21st Century Mechanics and Thermodynamics (part 1) by Hans Besseling. An attempt to present the theory in a direct and simple intuitive way. June 2005 1
Introduction It all began by staring at the nightly sky, filled with stars, slowly moving their positions. Occasionally a fast moving object, or even what looked like a rain of stars could be observed. Driven by curiosity mankind started the long quest of science, ultimately leading to the launching of human beings into space and their return on earth. Mechanics merged with thermodynamics enables us to simulate on the computerscreen all motion, here on earth and in the sky. 2
Agenda First the mathematical formulation of the mechanics of solid, undeformable bodies will be explained (Part 1, Part 2, Part3). The analysis of the strength of truss-structures was started by Eiffel at the end of the 19th century (Part4) With the introduction of deformation the concept of temperature and thermodynamics arises (Part 5, Part 6, Part 7). Emphasis will be laid on how the various concepts of the theory present themselves in a natural way, if one dares to leave the traditional approaches to the subject. 3
Basic hypotheses It was Galilëi (1638) who noted that for walking, or for falling objects, the observable effects were no different on a ship, gliding at constant speed through still water, from the effects on land. He concluded that the laws of nature were the same if formulated with respect to bodies moving at constant speed relative to each other. 4
Basic hypotheses continued From Galilëi we know that a general theory of motion of material bodies should hold not only for any initial position, but also for any initial velocity. Otherwise Galilëi s observations of relativity could not be true. The principle was mathematical-correctly formulated by Newton (1687). From Galilëi s work one might think he referred to horizontal initial velocities only. 5
Basic hypotheses continued We shall see that the equations of motion of the theory can be obtained from a conservation law by means of linear algebra. For a system of material bodies, not influenced from outside (a so-called closed system), we define energy as the quantity that is being preserved. 6
Overview First some mathematical concepts and methods have to be introduced. A grasp of differential calculus is a prerequisite. Newton s theory for the motion of point masses in a gravitational field is derived from the concept of relativity, due to Galilëi, more precisely formulated by Newton, and the hypothesis of conservation of energy. 7
Overview continued For rigid bodies of finite dimensions the equations of motion, due to Euler, are derived in Part 2. The experimental value of the gravitational constant G is given and the three physical dimensions, playing a role in mechanics, are discussed. As measuring units the meter, the kilogram and the second are introduced. 8
Mathematical preliminaries Physical theory aims at providing simulations of events, that can or could be compared with real experiments to determine the accuracy of the theory. Simulation of events in a quantitative way is always based upon an expression of the theory in terms of mathematical equations. 9
Mathematical preliminaries The values of the variables in the theory sometimes will be determined by one real number, but more often a variable has more than one component. Then all components have to be specified by real numbers to define the value of such a variable. A vector variable is our first example of a more-component variable. 10
Vector space Generally, one s first encounter with the concept of a vector is in the form of a geometrical representation of a vector in two or three dimensions. 11
Rectangular cartesian system 12
The position vector The position vector r gave for the observer the direction and the distance to the object, that was being observed. In a rectangular, cartesian coordinate system the vector can be defined numerically by its three components,. r, r, r x y z 2 2 2 1 2 x y z Its length is then given by r r r r. 13
Explanation Note that the notation for a vector is a bold latin character. The notation of a mathematical variable with a single value, like the length of a vector, is written as an italic latin character. The notation of a scalar multiplier is written as a greek character. 14
Inner product in a vector space A generalization of the concept of length is the inner product (denoted by o ). For the position vector we write r r r r r r 2 2 2 2 x y z In matrix algebra the notation is r x 2 r rx ry rz ry r z. 15
Scalar product in vector spaces If we have two vector spaces of the same dimension (= the number of components of each element), then we may form a scalar product (, ): rr, r r r r r r. x x y y z z In matrix algebra we write rr, r r r r r x x y z y r z. 16
Shorthand notations Symbols for a shorthand notation: (1) Meaning of is: for all (2) Meaning of is: element of (3) Meaning of is: there is an element 17
Properties of scalar product Just as for real numbers a and b holds a 0 if ab 0 b, so for elements a V1 and b V2 holds a 0 if a, b 0 b V. 2 18
Vector space continued The straight line with an arrowhead is a special example of the more general notion of a vector as an element of a real vector space, which is a triple (,,ƒ ) consisting of an additive commutative group, the field of real numbers and a function ƒ :, called scalar multiplication. 19
Explanation A group is a collection of elements with certain properties. For an additive group holds that the sum of two elements is again an element of the group. A vector space is a triple, where a triple is just what it says: three things: a group of elements, which can be added, a field specifying the kind of values the components may have, and the function defining the multiplication of an element by a single number, producing again an element of the group. 20
Rules satisfied in a vector space (1) ( u v) w u ( v w) u, v, w V. (2) u v v u u, v V. (3) 0 V such that u 0 u u V. (4) u V u V such that u ( u) 0 21
Explanation Adding two vectors means that the values of the corresponding components are being added. The adding of two vectors, originating from the same point and pictured by straight lines with arrowheads, can be done by the parallellogram rule. Show that the same result is obtained by adding the values of the cartesian components. 22
Scalar multiplication of vectors (1) ( u) ( ) u, (2) ( + ) u u u, (3) ( u v) u v, (4) 1 u u,, and uv, V. 23
Observations Notice that the scalar multiplication of a vector only affects its magnitude (length) and not its direction. Vectors in geometrical space may be added graphically by connecting tail to arrowhead of previous vector. If the sum ends with the arrowhead of the last vector reaching the tail of the first, the sum is zero. 24
Return to mechanics The following mathematical concepts have been introduced: -- Field of real numbers -- Vector space -- Scalar multiplication of vectors -- Inner product of elements of one vector space -- Scalar product of elements of different vector spaces Together with the rules from differential calculus these mathematical concepts are used to derive the equations of motion of point masses in a gravitational field from the basic hypotheses. 25
Physical dimensions We shall introduce variables with different physical dimensions: M for mass, L for length, T for time. We introduce standard units for measuring the variables: L is expressed in meters (m). M is expressed in kilogram (kg). T is expressed in seconds (s). Here we shall not dwell on the precise definition of these units. 26
Shorthand notation As a shorthand notation in mechanics differentiation of a variable with respect to time usually is indicated by putting a dot on top of the variable. So r 2 d r d d dk, r r r, K 2 dt dt dt dt 27
Inertial systems By differentiation with respect to time we obtain form the position vector r the velocity vector r. The relativity principle of Galileï is only valid in so-called inertial systems, moving with respect to each other at constant relative speeds. 28
Point masses A mathematical expression for the position, velocity and accelaration of an identifiable, so-called material point could be given by introducing the concept of a vector. For a body of finite dimensions additional mathematical concepts have to be introduced, but first we shall derive the equations of motion for material points, possessing a certain mass. 29
Kinetic energy of point masses The inner product of the velocity vector is a measure for the scalar quantity, called the kinetic energy : K 1 2 m r r The kinetic energies of point masses in a closed system may be added, but this can only make sense if the inner products of their velocities are first multiplied by a scalar factor, representing their relative mass ( ). m 30
Newton s first law Every body continues in its state of rest or of uniform motion in a straight line in so far it is not compelled by forces to change that state. This peculiar (forces are not yet defined) statement is Newton s first law. We prefer to state the hypothesis that the sum of the kinetic energies of point masses in a closed system remains constant in time if the masses do not influence each other. With Galilëi s principle of relativity it then follows that velocities remain constant. 31
Closed system of two masses Let us consider two point masses A and B, together establishing a closed system. If at any given initial time their velocities are given by ra and rb and if their relative masses are characterized by scalar multipliers m and m, then we have A r A d d 2 K= ( m r r m r r ) 0 r, r. dt dt A A A B B B A B m r, r m r, r 0 r, r. A A A B B B A B B 32
Two independent masses Apparently the two point masses A and B move at constant velocity, since the time derivatives of their velocities (called accelerations) are zero by the properties of the two scalar products. Note that we postulate that the initial velocities of all point masses are independently arbitrary. The relative mass of the two bodies does not come into play as long as they do not influence each other. 33
Two interacting point masses For the description of the interaction of two point masses A and B, constituting a closed system, we add a term to the sum K of their kinetic energies. The simplest intuitive hypothesis for this term is that it will be inversely proportional to the distance 1 2 ra rb ra rb and proportional to the relative masses m A and. m B 34
Total energy The term we add is the so-called potential energy. The total energy is the sum of the kinetic energies and the potential energy: E K P, E 1 2 ( m r r m r r ) A A A B B B GmAmB 1 ( ra rb ) ( ra rb ) 2 35
Gravitational Constant Note the introduction of the factor G in the extra term expressing the interaction. This factor has to be added since the kinetic energy and the potential energy may only be added if they have 2-2 the same physical dimension M L T. Check that the dimension of G must be M L T -1 3-2. 36
Explanation One could ask why for the potential energy a term has been added with a minus sign. The explanation is simply that otherwise it would later on follow from experiments that the gravitational constant would have a negative value. Introducing the minus sign here leads to a positive value for G. 37
Equations of motion d K P m r f, r dt A A AB A + m r f, r 0 r, r b B BA B A B m r f and m r f A A AB B B BA 38
Gravitational forces We observe GmA mb ( ra rb ) f f AB BA 3 ( ra rb ) ( ra rb 2 These are what Newton called forces, expressing the gravitational law. 39
Newton s second and third law The equations of motion, derived from the hypothesis of conservation of energy, are an expression for the second and third law put forward by Newton. However he introduced the forces as a concept independent of the gravitational law. Newton gave the forces the values, derived for them here, as the expression of the special case of gravitation. 40
Central forces Note that the two bodies attract each other by so-called central forces, acting along a straight line connecting them. The two forces are equal in magnitude and are opposite in direction, a characteristic property of all forces of interaction we shall encounter in mechanics. 41
Gravity phenomena on earth As a first check of our theory we may look at objects falling on earth. Since the mass of the earth is so much larger than the mass of a marble we may neglect the interaction of this marble with any other objects, moving in its vincinity. We have the marble, attracted by our gravitational law to the earth. Show that according to our equations everywhere on earth the acceleration of a falling marble is independent of its mass, just as it is observed. 42
The value of G Cavendish devised an experiment for the measurement of the force of mutual attraction of two masses (1797-1798). The experiment was repeated with increasing accuracy by Boys and finally by Hey and Chrzanowski (1942). 11 1 3 2 G (6.673 0.003)10 kg m s 43
The weight of a mass From the equations of motion it follows that acceleration of a mass is accompanied by forces acting on that mass. Apparently we may prevent acceleration due to gravity by annulling the gravitational force. Preventing a marble from falling we feel its weight. It is the gravitational force that we have to annul in order to prevent the marble from falling. The weight of a mass is its mass multiplied by the local acceleration due to gravity. 44
The nature of forces The question arises whether the action we exert on the marble to prevent it from falling is a vectorvalued function of the physical behaviour of the object we push against the marble. In other words are there other forces in nature besides the gravitational force, and may forces mathematically be added as vectors? By hypothesis our answer is affirmative, giving the equations of motion general validity. 45
Rigid bodies of finite dimension The observations with respect to the marble falling to earth suggest that the marble as well as the earth may be modelled as point masses: the earth with all its mass concentrated in its geometrical center, the marble with its mass also in its geometrical center. Our hypothesis with respect to the additivity of different forces acting on a mass enables us to extend our theory to the motion of all celestial bodies. 46
Energy of n celestial bodies The total energy of n point masses, n n n Gm 1 imj E mr r 2 i i i i 1 i j 1 j 1 ( ri rj ) ( ri rj ) 2 is independent of time. Hence E 0 ra, rb and ra, rb. It can be shown that the resulting equations simulate by computation with remarkable accuracy the motion of celestial bodies (e.g. The motion of the planets in our solar system). 47
Initial conditions The time derivative of the total energy of the system of n celestial bodies must be zero for any given initial configuration (i.e. for all position vectors of the n bodies) and for any given initial velocities. Show that the condition that the time derivative of the total energy be zero for all velocities leads to the following equations of motion. 48
Equations of motion of body i mr i i 1 j i Gm ( r r ) j i j ( r r ) ( r r ) i j i j 3 2 n Gm ( r r ) j i j. j i ( r r ) ( r r ) i j i j 3 2 49
Summary Instead of basing the equations of motion for point masses on Newton s laws, we derived these equations from Galilëi s principle of relativity and the hypothesis of conservation of energy. By the equations of motion we may simulate numerically the motion of celestial bodies. In doing so for fictitious systems we observe the chaotic motion if the number of bodies is chosen larger than two. 50
Computer simulations Here we quit the presentation of the theory for a moment and turn to the computer program Newton, which simulates the motion of n interacting point masses. In the file Planeten the positions and velocities of the planets of our solar system at some time are stored. So not only the motion of any system of point masses may be simulated, but also the motion of the planets around the sun. 51
Simulation program Newton for the motion of celestial bodies 52