Thus, we can reduce the dimension of. F or example, the analysis. Notice that, if we use the. By separa-. The same idea works for systems with several degrees of freedom,. Let us also note that the total energy of a Hamiltonian system might not. In fact, if F is a function on the phase space. In this case F is called an. Th us, the intersection. A rigid body and its three axes of symmetry. In fact, for an n -degree-of-freedom. W e will not prove this fact. How ever, it should be clear. They are prime examples of. While the notion of an energy surface is naturally associated with Hamil-.

Thus, just as for Hamiltonian systems, some of the dynamical. This is not the simplest way to solve the equations. More generally , for a smooth. Because a conservativ e force is the negative gradient of a potential, many authors. Prov e. A gradient system has no periodic orbits. If a gradient system. Equivalen tly, the equation for the angular velocity is. If A is diagonal with diagonal components moments of iner-. Find some inv ariant manifolds for this system.

As a physical. Are all three rotary motions Lyapunov. Do you observe an y other interesting phenomena associated with the. F or example, pay attention to the direction of the front cov er of the. Look for invariant quadric surfaces; that is, manifolds. F or example, show that the kinetic energy given by 1. The total angular momentum length of the angular. F or a complete mathematical description of rigid. One asp ect of this problem worth. To do so would require a functional relationship betw een the. F ortunately, this. Thus, the class that we will.

Recall that a manifold is supposed to be a set that is lo cally the same as.

## Differential Equations and Their Applications

R k must be a manifold. By this we mean that there is a smooth map G with domain. W and image graph g such that G has a smooth inverse. In fact, suc h a. Clearly, G is smooth. Open subsets and graphs of smooth functions are the prototypical exam-. How ever, these classes are too re-.

The unit circle T in the plane, also called the one-dimensional.

## Differential Equations with Applications to Industry

However, ev ery point of T i sc o n t a i n e di na neighbor-. In fact, each point in T is. A chart for a tw o-dimensional submanifold in R 3. Submanifolds of R n are subsets with the same basic property: Every point. T o formalize the submanifold concept for subsets of R n ,w em u s td e a l. Rather, we must allo w, as in the example provided by T , for graphs of func-. To o vercome. If W, G is a submanifold coordinate chart, then the map G is called a. I f S is a submanifold of R n , then, even though. As an example, let us show that T is a one-dimensional manifold.

Similarly , T is locally the graph of a smo oth function at points in the subsets. A simple but important result about submanifold coordinate charts is. If W, G is a submanifold coor dinate chart for a k -.

More over, the inverse of G is the restriction of a smo oth function that is. Hence, we also hav e. This proves that F is the in verse of G. If S is a submanifold, then we can use the submanifold coordinate charts. If S is a submanifold, then the op en subsets of S are all. If S is a submanifold of R n and if V is an open subset.

Suppose that S 1 is a submanifold of R m , S 2 is a subman-. Suppose that S 1 and S 2 are manifolds. A smooth func-. The function H is called the inv erse of F and is denoted by. Ho wever, we hav e determined. Indeed, the loc al repr esentative of the function F is given by. In the next. Thus, as a convenien t informality ,. Ho wever, there. In these cases, we will refer to the appropriate class of C r. Suppose that S is a k -dimensional submanifold. If W, G is a submanifold coordinate chart for a man-.

Informally , a set S together with a collection of subsets S is. If t wo. This abstract notion of a.

A linear subsp ace of R n is a submanifold. Let us suppose that S is the span of the k linearly indep endent vec-. W e will show that S is a k -dimensional submanifold. Let us denote the remaining set of standard. Hence, if t 1 , In fact,. Moreover, because the image of G has dimension k as a vector space,. As mentioned previously , linear subspaces often arise as inv ariant mani-. If S is.

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A complete. Howev er, the essential features of the. In other. F or example, consider the Volterraâ€”Lotk a system. In case a , b , c ,a n d d are all positive, this system models the interaction. Indeed, sup-. Let us now discuss level sets of functions. Recall that if H: It is instructive to outline a proof of this result because it provides our. For notational con venience let us. All other cases can be proved in a similar manner. W e are in a typical situation: W e have a function F: This calls for an application of the implicit function theorem.

A preliminary. Its deriv ative at. Of course, with respect to the usual. Suppose that F: Continuing with our outline of the proof of Proposition 1. Show that S n: Show that the surface of revolution S obtained by rotating. This manifold is homeomorphic to a two-dimensional torus T 2: Construct a homeomorphism. This exercise points out the weakness of our. I s C a one-dimensional. Show that the closed unit disk in R 2 is not a manifold. How should this concept be formalized?

W e have used, informally , the following proposition: If S is a manifold in. To make this proposition. Let us begin by considering some examples where the proposition on. Note that the vector assigned. F or this reason, we will view. Is S an inv ariant set? In particular, the orbit corresponding to this solution is contained. This requirement is not. Thus, the proposition on inv ariance is not valid. F or example, the following proposition is valid: Indeed, if S is compact, then it is bounded.

Thus, a. T o prove this requirement, recall from Euclidean geometry that a v ector. Moreov er, the. S 2 ; that is, at each base point on S 2 the corresponding principal parts of. Let us suppose that S is a k -dimensional submanifold of R n and G, W. W e have the following ob vious proposition: More precisely, suppose that W, G i s. By Proposition 1.

Thus, we see that S is just the image of a linear map from R k. If K is a second submanifold coordinate map at p ,s a y K: In fact, we hav e. The tangent bundle TS of a manifold S is the union of. Suppose that S 1 and S 2 are manifolds, and F: The derivative, also called the tangent map, of F is. However, it is also. W e simply use the local representatives of the function F and.

W e have already discussed this in ter-. However, because it is so. A comp act submanifold S of R n is an invariant man-. The idea of the proof is to change.

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In particular, we hav e that. In particular, for eac h. By the existence theorem, this initial value problem has a unique.

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Moreover, this. The solution remains in S as long. But S was assumed. By the extensibility theorem, if a solution on S does not. The second possibility is excluded by. If the solution approaches. Thus, it follows that all solutions on S exist for all time. State and prove a proposition that is analogous to Proposi-. We have men tioned several times the interpretation of the. This interpretation.

In fact, let. Prove that this is an equiv-. Finally , for manifolds S 1 and S 2 and a function F: The left panel depicts a heteroclinic saddle connection and a. The right panel depicts the phase portrait of the. Call a rest point isolated if it is the unique rest point in some open set. What is B?

Is there some H such that X H has exactly m rest points? If not,. What is its dimension? Is it true that for an open and dense. It is an interesting unsolved problem to. One of the key issues that must be resolved to determine the structural stabilit y. A heteroclinic orbit is an orbit that is contained in the stable manifold.

A basic fact from the theory of structural. Also, prove that X H cannot hav e a periodic orbit. Construct a homogeneous. Ho wever, if it is true that all heteroclinic. Moreover, this set is open and dense. A proof of these facts requires some work. Howev er, the main point is.

In fact, K can be. Of course, the conjecture that hetero-. There is an extensive and far-reaching literature on the subject of structural. The diagonal. The proof of Proposition 1. T o reiterate this result, suppose. In fact, if g: New coordinates can reveal unexpected features. As a dramatic. The follo wing precise statement of this fact, which is. Lemma 1. Let us. The action of the derivative of H 2 at the origin. In particular, DH 2 0 is the identit y, an in vertible linear transformation of. T o complete the proof we will use the inverse function theorem.

R n is a smooth function. Consider the function H: U and V contained in. R n , and a smooth function G: In view of the fact that F is continuous, the set U: Thus, by the uniqueness of the implicit solution,. By the inverse function theorem, there are t wo neighborhoods U and V of. The new coordinate,. The map g: The idea that a change of coordinates may simplify a given problem is a. Show that the implicit function theorem is a corollary of the. Prove that the function given by. Finally , show that the change of coordinates given by this birational. In this section we will consider the.

Howev er, the main purpose of this section. Perhaps the best way to understand the meaning of polar coordinates. W e have proved that the unit circle T is a. The wrapping function P: Clearly , P is smooth and surjective.

## (PDF) Ordinary Differential Equations with Applications

But P is not injective. In particular,. The function P is a covering map; that is, each point of T is con tained. Each suc h open set,. The image of a point of T under an angular coordinate map is called its. The polar wrapping function P: Howev er, all angular coordinate systems are compatible in the sense that. The totality. Find a collection of angular co ordinate systems that cover the.

Let us next consider coordinates on the plane compatible with the polar. The function P is a smooth surjective map that is not injective. Thus, P. Also, this function is not ac o v e r i n gm a p. How ever, P. A pol ar coo rdi n at e sy s te m on the punctured plane is. The collection of. T o obtain covering maps, the z -axis must be remov ed in the target plane in. Moreover, for spherical coordinates, the second variable m ust. Of course, because the expressions for the components of the Jacobian.

In general this is the best that we can do. In practice, perhaps the simplest way to change to polar coordinates is. A lso, if f is class C r , then the desingularized. Even if Proposition 1. The polar wrapping function factored through the phase cylinder. It is evident from formula 1. F or this reason, let us change the point of view one last time and.

The phase cylinder can be realized as a two-dimensional submanifold in. R 3 ; for example, as the set. F or this realization, the map Q: There is also a natural covering map R , from. In fact, by Exercise 1. Even though the phase cylinder can be realized as a manifold in R 3 , most. Prove the following statements. If F is the push forward to the. If F can be desingularized, then its desingularization retains the symmetry.

Supp ose that F is the push forward to the polar coordinate. Find the components of the push forward h of F to the phase cylinder realized. Show that the push forward of h to the Cartesian plane. We ha ve tacitly assumed that the. Howev er, this expression for the gradient of a function is correct only on Eu-. Recall that if G: In fact, if we work locally. Moreo ver, the. F rom this point of. Finally , the derivativ e of G may be interpreted as the. A Riemannian metric on. Of course, the usual inner product assigned in each tangent space of.

R n is a Riemannian metric for R n. Moreover, the manifold R n together with this. Riemannian metric is called Eu cl i dea n spa ce. Note that the Riemannian metric. For example, the norm of a v ector is the square root. It follows that the shortest distance. Thus, the geometry of Euclidean space is. Euclidean geometry , as it should be. Similarly , suppose that g is a. Roughly speaking, a curve. The gradient of G: The associated gradient. If the Riemannian. Consider the upper half plane of R 2 with the Riemannian metric.

The upper half plane. The geometry. Buy eBook. FAQ Policy. About this Textbook This textbook is a unique blend of the theory of differential equations and their exciting application to "real world" problems. Show all. Table of contents 5 chapters Table of contents 5 chapters First-order differential equations Braun, Martin Pages Second-order linear differential equations Braun, Martin Pages Systems of differential equations Braun, Martin Pages Qualitative theory of differential equations Braun, Martin Pages Here, the.

Euler-Bernoulli beam equation is used to model the roof rock between the pillars, which is the. The model predicts that the beam will break at the clamped. Many industrial mathematics problems contain an aspect of heat conduction. The authors show how an optimal heat balance formulation can be obtained by applying their. This new err or measure combined with the. Elastic rods are used in many industrial and engineering applications. This special.

The numerical method preserves. In an application to biological modeling, an article developing a mathematical model. The authors derive a coupled. Both computational work and special-. The authors also include models of more realistic bladder. The investigation of industrial mathematics problems sometimes leads to the. This special issue contains. The fractional variational iteration. Liouville derivative. Modeling in industrial mathematics problems with parabolic equations is very. This special issue also contains a survey paper in which the author investigates.

The author then goes on to. This special issue has covered both the theoretical and applied aspects of industrial. Papers contain the development of new mathematical models or well-known. It is this multidisciplinary nature of industrial mathematics that makes it a. W e are grateful to all the authors who have made a contribution to this special issue. Probabi lit y. Complex Analysis. Operation s.