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ANALYTICAL MECHANICS 1.The second is, by definition, the amount of time required for exactly 9,192,631,770 oscillations of a par- ticular atomic transition of the cesium isotope of mass number 133.In our initial study of the science of motion, mechanics, we shall assume that the physical space of ordinary experience is adequately described by the three-dimensional mathematical space of Euclid- ean geometry.VECTORS the particle is an idealization that does not exist-even an electron has a finite size but the idea is useful as an approximation of a small body, or rather, one whose size is relatively unimportant in a particular discussion.The position of a point in such a coordinate system is specified by three numbers or coordinates, x, y, and z. The coordinates of a moving point change with time; that is, they are functions of the quantity t as measured on our time scale.Physical Quantities and Units The observational data of physics are expressed in terms of certain fundamental entities called physical quantities-for example, length, time, force, and so forth.The meter is now defined as the distance occupied by exactly 1,650,763.73 wavelengths of light of the orange spectrum line of the isotope krypton 86.The basic type of coordinate system for our purpose is the Cartesian or rectangular coordinate system, a set of three mutually perpendicular straight lines or axes.The meter was formerly the distance between two scratches on a platinum bar kept at the International Bureau of Metric Standards, Sevres, France.Ve c t o r s In any scientific theory, and in mechanics in particular, it is necessary to begin with certain primitive concepts.A very useful concept in mechanics is the particle or mass point, an entity that has mass' but does not have spatial extension.We shall further assume that space and time are distinct and inde- pendent entities.The meter was formerly the platinum iridium also kept at the International Bureau.Fundamental Concepts.1.1.

ANALYTICAL MECHANICS

1. Fundamental
   Concepts.
   Ve c t o r s
   In any scientific theory, and in mechanics in particular, it
   is necessary to begin with certain primitive concepts. It is also necessary to
   make a certain number of reasonable assumptions. Two of the most basic
   concepts are space and time. In our initial study of the science of motion,
   mechanics, we shall assume that the physical space of ordinary experience is
   adequately described by the three-dimensional mathematical space of Euclid-
   ean geometry. And with regard to the concept of time, we shall assume that
   an ordered sequence of events can be measured on a uniform absolute time
   scale. We shall further assume that space and time are distinct and inde-
   pendent entities. Later, when we study the theory of relativity, we shall
   reexamine the concepts of space and time and we shall find that they are not
   absolute and independent. However, this is a matter to which we shall
   return after we study the classical foundations of mechanics.
   In order to define the position of a body in space, it is necessary to have
   a reference system. In mechanics we use a coordinate system. The basic
   type of coordinate system for our purpose is the Cartesian or rectangular
   coordinate system, a set of three mutually perpendicular straight lines or
   axes. The position of a point in such a coordinate system is specified by
   three numbers or coordinates, x, y, and z. The coordinates of a moving point
   change with time; that is, they are functions of the quantity t as measured
   on our time scale.
   A very useful concept in mechanics is the particle or mass point, an
   entity that has mass' but does not have spatial extension. Strictly speaking
   1 The concept of mass will be discussed in Chapter 2.
   1
   2 FUNDAMENTAL CONCEPTS. VECTORS
   the particle is an idealization that does not exist-even an electron has a
   finite size but the idea is useful as an approximation of a small body, or
   rather, one whose size is relatively unimportant in a particular discussion.
   The earth, for example, might be treated as a particle in celestial mechanics.
   1.1. Physical Quantities and Units
   The observational data of physics are expressed in terms of certain
   fundamental entities called physical quantities-for example, length, time,
   force, and so forth. A physical quantity is something that can be measured
   quantitatively in relation to some chosen unit. When we say that the length
   of a certain object is, say 7 in., we mean that the quantitative measure 7 is
   the relation (ratio) of the length of that object to the length of the unit (1 in.).
   It has been found that it is possible to define all of the unit physical quantities
   of mechanics in terms of just three basic ones, namely length, mass, and time.
   The Unit of Length
   The standard unit of length is the meter. The meter was formerly the
   distance between two scratches on a platinum bar kept at the International
   Bureau of Metric Standards, Sevres, France. The meter is now defined as
   the distance occupied by exactly 1,650,763.73 wavelengths of light of the
   orange spectrum line of the isotope krypton 86.
   The Unit of Mass
   The standard unit of length is the meter. The meter was formerly the
   platinum iridium also kept at the International Bureau.
   The Unit of Time
   The basic unit for measurement of time, the second, was formerly defined
   in terms of the earth's rotation. But, like the meter, the second is now
   defined in terms of a specific atomic standard. The second is, by definition,
   the amount of time required for exactly 9,192,631,770 oscillations of a par-
   ticular atomic transition of the cesium isotope of mass number 133.
   The above system of units is called the mks system? The modern
   ^ In this system there is a fourth unit, the coulomb, which is used to define electrical
   units.