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The subject of this book is the theory of differential equations and the calculus of variations.

It is based on a course of lectures which the author delivered for a number of years at the Physics Department of the Lomonosov State University of Moscow. Theorems of the Existence and Unioueness of Solution of the Equa. Nonhomogeneous Linear Equations with Constant Coefficients and 8.

Stability Under Constantly Operating Perturbations. First-Order Partial Differential Equations 2. Functionals Dependent on the Functions of Several Independent 6.

Variational Problems in Parametric Form 7. An Efernentary Problem with Moving Boundaries. Sufficient Conditions for an Extremum 3. Transforming the Euler Equations to the Canonical Form Direct Methods In Variational Problems 2. In the study of physical phenomena one is frequently unable to find directly the laws relating the quantities that characterize a phenomenon, whereas a relationship between the quantities and their derivatives or differentials can readily be established. Oue then obtains equations containing the unknown functions or vector functions under the sign of the derivative or differential.

Equations in which the unknown function or the vector function appears under the sign of the derivative or the differential are called differential equations. The following are some examples of differential equations:. The relation between the sought-for quantities will be found if methods are indicated for finding the unknown functions which are defined by differential equations. The finding of unknown functions defined by differential equations is the principal task of the theory of differential equations.

If in a differential equation the unknown functions or the vector functions are functions of one variable, then the differential equation is called ordinary for example, Eqs. I and 2 above. But if the unknown function appearing in the differential equation is a function of two or more independent variables, the differential equation is called a partial diOerential equation Eq. The order of a differential equation is the highest order of the derivative or differential of the unknown function.

A solution of a differential equation is a function which, when substituted into the differential equation, reduces it to an iOentity.

I has the solution where c ‘is an arbitrary constant.

Differential Equations and The Calculus Of Variations by Elsgolts, L

It is obvious that the differential equation 1. For a full determination, one must know the quantity of disintegrating substance. The procedure of finding the solutions of a differential equation is called integration calfulus the differential equation.


In the above case, it was easy to find an exact solution, but in more complicated cases it is very often necessary to apply approximate methods of integrating differential equations.

Just recently these approximate methods still led to arduous calculations. Today, however, highspeed computers are able to accomplish such work at the rate of several hundreds of thousands of operations per second. Consequently, the problem reduces to integrating this differential equation. We shall indicate an extremely natural approximate method for solving equation 1. We take the interval of time t For large values of n, within the limits of each one of these small intervals of time, the force F t, r, r changes but slightly the vector function F is assumed to be continuous ; therefore it may be taken, approximately, to be constant over every subinterval.

F 10, r0, r0. On this assumption, it is easy, from 1. Continuing this process, we get an approximate solution rn t to the posed problem with initial conditions for equation 1. It is intuitively clear that as n tends to infinity, the approxi- mate solution rn t should approach the exact solution.

Note that the second-order vector equation 1. Every vector equation in three-dimensional space may be replaced by three scalar equations by projecting onto the coordinate axes. Finally, it is possible to replace one second-order vector equation I. Phase space is the term physicists use for this space. The radius vector R t in this space has the coordinates rx, ry, r, vx, vy, v.

In this notation, 1. The solution of 1. If we apply the above approximate method to 1. And so, for t In this method, the desired solution R t is approximately replaced by a piecewise linear vector function, the graph of which is a certain polygonal line called Euler’s polygonal curve.

In applications, the problem for equation 1. Such a problem-unlike the problem with the. In other words, it is necessary to solve equation 1. Numerous problems in ballistics reduce to this boundary’-value problem. Obtaining an exact or approximate solution of initial-value problems and boundary-value problems is the principal task of the theory of differential equations, however it is often required to determine or it is necessary to confine oneself to determining only certain properties of solutions.

For instance, one often has to establish whether periodic or oscillating solutions exist, to estimate the rate of or decrease of solutions, and to find out whether a solution changes appreciably for small changes in the initial values. Let us dwell in more detail on the last one of these problems as applied to the equation of motion 1. In applied problems, the initial values r0 and r0 are almost always the result of measurement and, hence, are unavoidably determined with a certain error.

This quite naturally brings up the question of the effect of a small change in the initial values on the sought-for solution. If arbitrarily small changes in the initial values are capable of giving rise to appreciable changes in the solution then the solution determined by inexact initial values ro and ro usually has no applied value at all, since it does not describe the motion of the body under consideration even in an approximate fashion.


We thtis come to a problem, important in applications, of finding the conditions under which a small change in the initial values r0 and r0 gives rise only to a small change in the solution r t which they deter- mine.

A similar question arises in problems in which it is required to find the accuracy with which one must specify the initial values ro and r0 so that a moving point sh0uld-to within specified accuracy-take up a desired trajectory or arrive in a given region.

Just as important is the problem of the effect, on the elsgoltts, of small terms on the right-hand side of equation 1.

In certain cases, these small forces operating over a large interval of time are capable of distorting the solution drastically, and they must not be neglected. In other cases, the change in the solution due to the action of these forces is inappreciable, and if it does not exceed the required accuracy of computations, such small disturbing forces may be neglected.

We now turn to methods of integrating differential equations and the most elementary ways of investigating their solutions. An ordinary first-order differential equation of the first degree may, solving for the derivative, be represented as follows:.

Parte 1 de 4 JI. First-Order Dillerential Equations 2. Linear Differential Equations of the nth Order E.

Elsgolts – Differential Equations and the Calculus of Variations

Nonhomogeneous Linear Equations Euler’s Equations 7. Systems of Differential Equations I. Fundamentals 2. Finding Integrable Combinations 4. Theory of Stability I. Elementary Types of Rest Points 3. Lyapunov’s Second Method 4. V ariatlon and Its Properties.

Variational Problems with Moving Boundaries and Certain 1. Extremals with Corners 4. Variational Problems Involving a Conditional Extremum 2. The Ritz Method Chapter The following are some examples of differential equations: Repeating the same reasoning for the subsequent subinterva Is, we get Applying these formulas n times we arrive at the value R T.

First-order differential equations J. First-Order Differential Equations Solved for the Derivative An ordinary first-order differential equation of the first degree may, solving for the derivative, be represented as follows: Arquivos Semelhantes mikhailov – partial – differential – equations mikhailov – partial – differential – equations.

Calculus-single variable-Hughes-Hallet Calculus-single variable. Analytical and Numerical Methods.