Classical Mechanics Introduction

Philosopher Giving a Lecture on the Orrery (Joseph Wright of Derby, exhibited 1766)

Content

Introduction

The purpose of mechanics is to describe how objects change their position in space with time: motion. In physics the atomic hypothesis states:

All things are made of atoms - little particles that move around in perpetual motion, attracting each other when they are a distance apart, but repelling upon being squeezed into one another. Richard Feynman

Mechanics is described by the following major theories: classical mechanics, relativity and quantum mechanics. Relativity corrects classic mechanics for motions near the speed of light. Quantum mechanics extends and corrects classical mechanics and is used for describing the motions on the atomic scale.

Objects studied in classical mechanics are normally macroscopic objects, a factor $10^8$ larger than atoms, having finite size and being deformable (i.e. change shape). In classical mechanics we call these objects bodies. Classical mechanis uses idealized models ranging from simple to complex in order to study the behavior of a real object and a system of objects:

We will start with studying classical mechanics. The development of classical mechanics starts with the theory developed by Isaac Newton in his Philosophiae Naturalis Principia Mathematica (1687), commonly known as the Principia. Also known as Newtonian mechanics. More abstract theories, called analytic mechanics, are Lagrangian mechanics and Hamiltonian mechanics.

Classical mechanics may be divided into three subdisciplines, kinematics, dynamics, and statics. Kinematics is the study and description of the possible motions of material bodies. Dynamics is the study of the laws which determine, among all possible motions, which motion will actually take place in any given case. In dynamics we introduce the concept of force and mass. The central problem of dynamics is to determine for any physical system the motions which will take place under the action of given forces. Statics is the study of forces and systems of forces, with particular reference to systems of forces which act on bodies at rest.

Fundamental dimensions

Classical mechanics is based on 3 independent fundamental dimensions: space, time and mass.

Space

One aspect of motion is related to the position of an object in space. Two different objects are distinguished by their position, since we assume two objects can not occupy the same position simultaneous. Space is a concept derived from the observation of different objects and by defining spatial relations between them. The concept of space is defined in physics by a mathematical structure (geometry). Mathematically space is a geometry consisting of a set of points with a defined set of relations between these points.

In classical mechanics the concept of space used is called Galilean space. This structure is based on 3-dimensional Euclidean geometry, $\mathbb{E}^3$ and with following properties:

To measure the position of an object a reference object is chosen, together with a system of co-ordinate axes. The specific co-ordinates assigned to an object in space is relative to this system of co-ordinates.

Related to the concept of distance are the concepts: length, area and volume. The mathematics of calculus is used to give a precise meaning to these concepts in the case of smoothly curved shapes.

Time

Time defines an order and duration. Time is what we read from a clock. A clock is a moving system whose position can be read out. The motion of a clock is defined as the standard reference motion, used to define a standard sequence and a standard duration. A precise clock is a clock whose motion is as regularly as possible.

We call the classical mechanics view of time Galilean time. The structure of time used is based on the set of real numbers $\mathbb{R}$ and with following properties:

Space Time

A phenomenon or event is defined by its space and time coordinates $(x,t) \in \mathbb{E}^3 \times \mathbb{R}$. Motion is a sequence of space time co-ordinates: $(x(t),y(t),z(t))$. Geometrically this can be represented as a curve in a space-time diagram. The type of curve, or path of motion, is relative to the chosen reference system for space. For example the path of motion of a stone vertically dropped on the embankment from a moving train is a straight line for the observer in the train and a parabolic curve for the pedestrian on the embankment. So there is no such thing as an independently existing trajectory, but only a trajectory relative to a particular body of reference.

Mass

Mass is the subject of motion. It is associated with objects, it remains constant over time and measures the resistence to a change in motion as a scalar number. More specifically, inertial mass, is a quantitative measure of an object's resistance to acceleration.

Derived concepts

From the fundamental quantities there are derived quantities that are used to describe and explain the motion of objects. These quantities will be studied later in great detail but are summarized here.

Galilean relativity

Now in virtue of its motion in an orbit round the sun, our earth is comparable with a railway carriage travelling with a velocity of about 30 kilometres per second. If the principle of relativity were not valid we should therefore expect that the direction of motion of the earth at any moment would enter into the laws of nature, and also that physical systems in their behaviour would be dependent on the orientation in space with respect to the earth. For owing to the alteration in direction of the velocity of revolution of the earth in the course of a year, the earth cannot be at rest relative to the hypothetical system K0 throughout the whole year. However, the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical non-equivalence of different directions. This is a very powerful argument in favour of the principle of relativity. Einstein, 1916

In physics often the word relative is used as opposite for absolute. We define a quantity as relative if its measurement depends on the reference frame from which the observation is done. It is important to know which quantities are relative and which absolute since we want to write the laws of nature in a form which doesn't depend on the particular reference frame used. Therefore we need quantities that remain invariant or laws that are symmetrical. Professor Herman Weyl has given this definition of symmetry: a thing is symmetrical if one can subject it to a certain operation and its appears exactly the same after the operation.

By observation it has been determined that the laws of physics are symmetrical for translational displacement and rotation about an angle. It doesn't matter from which position or from which angle we observe phenomena. If we find a law in one reference frame it holds also in any other reference frame which differ from that one by a translation and or rotation of its axes.

Quantities that have an orientation in space, i.e. have a direction besides a magnitude, are represented by the mathematical object of a vector. A vector is an object independent of the reference system, however its coordinates are not. For this reason physicists try to write the laws of nature in vector notation. For practical calculations however one need still a co-ordinate representation of a vector.

Galileo was the first who studied the relativity of velocity. He used a thought experiment, Galileo's ship, to analyze this question. One observer in a ship cabin at rest and one observer in a ship cabin moving at constant speed and direction. Both observers have no windows to look outside and only observe motion within their cabin. Then he concluded that none of them could tell whether he was moving or at rest. Physicist formulated this into the following hypothesis.

Hypothesis: Galilean Relativity

Any two observers moving at constant speed and direction, (i.e. uniform rectilinear motion), with respect to one another will obtain the same results for all mechanical experiments.

It has been shown by experiments that the Galilean relativity hypothesis holds for the laws of classical mechanics. This idea of relativity gave rise to the definition of an inertial reference frame.

To study natural phenomenon one need a coordinate system for measurement of position. A coordinate system should be attached to some physical body. A body is always in motion relative to the other bodies. If we choose our reference frame such that the body to which it is attached is in uniform rectilinear motion then the motion of objects we study in such a reference frame can not be influenced by the motion of the reference frame itself. This follows from the hypothesis of Galilean Relativity, which states that observers in uniform motion obtain the same results for all mechanical experiments and therefore the results can not depend on the motion of the reference frame in uniform rectilinear motion. The explanation of the motion can be attributed only to the interaction of objects within the reference frame itself. Without such an interaction an object observed from in reference frame in uniform rectilinear motion would also be in a state of uniform rectilinear motion relative to the reference frame. These type of reference frames, whose motion, does not influence the observation of the motion of the objects within the reference frame, are called inertial reference frames.

Inertial reference frame

An inertial reference frame is a co-ordinate system for space and time measurement with the property that all laws of classical mechanics hold good. An observer occupying an inertial reference frame is called an inertial observer.

Based on the postulate of Galilean Relativity we can conclude that all coordinate systems in uniform rectilinear motion with respect to an inertial one are themselves inertial. How do we select an inertial reference frame ? Any reference frame that is at rest or is moving with constant velocity, e.g. no acceleration ($\vec{a}=0)$, with respect to the average position of the fixed stars can be chosen as an inertial reference frame. A reference frame attached to the earth can be considered to be inertial for most partical purposes in classical mechanics.

If we use an inertial reference frame all accelerations observed can be attributed to a net force exerted by a specific object in the environment and each net force is associated with an equal but opposite net force. This follows directly from Newtons second and third laws. The type of forces we observe are limited to the forces we know: (1 gravity (2 electomagnetic (such as the tension in a rope, force of friction, spring forces, normal forces) or (3 nuclear.

However it is also possibe to use in specific situations, non-inertial reference frames, such as a rotating frame. The price we have to pay is that we can no longer attribute all observed accelerations to a force exerted by a specific object in the environment. In order to explain the motion we have to introduce, pseudo or inertial forces. For these inertial forces, there is also no reaction force applicable.

In the past Aristotle thought that for a body to move a force was needed. If this was $F=mv$ would be a law of physics. But this would imply that the laws of physics depend on the speed of the reference frame. So a prediction following from Galileo's principle of relativity is that free objects will move in straight lines at constant speed. Galileo performed a series of experiments in which he determined that frictionless motion would indeed be in a straight line at constant speed. Consider a ball rolling in a smooth bowl. The ball rolls from it's release point to the opposite end and back to a certain place slightly below the initial point. As the surfaces of the bowl and ball are made smoother and smoother the ball returns to a point closer and closer to the initial one. In the limit of zero friction, he concluded, the ball would endlessly go back and forth in this bowl. From this hypothesis follows that the velocity of a body tends to remain unchanged (a property called inertia).

Hypothesis: Law of inertia

Every body remains being at rest or being in uniform rectilinear motion unless it is compelled to change that state by forces impressed on it.

An inertial reference frame can now also be defined as a reference frame where the law of inertia holds. In contrast with a non-inertial reference frame where the law of inertia does not hold and the motion of bodies cannot be explained by the laws of classical mechanics.

Although the measurement of velocities is relative, classical mechanics is still based on the concepts of absolute space and time. All rods and clocks are synchronized and therefore always measures the same length and time independent of the observer or velocity of rods and clocks. Therefore length, duration and simultaneity of events will be agreed upon by any two inertial observers.