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Unformatted text preview: Physics 53 Energy 1 What I tell you three times is true. Lewis Carroll The interplay of mathematics and physics The mathematization of physics in ancient times is attributed to the Pythagoreans, who taught that everything true is contained in numbers. But the introduction of algebraic equations as a way of stating laws of nature dates from the time of Galileo. In Isaac Newton science had both a great innovator in mathematics and a great analyst and experimenter on natural phenomena. Because he was (independently of Leibniz) the discoverer of calculus, he was able to think of physical processes in terms of rates, inFnitesimal changes, and summations of inFnitesimals into Fnite quantities by means of integrals. Although he published relatively little of his thinking on these matters his famous book, the Principia , contains no reference to calculus, giving geometrical arguments and proofs instead it seems clear that he thought analytically, in the modern mathematical sense of that word. In the 18 th century great advances were made in mathematical analysis, and many of these were applied to indeed, arose from problems in physics. New mathematical formulations were found for the content of Newtons three laws of motion, making easier the solutions of many physical problems. The work of Euler and Lagrange in particular gave important new insights, and led to the emergence in the 19 th century of what is probably the most important single concept in science, that of energy . In this part of the course we deal with energy as it applies to the mechanics of particles and systems of particles. Later we extend the concept to uids and thermal systems, and in the next course we discuss the energy associated with electromagnetic Felds. The scalar product In our discussion of energy we will need to use the product of two vectors that results in a scalar. Consider two vectors, A = ( A x , A y , A z ) and B = ( B x , B y , B z ) . Multiplying one component of A by one component of B gives a set of 9 pairs. This set as a whole is not very useful because it does not transform simply when we rotate the coordinate axes. PHY 53 1 Energy 1 Some combinations of the 9 do have simple transformation properties, however. The simplest of these is A x B x + A y B y + A z B z . It can be shown to be unchanged by rotation of the axes, so it is a scalar, and is therefore called the scalar product . The standard notation for it uses the dot multiplication sign between the two vectors: Scalar product A B = A x B x + A y B y + A z B z Because of the notation, it is often called the dot product. We can rewrite this formula in terms of the magnitudes and relative direction of the two vectors. Let the vectors be as shown in arrow representation. Then it is easy to derive a very useful formula: Scalar product formula A B = AB cos Some properties of the scalar product A B that follow from this: A B is positive if < / 2...
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