Advanced Classical Mechanics I. PH610/PH710. Interpretation of Newton's Laws of Motion (not discussed in the Textbook) Instructor: Ryoichi Kawai

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1 Advanced Classical Mechanics I. PH610/PH710 Interpretation of Newton's Laws of Motion (not discussed in the Textbook) Instructor: Ryoichi Kawai Department of Physics University of Alabama at Birmingham 2009 Fall Semester

2 Evolution of Classical Mechanics Galileo Galilei ( ) Johannes Kepler ( ) René Decartes ( ) Laws of Motion Isaac Newton ( ) Lagrangian Joseph L. Lagrange ( ) Piérre S. Laplace ( ) Hamiltonian and Hamilton-Jacobi Theory Carl G. J. Jacobi ( ) William R. Hamilton ( ) Mach Principle Ernst Mach ( ) Chaos Henri Poincaré ( ) Relativity Albert Einstein ( ) Andrei N. Kolmogorov ( ) Edward N. Lorenz (1917- ) Yakov G. Sinai (1935- )

3 Newton's Laws of Motion I. Every body continues in its state of rest or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it. II. The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. III.To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal and directed to contrary parts. Sir Isaac Newton ( ) Philosophiae naturalis principia mathematica (1687) Historical Information

4 Halliday-Resnick-Walker Fundamentals of Physics First Law: If no force acts on a body, then the body's velocity cannot change; that is, the body cannot accelerate. Second Law: The net force on a body is equal to the product of the body's mass and the acceleration of the body. (F=ma) Third Law: When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction. David Morin Introductory Classical Mechanics First Law: A body moves with constant velocity (which may be zero) unless acted on by a force. Second Law: The time rate of change of the momentum of a body equals the force acting on the body. Third Law: The forces two bodies apply to each other are equal in magnitude and opposite in direction.

5 Critiques of Newston's Laws (Kirchhoff, Mach, Hertz, Clifford, Poincaré,...) Is the Newton Equation F=ma useful? Quantities defined by Newton a= d 2 x dt 2 well defined and measurable. m = quantity of matter (mass) defined as density times volume. mv=m dx dt, defined as motion (momentum). F = force, not defined. Neither m nor F is clearly defined. F=ma is useless!

6 Ernest Mach's view of Newton's laws E. Mach, Science of Mechanics (Lipzig, 1874) Mach argued that Newton's laws can be stated in a single law. When two compact objects act on each other, they accelerate in opposite directions, and the ratio of their accelerations is always the same. Ernst Mach ( ) Note that there is no mass nor force in this statement. Pick a standard object A and its mass is defined as ma=1 kg. Collide another object B with A. Then, the mass of B is defined as Force is then defined as a B a A = m A m B F =m A a A =m B a B

7 Modern formulation of Newton's Laws Principle 1 There exist certain frames of reference, called inertial frames, with the following properties: Every isolated particle moves in a straight line in such a frame. 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. Principle 2 Consider two particles 1 and 2 isolated from all other matter, but not from each other. Observe them from an inertial frame. There exists a constant 12 >0 and a constant vector K independent of the time such that v 1 t 12 v 2 t =K where v j is the velocity of particle j. Moreover, 12 is always the same number for particles 1 and 2 but K depends on particular motion. Similar relation holds between particle 2 and 3, and also between 3 and 1. Then, =1

8 Definition of mass =1 can be satisfied by using positive constants mj such that ij = m j m i Momentum conservation v 1 t 12 v 2 t =K m 1 v 1 t m 2 v 2 t =P 12 Definition of force m 1 v 1 t m 2 v 2 t =P 12 m 1 a 1 t m 2 a 2 t =0 F 12 F 21 =0 where force is defined as F 12 =m 1 a 1 Newton's 3 rd law Newton's 2 nd law Suggested readings: L. Eisenbud, On the Classical Laws of Motion, Am. J. Phys. 26 (1958) 144. J. V. José and E. J. Saletan, Classical Dynamics, Sec. 1.2 (Cambridge Univ. Press, 1998)