically along the trajectory, and let x s 0 and x s 1 be any two points on the trajectory. Then the definition of Eq. Although this definition of l seems to depend on the parameters, it actually does not see Problem The trajectory can be parameterized by the time t or even by l itself, and the result would be the same.
To see this, consider Fig. The tangent vector at the point x on the curve is in the direction ofT. The chord vector. In this limit the vector T can be expressed as T. X which is of unit length and parallel to T. Then 1. The curvature K of the trajectory, the inverse of the radius of curvature p see Problem II is defined by 1.
When these expressions for 1. The acceleration lies in the plane formed by T and n, called the osculating plane. The unit vector B normal to the osculating plane , Fig. Smce 7 is parallel ton and n is a unit vector, the rate of change of B is parallel ton. Of course, the first and the second time derivatives of x do not exhaust all possible properties of the motion; for instance one could ask about derivatives of higher order.
We therefore now turn from the mathematical :reatment of the motion of point particles to a discussion of the physical principles that Jetermine that motion. Moreover, as we will show later, results about the motion of extended objects can be derived from axioms about idealized point particles.
These axioms, as we have said, are ultimately justified by experiment, so we will try to state them in terms of idealized "thought experiments. The first is that the small objects we speak of, even though they are to approximate points, may not get too small. As is well known, at atomic dimensions classical mechanics breaks down and quantum mechanics takes over.
That breakdown prescribes a lower limit on the size of the objects admitted in our discussions. The other restriction is on the magnitudes of the velocities involved: we exclude speeds so high that relativistic effects become important. Unless otherwise stated, these restrictions apply to all the following discussion. Our treatment starts with the notion of an isolated particle. An isolated particle is a sufficiently small physical object removed sufficiently far from all other matter.
What is meant here by "sufficiently" depends on the precision of measurements, and the statements we are about to make are statements about limits determined by such precision. They are true for measurements of distance whose uncertainties are large compared to characteristic dimensions of the object "in the limit" of large distances or of small objects and for measurements of time whose uncertainties are large compared to characteristic times of changes within the object "in the limit" of long times.
These distances and times, though long compared with the characteristics of the object, should nevertheless be short compared to the distance from the nearest object and the time to reach it. When isolation is understood in this way, the accuracy of the principles stated below will increase with the degree of isolation of the object. It follows that there are two ways to test the axioms: the first is by choosing smaller objects and removing them further from other matter, and the second is by making cruder measurements.
We say this not to advocate cruder measurements, but to indicate that the relation between theory and experiment has many facets. Not only are the axioms actually statements about the result of experiment, but the very terms in which the axioms are stated must involve detailed experimental considerations, even in an "axiomatized" field like classical mechanics.
Similar considerations apply to the two restrictions described above. The detection of quantum or relativistic effects depends also on the accuracy of measurement. The first of these quantifies the notion of time [see the discussion after Eq. The second states conservation of momentum for two-particle interactions, which is equivalent to Newton's third law. Together they pave the way for a statement of Newton's second law. Both principles must be understood as statements about idealized experiments.
They are based on years, even centuries, of constantly refined experiments. We continue to assume that physical space is three dimensional, Euclidean, and endowed with the usual Euclidean metric of Eq. Property A Every isolated particle moves in a straight line in such a frame.
Property B If the notion of time is quantified by defining the unit of time so that one particular isolated particle moves at constant velocity in this frame, then every other isolated particle moves at constant velocity in this frame. This defines the parameter t to be linear with respect to length along the trajectory of the chosen isolated particle, and then Property B states that any other parameter t, defined in the same way by using some other isolated particle, will be linearly related to the first one.
Buried in this statement is the nonrelativistic idea of simultaneity. Although it is not practical to measure time in terms of an isolated free particle, it will be seen eventually that the laws of motion derived from the two principles imply that the rotation of an isolated rigid body about a symmetry axis also takes place at a constant rate and can thus be used as a measure of time.
In practice the Earth is usually used for this purpose, but corrections have to be introduced to take account of the fact that the Earth is not really isolated in terms of the measuring instruments being used or completely rigid. The most accurate modem timing devices are atomic and are based on a long chain of reasoning stretching from the two principles and involving quantum as well as classical concepts.
The existence of one inertial frame, as postulated in Principle 1, implies the existence of many more, all moving at constant velocity with respect to each other.
Two such inertial frames cannot rotate with respect to each other, for then a particle moving in a straight line in one of the frames would not move in a straight line in the other. The transformations that connect such inertial frames are called Galileian; more about this is discussed in later chapters. REMARK: Since inertial frames are defined in terms of isolated bodies, they cannot in general be extended indefinitely. In other words, they have a local character, for extending them would change the degree of isolation.
Suppose, for instance, that there exist two inertial frames very far apart. If they are extended until they intersect, it may tum out that a particle that is free in the first frame is not free in the second.
Considerations such as these play a role in physics, but are not important for the purposes of this book Taylor, In general they will not move at constant velocities: their proximity leads to accelerations the particles are said to interact.
Moreover, although K depends both on the inertial frame in which the motion is being observed and on the particular motion, t-t 12 does not: t-t 12 is always the same number for particles I and 2. Even if the motion is interrupted and a new experiment is performed, even if the interaction between the particles is somehow changed say from gravitational to magnetic, or from rods of negligible weight to rubber bands , the same number t-t 12 will be found provided the experiment involves the same two particles 1 and 2.
If a similar set of experiments is performed with particle 1 and a third particle 3, a similar result will be obtained, and the same is true for experiments involving particles 2 and 3. Existence of Mass. It follows from 1. The m, are, of course, the masses of the particles. It should now be clear that Principle 2 states the law of conservation of momentum in two-particle mteractions. The masses of the particles are not unique: it is the ft, 1 that are determined by experiment, and the ft, 1 are only the ratios of the masses.
But given any set of m 1 , any other set is obtained from the first by multiplying all of the m 1 by the same constant. What is done in practice is that some body is chosen as a standard say, I cm 3 of water at 4"C and atmospheric pressure and the masses of all other bodies are related to it.
The important thing is that once such a standard has been chosen, there is just one number m, associated with each body, independent of any object with which it is interacting. The vectors on the right-hand sides of Eqs. They [or rather their analogs obtained by differentiating 1. In fact Eqs. Indeed, let the force F acting on a particle be defined by the equation 1.
Now tum to the first of Eqs. The experiment it describes involves the interaction of particles I and 2: the acceleration a 1 of particle 1 arises as a result of that interaction. Then the first of Eqs. Actually it is an implicit statement of Principle 1 and was logically necessary for Newton's complete formulation of mechanics. Although such questions would seem to arise in understanding Newton's laws, we should remember that it is his formulation that lies at the basis of classical mechanics as we know it today.
The two principles as we have given them are, in fact, an interpretation of Newton's laws. This interpretation is due originally to Mach Our development is closely related to work by Eisenbud It is interesting that Eq. Why, one may ask, has this definition been so important in classical mechanics? As may have been expected, the answer lies in its physical content. It is found empirically that in many physical situations it is rna that is known a priori rather than some other dynamical property.
That is, the force is what is specified independent of the mass, acceleration, or many other properties of the particle whose motion is being studied. Moreover, forces satisfy a superposition principle: the total force on a particle can be found by adding contributions from different agents. It is such properties that elevate Eq. For an interesting discussion of this point see Feynman eta!.
REMARK: Strangely enough, in one of the most familiar cases of motion, that of a particle in a gravitational field, it is not the force rna that is known a priori, but the acceleration a.
There are theorems stating that, with certain unusual exceptions see, e. In this book we will deal only in situations for which solutions exist and are unique. For many years physicists took comfort in such theorems and concentrated on trying to find solutions for a given force with various different initial conditions. Recently, however, it has become increasingly clear that there remain many subtle and fundamental questions, largely having to do with the stability of solutions.
To see what this means, consider the behavior of two solutions of Eq. One then asks how the separation ox t between the solutions behaves as t increases: will the trajectories remain infinitesimally close, will their separation approach some nonzero limit, will it oscillate, or will it grow without bound? Then the solutions of the differential equation are called stable if Dx t approaches either zero or a constant of the order of ox to and unstable if it grows without bound as t increases.
Stability in this sense is highly significant because initial conditions cannot be established with absolute precision in any experiment on a classical mechanical system, and thus there will always be some uncertainty in their values. Suppose, for example, that a certain dynamical system has the property that it invariably tends to one of two regions A and B that are separated by a finite distance.
In general the final state of such a system can in principle be calculated from a knowledge of the initial conditions, and therefore each initial condition can be labeled a orb, belonging to final state A or B. If there exist small regions that contain initial conditions with both labels, then in those regions the system is unstable in the above sense. In fact there exist dynamical systems in which the two types of initial conditions are mixed together so tightly that it is impossible to separate them: in every neighborhood, no matter how small, there are conditions labeled both a and b.
Then even though such a system is entirely deterministic, predicting its end state would require knowing its initial conditions with infinite precision. Such a system is called chaotic or said to exhibit chaos. An everyday example of chaos, somewhat outside the realm of classical mechanics, is the weather: very similar weather conditions one day can be followed by drastically different ones the next.
It turns out that exactly the opposite is true: most classical mechanical systems are unstable in the sense defined above. The most common exceptions are systems that can be reduced to a collection of one-dimensional ones.
One-dimensional systems are discussed in Section 1. For many reasons, the leading early contributors to classical mechanics were concerned mainly with stable systems, and for a long time their concerns continued to dominate the study of dynamics. In recent years, however, significant progress has been made in understanding instability, and later in this book, especially in Chapter 7, we will discuss specific systems that possess instabilities and in particular those solutions that manifest the instabilities.
Now consider another observer looking at the same particle at the same time, and suppose that in the second observer's frame the position vector of the particle is y, with coordinates y,. It is clear that there must be some transformation law, a sort of dictionary, that translates one observer's coordinates into the other's, or they could not communicate with each other and could not tell whether they are looking at the same particle. This dictionary must give the x, in terms of the y, and vice versa: it should consist of equations of the form 1.
Note that the transformation law in general depends on the time t. If the j; functions are known, Eq. Indeed, one obtains don't forget the summation convention 1. In that case the last three terms of 1. This is interpreted see the Remark later in this section to mean that the acceleration is a vector, that the transformation law for its components is the same as that for the components of the position vector.
This is also true if one demands that the acceleration seen by one observer should be determined entirely by the acceleration i. Before we prove the converse, we make a comment. It is actually relative position vectors that transform like accelerations and forces.
A relative position is the difference of two position vectors, the vector that goes from one particle to another. Let there be two particles, with position vectors x and x' in one frame and y and y' in the other. But it is also seen that when the frames are inertial, so that the j; 1 are constants, the transformation law for the relative positions is of exactly the same form as for the acceleration. Incidentally, this is true also for relative velocities, which can be defined either as the time derivatives of relative positions or, equivalently, as the difference of the velocities of two particles.
Furthermore, there is a transformation law by which the components of a vector in one frame can be found from 1. Moreover, for vector equations to be frame independent i. D We now return to the proof of the converse. If the acceleration is a vector, its transformation law must be linear and homogeneous, which means that each of the last three r.
But that implies that all the second derivatives of the j; x, t must vanish and hence that the j; x, t must be linear in the xk and in t, that is, of the form 1. This proves the converse: it establishes the transformation law of the position if acceleration is a vector. This also exhibits the intimate connection between '. The result may be summarized as follows: If the acceleration is a vector in one frame, then it is a vector in and only in relatively mertial frames.
If force is a vector and Newton's laws are valid in one inertial frame, then. Stated in terms of the observers for after all, it is the observers who determine the frames, not the other way around , if Newton's laws are to be valid for different observers, those observers can be moving with respect to each other only at constant velocity. The transformation defined by the j; functions of 1. We now extend this statement of the general problem: to find not only x t but also x t.
Once x t is known, of course, x t is easily found by simple differentiation; so this extension of the problem may seem redundant. Let It be so; in due course the reason for it will become clear. Solving 1. Often, however, one is not interested in all of the details of the motion, but in some properties of it, like the energy, or the stable points, or. Then it becomes superfluous actually to find x t and x t. These objects, functions of x and x, are called dynamical variables. Many dynamical variables are quite important.
In Section 1. An important property of the extended definition of momentum is evident from Eq. In terms of forces, this means that the total momentum of the two-particle system is constant if the sum of the external forces acting on it is zero. Similarly, if the total force on one particle is zero, then according to 1. In fact it is the constancy of momentum, its conservation in this kind of situation, that singles it out for definition. All this assumes, as usual, that all measurements are performed in inertial systems.
Otherwise, Eq. In general S may be any moving point, but we shall restrict it to be an inertial point, that is, a point moving at constant velocity in any inertial frame more about this in Section 1. The time derivative of 1. In particular, if the torque is zero, the angular momentum is conserved. It is only when x and ' are known as functions oft that F can be written as a function oft alone and Eq. But x and v are not known as functions of the time until the problem is solved, and when the problem is solved there is nothing to be gained from 1.
Thus Eq. Often, however, F is a function of x alone, with no v or t dependence. This equation doesn't give x t in terms of the initial conditions, but it does give the magnitude v t of the velocity, provided the integral on the left-hand side can be performed. But there are two problems with performing the integral.
The first involves the upper limit x t. Recall that the problem is to find x t , so x t is hardly useful as an upper limit. What Eq. To find v as a function oft requires finding x as a function oft some other way.
Since the time t actually plays no role in Eq. But in spite of these difficulties, Eq. The usefulness of Eq. Equation 1.
In words, the work done by the force F along the trajectory is equal to the change in kinetic energy between the initial point x 0 and the final point x of the trajectory. This path dependence reduces the practical value of the theorem, because often only some of the points of a trajectory are known without the trajectory being known in its entirety. Fortunately, however, it turns out that for many forces that occur in nature the integral that gives W is not path dependent; it depends on nothing but the end points.
Forces for which this is true are called conservative. Conservative forces are common in physical systems, so we now study some of their properties. Let F be a conservative force it should be borne in mind that in all of these equations F is a function of x , and let C 1 and C 2 be any two paths connecting two points x 1 and x 2 , as shown in Fig.
Then, by the definition of a conservative force, 1 1 1 -1 rJ. Since the left-hand side of 1. The argument can be reversed to show that Eq. In other words, if F is conservative, there exists a function V satisfying Eq.
This important function V is called the potential energy of the dynamical system whose force is F. Up to now, a one-particle dynamical system has been characterized by its vector force function F. That is, given F, the dynamical problem is defined: solve 1. From now on, if the force is conservative, the characterization can be simplified: all that need be given is the single scalar function V, and F is easily obtained from V via 1 1. This reduction from the three components of a vector function to a single function simplifies the problem considerably.
Later, when we deal with systems more complicated than one-particle systems, it will be seen that the simplification obtained from a potential energy function can be even greater. Let us return to Eq. We have derived conservation of energy for any dynamical system consisting of a single particle acted upon by a conservative force. Because the total energy E is a sum of T and V, and because V is defined only up to an additive constant, E is also defined only up to an additive constant.
This constant is usually chosen by specifying V to be zero at some arbitrarily designated point. It is clear that conservation of energy does not solve the problem of finding x t in conservative systems, but it is a significant step. We will see in what follows that in one-dimensional systems it leads directly to a solution. It is important to note that the three dynamical variables p, L, and E that have been defined are associated with conservation laws.
Historically, it is their conservation in many physical systems that has led these functions of x and :X to be singled out for special consideration and definition. For example, the arena for the single-particle dynamical systems we have been discussing is threedimensional Euclidean space. In another way, the arena for a single particle constrained to remain on the surface of a sphere is the two-dimensional spherical surface.
The dimension of the arena is called the number of degrees of freedom of the system or simply the number of its freedoms. It is the number of functions of the time needed to describe fully the motion of the system. The simplest of all arenas is that for a particle constrained to move along a straight line. This is clearly a one-freedom system there are other one-freedom systems; e. One-freedom systems are of particular interest not only because they are the simplest, but also because often a higher-freedom problem is solved by breaking it up into a set of independent one-freedom problems and then solving each of those by the technique we will now describe.
This technique reduces the problem to performing a one-dimensional integration or, as is said, to quadrature. In one dimension all forces that depend only on x are conservative. Thus F is the negative deriYative of a potential energy function V x defined up to an arbitrary additive constant, and Eq. This equation is obtained by mtegrating 1.
Classical dynamics a contemporary approach pdf Classical dynamics of particles and systems solutions pdf dynamics of particles and systems solutions pdf This site contains solutions Jose and Saletan Solution Manual: Classical dynamics a contemporary approach pdf download?
Lagrangian and Hamiltonian dynamics, canonical transformations, the Hamilton-Jacobi equation, perturbation methods, and rigid bodies. A Contemporary Approach By? The authors cover all the material Classical dynamics a contemporary approach pdf -? They also deal with more advanced topics such claxsical the relativistic Kepler problem, Liouville and Darboux theorems, and inverse and chaotic scattering.
My library Help Advanced Book Search. Cambridge University Press Amazon. A Contemporary Approach Jorge V. The book contains many worked examples and over homework exercises. It will be an ideal textbook for graduate students of physics, applied mathematics, theoretical dynamica, and engineering, as well as a useful reference for researchers in these fields.
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