Integrals of Motion

3 Key excerpts on "Integrals of Motion"

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  Modern Astrodynamics

  Fundamentals and Perturbation Methods

  • Victor R. Bond, Mark C. Allman(Authors)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  Since the approach taken is to derive the fundamental integrals, we next give the classical definition of an integral of the motion and briefly discuss the concept of a completely solvable or integrable system. We then develop the fundamental integrals and relationships among them. Throughout we minimize reliance on geometry, stressing a somewhat rigid mathematical development. THE TWO-BODY PROBLEM 13 Figure 2.1 Two points of mass relative to fixed origin O. 2.2 Physics of the Two-Body Problem We consider two points of mass, nt andm2, located at positions r andr2 with respect to a fixed origin O. By a fixed origin we mean a point that is fixed to an inertial frame, which we defined in §2.1. We assume that these two masses interact by a force that is a function only of the relative distance between the two masses and that is directed along the position of mt with respect to m (fig. 2.1). This force can be expressed as where and the unit vector is r = r 2 - r. „ 1 r = -r . r Applying Newton's second law of motion to m% gives mih = -F , and Newton's second and third laws of motion to mi yield ifi = F. (2.1) (2.2) (2.3) (2.4) (2.5) 14 CHAPTER 2 Now obtain the differential equation of motion of m 2 relative to m i by subtract-ing (2.5) from (2.4) after dividing by their masses. The result is r 2 -f i = F F. m 2 mi Using equations (2.1) and (2.2) this becomes ) (2.6) f = -( — + — )Fr. m 2 m) In classical mechanics the reciprocal of the factor 1 1 _ m + m 2 rri2 m mtn 2 is called the reduced mass. We will not bother giving it a symbol here. Now we invoke Newton again. This time we specify that the magnitude of the force F is (2.7) That is, the force is that of Newton's universal law of gravitation. G is a pro-portionality factor called the Universal Gravitational Constant. Using equation (2.7) in equation (2.6) we obtain .. _ mi +m 2 ^mm 2 A r — & - r.

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  Dynamics

  Theory and Application of Kane's Method

  • Carlos M. Roithmayr, Dewey H. Hodges(Authors)
  • 2016(Publication Date)
  • Cambridge University Press

    (Publisher)

  Indeed, this is what happens in reality. The reason for this discrepancy between predicted and actual motions is that certain physically unavoidable, dissipative effects, such as frictional resistance to rotation, have been left out of account in the analysis. Problem 14.12 deals with a more realistic approach. 9.6 NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS OF MOTION 265 When the expressions for u 1 , u 2 , and u 3 available in Eqs. ( 25 ), ( 13 ), and ( 24 ), respectively, are substituted into Eqs. ( 3 )–( 5 ), it can be seen that q 1 , q 2 , and q 3 are governed by a set of coupled, nonlinear differential equations with time-dependent coefficients. Generally, solutions of such sets of equations can be found only by nu-merical integration. 9.6 NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS OF MOTION When the dynamical (see Sec. 8.1 ) and/or kinematical (see Sec. 3.4 ) differential equa-tions governing the motion of a system S cannot be solved in closed form (see Sec. 9.5 ), one can use a numerical integration procedure to determine for time t > t 0 , or for time t < t 0 , the values of the dependent variables appearing in the equations, provided that the values of these variables are known for t = t 0 , t 0 being a particular value of t . To this end, one can employ any one of many computer programs † applicable to the solution of a set of ν first-order differential equations of the form dx i dt = f i ( x 1 ,. . . , x ν ; t ) ( i = 1 ,. . . ,ν ) (1) where x 1 ,. . . , x ν are unknown functions of t , and f 1 ,. . . , f ν are known, generally non-linear functions of x 1 ,. . . , x ν , and t . In this form, the ordinary differential equations are referred to as explicit . Sometimes the differential equations of motion of a system are available precisely in the form of Eqs.

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  Available until 25 Jan |Learn more

  Classical Mechanics

  • Tai L. Chow(Author)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  If the differential equation of motion is an equation with separable variables, rather than introducing integration constants, we can evaluate the definite integrals on both sides of the equation over the appropriate range. (d) Determine the required quantities and analyze the obtained results. A particular problem is solved by integrating the differential equations of motion and evaluating the integration constants. In order to be able to analyze the solution, it should be carried out in the most general form, inserting the numerical data only in the final results. In general, the force function may be a function of position, velocity, or time. A problem where the applied force is a function of all three variables simultaneously is difficult to solve. The prob-lems that we shall consider, therefore, generally fall into one of the following four categories: 1. The applied force is constant. 2. The applied force is time dependent. 3 58 Classical Mechanics © 2010 Taylor & Francis Group, LLC 3. The applied force is velocity dependent. 4. The applied force is position dependent. 3.2 MOTION UNDER CONSTANT FORCE We consider the motion of a projectile under the action of the gravitational force near the Earth’s surface. Although the gravitational acceleration varies with locality, in a small local region, it is constant to a good approximation both in magnitude and direction. Example 3.1: Motion of a Projectile A particle of mass m is projected with an initial speed v 0 at an angle α with the horizontal. Find the following: (a) The position vector of the particle at any time (b) The time to reach the highest point (c) The maximum height reached (d) The time of flight (e) The range (f) The equation of the trajectory (g) The parabola of safety (region of space in which the projectile can reach) Solution: We assume the air resistance is negligible; then the force acting on the projectile after its firing is the gravitational force mg .

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