Classical Mechanics.
"The study of the motion of bodies under the action of physical forces"
Table Of Contents.
1. Newton's System.
2. Lagrange's System.
3. Energy is Consistent.
4. The Beautiful Truth.
Foreword.
Due to the nature of classical mechanics, it doesn't make sense to suck the maths out of this section and put it in Maths in Physics. This chapter is therefore quite maths heavy!
1. Newton's System.
Aristotle was a Greek philosopher from 384-322 BC. He is often considered the first physicist - the word physics comes from the title of one of his works.
Aristotle believed the world was split into 5 spheres - 4 were of the mortal world and called the elements: fire, water, wind and earth. All material objects could be broken down into these components which is not dissimilar from Democritus' model of atoms. The fifth sphere was the aether which made up the heavens - he proposed it was weightless and, in his words, incorruptible.
Unfortunately this entirely nonsense.
Aristotle did lay the groundworks for classical mechanics though:
Aristotle proposed the idea of forces. He observed the workings if the world around him and concluded that "The velocity of an object is always proportional to the force applied to it."
This model of action and reaction went unchallenged for 2000 years until a young English physicist published a groundbreaking paper: Philosophiæ Naturalis Principia Mathematica. His name? Isaac Newton.
In this paper, Newton took Aristotle's law of forces and altereted into his three laws:
Law 1.
"An object in motion stays in motion unless acted on by an external force. An object at rest stays at rest unless acted on by an external force."
This law is often thought of as common sense - if I don't touch an object it doesn't move, big deal! However, at the time this was revolutionary because it formally introduced friction. Moving objects in Newtons time meant clunky wooden carriages which were incredibly inefficient. Objects with a high friction coefficient, such as these carriages, would stop almost immediately if not acted upon by a force (a pulling horse).
If you didn't know that friction was a force secretly acting on the carriage you would assume the exact opposite of Newton's first law: "Objects in motion become stationary unless acted on by an external force." Never underestimate the power of the first law!
Law 2.
"The acceleration of an object is directly proportional to the force applied to it and inversely proportional to the mass of the object. F = MA."
The beady eye reader notices here that this is almost identical to Aristotle's original equation of motion. The only difference is that Newton showed that it was the acceleration that changed as force changed, not velocity. He also formalised the constant of proportion as being the mass of the object - something Aristotle never realised.
F = MA is perhaps the most important equation in all of classical mechanics because it links three of the most fundamental measurements: force, acceleration and mass. As we discuss later in more depth, if you have the acceleration of an object as a function of time you can use some maths jiggling to produce displacement and velocity. We can therefore add these fundamental measurements to the list.
Law 3.
"Everyone action has an equal and opposite reaction."
Imagine placing a book on a desk. What stops the book falling through the desk? It has a weight acting down and no sideways forces. Newton realised that because the book doesn't fall through the desk, the vertical forces on the book must balance - there must be some force acting upwards that counters the weight. We call this force the reaction force.
Newton then showed that all forces have an associated reaction force that is of equal magnitude to the original force but acts in the opposite direction.
This principle allows rockets to fly. All a rocket is a metal tube that ejects burning fuel; the ejection of this fuel causes a reaction force that pushes the rocket up.
An interesting thought experiment for the reader: can you think of a mode of transport that does not involve the principle of reaction forces? For example walking is merely repeated reaction forces to weight off the floor. (If you find one please make your way to Sweden for a Nobel prize).
3. Lagrange's System.
Joseph-Louis Lagrange was an Italian mathematician who lived from 1736 to 1813. In the field of mathematics he applied differential calculus to probability, invented a method of solving differential equations called 'variation of parameters' and proved that all natural numbers are can be expressed as the sum of 4 square numbers.
These achievements had already put his name on the world stage but Lagrange, like all great scientists was not content to stop there. His great contribution to science came from his work on the principle of least action - from which sprung a new system of maths (unimaginatively) named the Lagrangian.
The fundamental conceptual idea behind the Lagrangian is as follows:
For every system, there exists a unique mathematical value called the Lagrangian. It is defined as:
L = T - U.
L - the Lagrangian.
T - kinetic energy of the system (the T comes come the French travail mécanique meaning mechanical work).
V - potential energy (sometimes represented by a U).
For most systems, this potential energy means gravitational potential energy which depends on the positions of all the particles in the system. For real life situations, we always approximate this because measurements are never precise enough to know the position of all individual particles.
The Lagrangian is not very exciting on it's own - really it's just a number. However, when it is combined with the principle of least action exciting conclusions can be drawn.
The Principle of Least Action.
Paths of Descent.
In 1696, Johan Bernoulli released a mathematical challenge to the world, a problem today known as the path of descent. Imagine two free points in space A and B such that A is higher than B as seen to the right.
The question seems simple at first: what is the path a dropped ball should take to minimise the time of descent. Picture building a frictionless slide between A and B for the ball to roll down.
If A was directly vertically above B, this would be trivial: simply let the ball freefall for the fastest time. A straight line is not the answer for the question though because although distance is minimised, the ball does not accelerate very much and so it's speed down the "slide" is slower.
The alternative is to maximise acceleration and not care about path distance. However, this is still not the correct answer. These two opposite slides are shown below:
Bernoulli had cheated slightly in his own competition because he had found a solution before he posted the problem - he wanted to prove he was better than the English upstart Newton. The story goes that Newton also submitted a less elegant solution but didn't sign his name. Bernoulli recognised it as Newtons, saying "I recognise the Lion by his claws", which would be a strong contender for best physics quote of all time.
Bernoulli's solution to his own problem is as follows:
Bernoulli's Solution.
He first took inspiration from observing how light interact at medium boundaries. When waves travel into a different medium , they change speeds and if they are not normally incident this also causes a change in direction. This change in direction is called refraction and it is governed by a very simple law.
Snell's law states that sin(0i) / sin(0r) is always a constant n. This n is called the refractive index and can also be expressed as the ratio between the speed of light in the medium and the speed of light in a vacuum.
Pierre Fermat showed that Snell's law could be produced from the idea that when light refracts though an object, it takes the least possible time to do so. It takes a time minimising path.
Bernoulli knew of Fermat's works and made the connection between the path of optimisation chosen by refracting light and the path needed for a ball dropping between A and B. He made a very clever analogy between gravity and light:
Rather than use a ball falling down due to gravity, Bernoulli pictured layers and layers of glass each with different density's. The top layer of glass was most dense - light travels very slowly through this layer. The next layer down was slightly less dense and the next after that slightly less dense again. This pattern continues down through the glass. Bernoulli then conjectured that each layer of glass was infinitely thin and there were an infinite number of layers.
If light was shone into the top layer, it would refract (obeying Snell's law) into the second law which would then refract into the third and so on. This set up is pictured on the left.
In each layer, light also travels at different speeds, fastest in the bottom layers and slowest in the top. Bernoulli realised that if the layers were infinitely thin, the change in density would be continuous and the light would smoothly accelerate. In order to map this to the model of a ball falling, this acceleration would need to equal g.
To map this accurately, think of the energy transfers as the ball falls. It starts with gravitational potential energy that is the converted into kinetic energy. GPE is dependent on the distance from the top of the glass, we will call this d. We can say that:
1/2mv^2 = -mgd
This can be rewritten as v = (2mgd)^1/2.
In other words, the velocity of the ball, v, is directly proportional to the square root of the distance it has fallen, d. To write this simply:
V ∝ d^1/2.
Let's define the width of the first layer of glass as d1 and the second, d2, and so on. We also define the angle of incidence in each layer and velocity through the same metric but using 0 and v respectively.
We can put this into snell's law:
Sin(01) / Sin(02) = v1 / v2.
Which becomes:
Sin(01) / v1 = Sin(02) / v2.
By subbing in our proportionality equation from earlier:
Sin(01) / d1^1//2 = Sin(02) / d2^1/2
This also holds true for the next layer down and so the equation can be rewritten for 02, d2 and 03, d3. By equating all of these expressions for the lower layers,Bernouili found that:
Sin(0n) / dn^1/2 must equal some constant K.
Bernoulli spotted that Sin(0n) / dn^1/2 = k, is the equation of a cycloid. A cycloid is the path traced out by a point on a moving wheel as seen left.
3. Energy is Consistent.
proving coservation of Energy.
For a single particle system, the total energy is defined as:
E = T + V.
The energy change for a system over time is therefore:
Ė = dT/dt + dV/dt
T = 1/2mv^2
Ignoring relativistic effects (see the special relativity chapter), we assume mass to be a constant but v is free to change - v is almost always the factor that changes kinetic energy.
dv^2/dt = dv x v/dt.
We can use a clever differential trick called the product rule here. The product rule says if you have a function (A(x)) that is the product of two smaller functions (B(x)and C(x)), where both B and C are functions of a common variable (x), then:
dA/dx = A.dB/dx + B.dA/dx.
In our case A is v^2 and B and C are both v. Therefore:
dv^2/dt = v.dv/dt + v.dv/dt.
= 2(v.dv/dt)
dv/dt is of course just a so:
dv^2/dt = 2va.
We can't forget those constants though!
T = 1/2mv^2
So dT/dt = mva.
Turns out this result comes out in physics a lot so it's a good one to have stored upstairs.
Retuning to our equation for energy change (Ė = dT/dt + dV/dt), we need to use an even cleverer trick to differentiate the potential energy, V.
One of the fundamental principles behind classical mechanics is the potential energy principle. This sates:
"All forces come from unique potential energy functions. These functions are denoted by V(x)."
The exact relationship is defined as:
F(x) = - d V(x) / dx.
Important Note: the above equation assumes the force causes the potential energy to decrease. For forces that act against potential energy and increase it, the minus sign on the right side is removed.
This equation becomes logical when you remember that the right side (excluding the minus sign) merely shows the rate of change of potential energy. In words, this equation says "when a force is applied, potential energy changes at a rate dependent on the force."
We have run into a barrier in the form of an annoying variable. We can only apply the potential-energy principle if we are differentiating V(x) with respect to x. We are doing dV(x) / dt though.
The secret to dodging this barrier is to use another differential trick called the chain rule. This states:
dA/dB can be expressed as dA/dC x dC/dB, where C is another function.
The chain rule can also be used for cascades of new functions where each new function is defined based on the last one.
For our case, we will use C = x (for displacement). So:
dV/dt = dV/dx x dx/dt
Recognise these terms now?
We use the potential-energy principle to convert dV/dx into -F(x).
dx/dt is just v.
With the end now in sight, we will combine these to show:
dV/dt = -F(x) x v.
Returning to Ė = dT/dt + dV/dt, we have no defined both terms. The final stretch is to combine them:
Ė = mva + -F(x) x v.
Ė = v(ma - F(x))
Newton's second law shows us that F = ma so ma - F(x) = 0.
Therefore (phew):
Ė = 0
Et voila!!! The energy change of the system is 0, in other words energy has been conserved
Proving F = MA from E-L.
4. The Beautiful Truth.
At this point, I have to confess something. In section 1 and 2 I introduced the Newtonian and Lagrangian systems as being totally different and independent. That was a lie but don't worry, the truth is much more beautiful.
The truth is that the Lagrangian and Newtonian mechanics boil down to exactly the same thing and we can prove it:
Taking the Euler - Lagrange equation to be:
d / dt (δL / δv) - δL / δx = 0.
We can force F = ma (Newtons second law) to drop out this equation by using the potential energy principle.
Returning to the Euler-Lagrange equation, δA / δB is a partial differential. It is very similar to a normal differential formula with one subtle difference. Let's take A = B^2 G^4 H^11. Because we are doing a partial differentiation respect to B, we assume all other variables are constants. So we treat G^4 H^11 as though it was 1. This means we are just left with B^2 which differentiates to 2B.
Therefore, δB^2 G^4 H^11/ δB = 2B.
δL / δv is an expression for a partial differential on the Lagrangian where we take all variables that aren't v as constants.
L = T - V.
L = 1/2 m v^2 - V(x)
We can therefore say:
δL / δv = mv.
Because V(x) has no value of v and so is treated like a constant, it disappears.
d / dt is just a classic differentiation of all variables with respect to time.
dmv / dt is the rate of change of mv:
dmv / dt = ma
The other term in the equation is δL / δx. This is another partial differential on the Lagrange but with respect to displacement, x, not velocity.
δL / δx = δ1/2 m v^2 - V(x) / δx
= 0 - - F(x) (using the potential energy principle from the energy conservation proof)
= F(x)
The Euler-Lagrange equation is set to 0, and we now have expressions for all the terms in it, so:
ma - F(x) =0
Or F(x) = ma
This can be rewritten in dot form as:
F(x) = mx^..
Look familiar? From this result we can see that the E-L equation is nothing more than a very fancy rewriting of F = ma.