Download 1 - Ryoichi Kawai

Classical Mechanics I.
PH461/PH561
Newton's Laws of Motion
(Addition not discussed in the Textbook)
Instructor: Ryoichi Kawai
Department of Physics
University of Alabama at Birmingham
2009 Fall Semester
Evolution of Classical Mechanics
Galileo Galilei
(1564-1642)
Johannes Kepler (1571-1630)
René Decartes (1596-1650)
Isaac Newton (1642-1727)
Hamiltonian and Hamilton-Jacobi Theory
Lagrangian
Joseph L. Lagrange
Piérre S. Laplace
Laws of Motion
(1736-1827)
(1749-1827)
Carl G. J. Jacobi (1804-1851)
William R. Hamilton (1805-1865)
Mach Principle
Ernst Mach (1838-1916)
Chaos
Henri Poincaré
(1854-1912)
Relativity
Albert Einstein (1897-1955)
Andrei N. Kolmogorov (1903-1987)
Edward N. Lorenz
(1917- )
Yakov G. Sinai
(1935- )
••••
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
(1643-1727)
Philosophiae naturalis principia mathematica (1687)
Historical Information http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html
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.
Critiques of Newston's Laws
(Kirchhoff, Mach, Hertz, Clifford, Poincaré, ...)
Is the Newton Equation F=ma useful?
Quantities defined by Newton
d2 x
a= 2
dt
well defined and measurable.
m = “quantity of matter” (mass) defined as density times volume.
mv=m
dx
, “defined as motion” (momentum).
dt
F = “force”, not defined.
Neither m nor F is clearly defined.
F=ma is useless!
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
(1938-1916)
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
aB mA
=
a A mB
Force is then defined as
F =m A a A =m B a B
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 vj 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,
12 23  31=1
Definition of mass
12 23  31=1
mj
can be satisfied by using positive constants mj such that ij =
mi
Momentum conservation
v 1 t 12 v 2 t = K
m1 v 1 t m2 v 2 t = P 12
Definition of force
m1 v 1 t m2 v 2 t = P 12
m1 a 1 t m2 a 2 t =0
where force is defined as F 12 =m1 a 1
F 1 2  F 2 1 =0
Newton's 3rd law
Newton's 2nd 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)