The Key to Newtons Dynamics: The Kepler Problem and the Principia

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The Key to Newton's Dynamics - The Kepler Problem and the Principia (Electronic book text)

By a simple revision of Newton's diagram for Proposition 6 of the third edition of the Principia, one can see directly how the mathematics of uniform circular motion have been employed to solve the Kepler problem of elliptical planetary motion in Proposition Newton strove initially to build his dynamics on the linear kinematics of Galileo; and, in this utilization of uniformly accelerated linear motion to solve more complicated problems, he can be seen as revolutionary. But he could not escape completely from the coils of celestial circularity, and in his utilization of uniform circular motion to solve problems of elliptical motion, he can be seen as reactionary.

The key to understanding Newton's mature dynamics resides in the discussion of the alternate dynamics ratio, as presented here in section six. Edit this record.

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Mark as duplicate. Find it on Scholar. Request removal from index. Revision history. From the Publisher via CrossRef no proxy dx. Configure custom resolver. John Herivel.

Whiteside - - History of Science 5 1 Bruce Brackenridge - - Annals of Science 47 1 Eloge: Bruce Brackenridge, — Loup Verlet - - History of Science 34 3 Curvature in Newton's Dynamics. This is usually only applied to scalars, how- ever. In vectors, if the time derivative of a vector is the zero vector, then that vector does not change magnitude or direc- tion. In other words, the angular momentum vector of a planet is a constant vector: where F is the impressed force and r is the lever arm over which the torque acting.

The vector r begins at the axis of rotation and ends at the point where the impressed force is acting.

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The torque vector r indicates in which direc- tion the body tends to rotate. This is the core concept of Kepler's 2nd law. But while torque is usually applied to rigid bodies, such as wheels and levers, it can also be applied in celestial mechan- ics. The concept of torque can be applied to any body with respect to a fixed point in space. The vector between this fixed point and the body then becomes the lever arm, al- though it is by no means a solid one.

What is the mathematical expression for angular momentum, though? Here, however, the impressed force will be gravity. Our fixed reference point will be the Sun itself. This makes perfect sense: if you can only pull radially on bucket as the Sun can only pull radially on a planet , you won't be giving the bucket rotating about an axis a tendency to speed up or slow down in its rotation. Making this substitution and also exploiting the fact that the cross product is associative with respect to scalar factors , we find that, 4. Therefore 0 d dtL must also be the zero vector: 0.

This ray is simply the vector r that we've been using. And we shall continue to use it; r, remember, is defined as the vector from the Sun to the planet. They are called polar vectors, or ordinary vectors. But a vector product of two polar vectors such as , will not change sign under inversion. Such vectors are called axial vectors or pseudovectors. The scalar product of a polar vector and a pseu- dovector is a pseudovector; it changes sing under inversion, where a scalar vector does not. This law can be devel- oped easily using polar basis vectors.

The three vectors r , , and form a triangle. The area of this triangle closely approximates the area swept out by the vector r during that short time. Polar coordi- nates are useful for dealing with motion around a central point — just the case we have with planets moving around the Sun. If we let taking the limit of both sides of Eq. Therefore we need make no change of notation. Our definition of r as a polar basis vector merely quantifies our work in the plane of the orbit. This makes it easy to talk about the com- ponent of a vector along r, the radial direction, and the com- ponent along , the transverse direction.

But we know that the mass of the planet is constant, and we also know from our work earlier that the angular momentum vector is constant and thus its magni- tude certainly is. Therefore, the time derivative of area swept out by this ray is constant. In other words, no matter where on the orbit the planet is, its ray still sweeps out the same amount of area. This is Kepler's second law.

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We shall require them later in the proof anyway. But this looks like something we've already dealt with! Solving for r and applying the chain rule, we find that, 7 8. But we're not finished. This doesn't tell us much about the shape of a planet's orbit, although all the pieces are there.

Newton and Kepler's Laws

This is the general solution, and it could be an orbit of any of the possible shapes though we can't be sure what they are yet or any of the possible orientations. We're interested in knowing the shape, of course, so we want to restrict the pos- sible orientations. To do that, we'll take a special case. At this point, r, the position vector of the planet, will have only a component in the positive x-axis. We'll also assume that the planet orbits the Sun counter- clockwise, through increasing measures of angles.

If this is the case, then the velocity v at the instant of perihelion should be orthogonal to the position vector r, and it should have only a component in the positive y—axis. It states that the square of the orbit is proportional to the cube of the semimajor axis. The constant of proportionality is independent of the individual planets; in other words, each planet has the same constant of proportionality.

Solving for r gives, Multiplying both sides by dt gives, 2 2. A 65 The focus—directrix distance should be a constant, which p is: L, G, m, and M are all individually constant; therefore the expression 2 L GMm ust also be constant. Therefore, New- ton's laws of motion and universal gravitation dictate that the orbits of planets follow conic sections. This is Kepler's first law. An ellipse is only one type of conic section. One question might be — why is an ellipse allowed while the other conic sections are not?

When Kepler said planet, he meant a body that repeatedly returns to our skies.

Newton's Laws revisted

The curve repre- senting the orbit is closed — it must repeatedly retrace itself. The other two conic sections — the parabola and hyperbola — are open curves and correspond to a position where the body has sufficient velocity to escape from the Sun's gravity well. The body would approach the Sun from an infinite distance, round the Sun rapidly, and then recede away into the infinite abyss, never to be seen again. There- fore, the square of the period of a planet is proportional to the cube of the length of the semimajor axis, and this propor- tionality is the same for all planets.

This is Kepler's 3rd law. Asimov, I. Armitage, A. Quod erat demonstrandum. And the intersection of these two planes gives a direction space. Byron, Frederick W. This is known as the autumnal equinox AE. Cohen, I.

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Hawking, S. Heese, M. Hall A. Chandrasekhar, S. Christianson, Gale E.

About the Author

Heath, T. Manual, A Portrait of Isaac Newton, Holton, G. Donahue, W. Kepler, , French translation, J. Kepler, J. Kepler, , translated by C. Wallis, Prometheus Books, Emch, G.

Motz, L. Weaver, The Story of Physics, Plenum, Years later it was discovered that much of his erratic behavior may have been caused by mercury poisoning. Recent samples of his hair showed he had forty times the level of mercury considered normal. Newton discovered the origin of color. He discovered the nature of gravity.

He invented calculus.