"So few went to hear him, and fewer understood him, that oftimes he did, for want of hearers, read to the walls. He usually stayed about half an hour; when he had no auditors he commonly returned in a quarter of that time."
Appraisal of a Cambridge lecturer in classical mechanics, circa 1690
The fundamental principles of classical mechanics were laid down by Galileo and Newton in the 16 and 17 centuries. In 1686, Newton wrote the Principia where he gave us three laws of motion, one law of gravity and pretended he didn’t know calculus. Probably the single greatest scientific achievement in history, you might think this pretty much wraps it up for classical mechanics. And, in a sense, it does. Given a collection of particles, acted upon by a collection of forces, you have to draw a nice diagram, with the particles as points and the forces as arrows. The forces are then added up and Newton’s famous “” is employed to figure out where the particle’s velocities are heading next. All you need is enough patience and a big enough computer and you’re done.
From a modern perspective this is a little unsatisfactory on several levels: it’s messy and inelegant; it’s hard to deal with problems that involve extended objects rather than point particles; it obscures certain features of dynamics so that concepts such as chaos theory took over 200 years to discover; and it’s not at all clear what the relationship is between Newton’s classical laws and quantum physics.
The purpose of this course is to resolve these issues by presenting new perspectives on Newton’s ideas. We shall describe the advances that took place during the 150 years after Newton when the laws of motion were reformulated using more powerful techniques and ideas developed by some of the giants of mathematical physics: people such as Euler, Lagrange, Hamilton and Jacobi. This will give us an immediate practical advantage, allowing us to solve certain complicated problems with relative ease (the strange motion of spinning tops is a good example). But, perhaps more importantly, it will provide an elegant viewpoint from which we’ll see the profound basic principles which underlie Newton’s familiar laws of motion. We shall prise open “” to reveal the structures and symmetries that lie beneath.
Moreover, the formalisms that we’ll develop here are the basis for all of fundamental modern physics. Every theory of Nature, from electromagnetism and general relativity, to the standard model of particle physics and more speculative pursuits such as string theory, is best described in the language we shall develop in this course. The new formalisms that we’ll see here also provide the bridge between the classical world and the quantum world.
There are phenomena in Nature for which these formalisms are not particularly useful. Systems which are dissipative, for example, are not so well suited to these new techniques. But if you want to understand the dynamics of planets and stars and galaxies as they orbit and spin, or you want to understand what’s happening at the LHC where protons are collided at unprecedented energies, or you want to know how electrons meld together in solids to form new states of matter, then the foundations that we’ll lay in in this course are a must.
In the rest of this section, we’ll take a flying tour through the basic ideas of classical mechanics handed down to us by Newton. More details can be found in the lectures on Dynamics and Relativity.
We’ll start with a single particle.. A particle is defined to be an object of insignificant size. e.g. an electron, a tennis ball or a planet. Obviously the validity of this statement depends on the context: to first approximation, the earth can be treated as a particle when computing its orbit around the sun. But if you want to understand its spin, it must be treated as an extended object.
The motion of a particle of mass at the position is governed by Newton’s Second Law or, more precisely,
(1.1) |
where is the force which, in general, can depend on both the position as well as the velocity (for example, friction forces depend on ) and is the momentum. Both and are 3-vectors which we denote by the bold font. Equation (1.1) reduces to if . But if (e.g. in rocket science) then the form with is correct.
General theorems governing differential equations guarantee that if we are given and at an initial time , we can integrate equation (1.1) to determine for all (as long as remains finite). This is the goal of classical dynamics.
Equation (1.1) is not quite correct as stated: we must add the caveat that it holds only in an inertial frame. This is defined to be a frame in which a free particle with travels in a straight line,
(1.2) |
Newtons’s first law is the statement that such frames exist.
An inertial frame is not unique. In fact, there are an infinite number of inertial frames. Let be an inertial frame. Then there are 10 linearly independent transformations such that is also an inertial frame (i.e. if (1.2) holds in , then it also holds in ). These are
3 Rotations: where is a orthogonal matrix.
3 Translations: for a constant vector .
3 Boosts: for a constant velocity .
1 Time Translation: for a constant real number
If motion is uniform in , it will also be uniform in . These transformations make up the Galilean Group under which Newton’s laws are invariant. They will be important in section 2.4 where we will see that these symmetries of space and time are the underlying reason for conservation laws. As a parenthetical remark, recall from special relativity that Einstein’s laws of motion are invariant under Lorentz transformations which, together with translations, make up the Poincaré group. We can recover the Galilean group from the Poincaré group by taking the speed of light to infinity.
We define the angular momentum of a particle and the torque acting upon it as
(1.3) |
Note that, unlike linear momentum , both and depend on where we take the origin: we measure angular momentum with respect to a particular point. Let us cross both sides of equation (1.1) with . Using the fact that is parallel to , we can write . Then we get a version of Newton’s second law that holds for angular momentum:
(1.4) |
From (1.1) and (1.4), two important conservation laws follow immediately.
If then is constant throughout the motion
If then is constant throughout the motion
Notice that does not require , but only . This means that must be parallel to . This is the definition of a central force. An example is given by the gravitational force between the earth and the sun: the earth’s angular momentum about the sun is constant. As written above in terms of forces and torques, these conservation laws appear trivial. In section 2.4, we’ll see how they arise as a property of the symmetry of space as encoded in the Galilean group.
Let’s now recall the definitions of energy. We firstly define the kinetic energy as
(1.5) |
Suppose from now on that the mass is constant. We can compute the change of kinetic energy with time: . If the particle travels from position at time to position at time then this change in kinetic energy is given by
(1.6) |
where the final expression involving the integral of the force over the path is called the work done by the force. So we see that the work done is equal to the change in kinetic energy. From now on we will mostly focus on a very special type of force known as a conservative force. Such a force depends only on position rather than velocity and is such that the work done is independent of the path taken. In particular, for a closed path, the work done vanishes.
(1.7) |
It is a deep property of flat space that this property implies we may write the force as
(1.8) |
for some potential . Systems which admit a potential of this form include gravitational, electrostatic and interatomic forces. When we have a conservative force, we necessarily have a conservation law for energy. To see this, return to equation (1.6) which now reads
(1.9) |
or, rearranging things,
(1.10) |
So is also a constant of motion. It is the energy. When the energy is considered to be a function of position and momentum it is referred to as the Hamiltonian . In section 4 we will be seeing much more of the Hamiltonian.
Example 1: The Simple Harmonic Oscillator
This is a one-dimensional system with a force proportional to the distance to the origin: . This force arises from a potential . Since , momentum is not conserved (the object oscillates backwards and forwards) and, since the system lives in only one dimension, angular momentum is not defined. But energy is conserved.
Example 2: The Damped Simple Harmonic Oscillator
We now include a friction term so that . Since is not conservative, energy is not conserved. This system loses energy until it comes to rest.
Example 3: Particle Moving Under Gravity
Consider a particle of mass moving in 3 dimensions under the gravitational pull of a much larger particle of mass . The force is which arises from the potential . Again, the linear momentum of the smaller particle is not conserved, but the force is both central and conservative, ensuring the particle’s total energy and the angular momentum are conserved.
It’s easy to generalise the above discussion to many particles: we simply add an index to everything in sight! Let particle have mass and position where is the number of particles. Newton’s law now reads
(1.11) |
where is the force on the particle. The subtlety is that forces can now be working between particles. In general, we can decompose the force in the following way:
(1.12) |
where is the force acting on the particle due to the particle, while is the external force on the particle. We now sum over all particles
(1.13) | |||||
where, in the second line, we’ve re-written the sum to be over all pairs . At this stage we make use of Newton’s third law of motion: every action has an equal and opposite reaction. Or, in other words, . We see that the first term vanishes and we are left simply with
(1.14) |
where we’ve defined the total external force to be . We now define the total mass of the system as well as the centre of mass
(1.15) |
Then using (1.11), and summing over all particles, we arrive at the simple formula,
(1.16) |
which is identical to that of a single particle. This is an important formula. It tells that the centre of mass of a system of particles acts just as if all the mass were concentrated there. In other words, it doesn’t matter if you throw a tennis ball or a very lively cat: the center of mass of each traces the same path.
The total momentum is defined to be and, from the formulae above, it is simple to derive . So we find the conservation law of total linear momentum for a system of many particles: is constant if vanishes.
Similarly, we define total angular momentum to be . Now let’s see what happens when we compute the time derivative.
(1.17) | |||||
(1.18) |
The last term in this expression is the definition of total external torque: . But what are we going to do with the first term on the right hand side? Ideally we would like it to vanish! Let’s look at the circumstances under which this will happen. We can again rewrite it as a sum over pairs to get
(1.19) |
which will vanish if and only if the force is parallel to the line joining to two particles . This is the strong form of Newton’s third law. If this is true, then we have a statement about the conservation of total angular momentum, namely is constant if .
Most forces do indeed obey both forms of Newton’s third law: and is parallel to . For example, gravitational and electrostatic forces have this property. And the total momentum and angular momentum are both conserved in these systems. But some forces don’t have these properties! The most famous example is the Lorentz force on two moving particles with electric charge . This is given by,
(1.20) |
where is the velocity of the particle and is the magnetic field generated by the particle. Consider two particles crossing each other in a “T” as shown in the diagram. The force on particle 1 from particle 2 vanishes. Meanwhile, the force on particle 2 from particle 1 is non-zero, and in the direction
(1.21) |
Does this mean that conservation of total linear and angular momentum is violated? Thankfully, no! We need to realise that the electromagnetic field itself carries angular momentum which restores the conservation law. Once we realise this, it becomes a rather cheap counterexample to Newton’s third law, little different from an underwater swimmer who can appear to violate Newton’s third law if we don’t take into account the momentum of the water.
The total kinetic energy of a system of many particles is . Let us decompose the position vector as
(1.22) |
where is the distance from the centre of mass to the particle . Then we can write the total kinetic energy as
(1.23) |
Which shows us that the kinetic energy splits up into the kinetic energy of the centre of mass, together with an internal energy describing how the system is moving around its centre of mass. As for a single particle, we may calculate the change in the total kinetic energy,
(1.24) |
Like before, we need to consider conservative forces to get energy conservation. But now we need both
Conservative external forces:
Conservative internal forces:
where . To get Newton’s third law together with the requirement that this is parallel to , we should take the internal potentials to satisfy with
(1.25) |
so that depends only on the distance between the and particles. We also insist on a restriction for the external forces, , so that the force on particle does not depend on the positions of the other particles. Then, following the steps we took in the single particle case, we can define the total potential energy and we can show that is conserved.
Let us return to the case of gravitational attraction between two bodies but, unlike in Section 1.2.4, now including both particles. We have . The potential is . This system has total linear momentum and total angular mometum conserved, as well as the total energy .