Monday, December 28, 2015

Classical Mechanics Primer

There are at least two senses in which the term “Classical Physics” is used. First, classical may be used to mean “all physics that is not quantum physics”. The second sense is more similar to what most people usually have in mind, and basically refers to all physics which is not “modern”—that is, physics before General Relativity and Quantum Mechanics.  Physicists always mean to use the first sense of the term, and laypeople usually switch from one to the other, sometimes without noticing. From this point on, I’ll be using the physicist’s sense unless noted otherwise.

Now, the expression “not quantum” needs some unpacking. As an example, consider a toaster with a knob that you can turn at your leisure over some period of time, t. If you were to plot the energy flowing through the toaster’s resistance as you turn the knob, the graph that you will get might look something like this:
In Classical Physics, there is no limit to the number of times that you could zoom into any section of this plot, and every single time you would see a smooth, continuous line like the figure above. If you moved along the curve, you could always ride the line like a smooth ramp to any value of E that you liked, and in doing so you would pass through all points in between.

This is not the case in Quantum Mechanics, where many physical quantities are said to be quantized.  In the toaster example above, the plot will look the same initially, but after zooming in a finite number of times, eventually you will reach a plot that looks more like this:
This is what makes Quantum Mechanics so weird. And so, Classical Physics includes all the branches of physics where this effect is not taken into account, regardless of when said branch developed. This includes theories such as Classical Mechanics, Electromagnetism, Thermodynamics, Fluid Mechanics, and General Relativity, among others. Usually, classical theories are an excellent approximation for all kinds of real world phenomena, even though the real world is actually quantized.



Newton’s Three Laws of Motion are as follows, though you may find some different wording depending on who you ask:
1. Objects move at constant velocity as long as there is no net force acting upon them. Velocity is not the same as speed, as it also includes direction and is therefore a vector quantity (usually notated in bold). Also, zero is a perfectly valid value for a constant velocity.
2. If there is a net force acting upon an object this will cause it to accelerate, according to the relation \(\mathbf{F} = m\boldsymbol{a}\).
3. Every action produces a reaction of equal magnitude and opposite direction. So if a certain object A pushes or pulls on B, B will answer with a push or pull of its own. This is usually notated like this: $$\mathbf{F_{AB}} = -\mathbf{F_{BA}}$$.
The whole of Classical Mechanics emerges from these three laws, though by far the one used the most in calculations is the second one. At the high school level, this usually means doing lots of trigonometry for setups such as the following, which may distress some readers:
A typical problem in mechanics is to figure out the equation of motion of a body, given some initial conditions. We begin with some information about the object’s starting position, forces and constraints it is subjected to, and work our way up to the equation that will tell us how the object will actually move thereafter. In the case of Newton’s Second Law, all solutions are essentially variations on solving the differential equation $$ \mathbf{F} = m\mathbf{a} = m\frac{\mathrm{d} \mathbf{v}}{\mathrm{d} t}= m\frac{\mathrm{d} }{\mathrm{d} t}\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=m\mathbf{\ddot{x}}, $$ which would yield x as a function of t, or x=x( t ).

Unfortunately, most real world problems are not as clear cut as blocks on inclined planes, and Newton’s Laws are utterly impractical in their original formulation for solving anything but the most simple physical systems. Fortunately, there exist a couple of alternative approaches that manage to elegantly generalize Newton’s Laws so that they’re applicable to nearly any classical system one can think of, at the cost of introducing some abstraction. These approaches are absolutely equivalent to Newton’s Laws, and indeed incorporate them at their core, but their mathematical appearance is quite different.

Lagrangian Mechanics


If we think of space in three dimensions, then the position of a particle at any given moment can be described by a position vector from the origin, with individual components in the x, y, and z directions: \(\mathbf{r}=(x,y,z)\). The path that the particle will take over time will once again depend on the forces, constraints, and initial conditions it is subjected to. So, what we’re looking for is the way that \(\mathbf{r}\) will evolve over time, \(\mathbf{r}= \mathbf{r}(t)\).

However, working with cartesian coordinates (x,y,z) doesn’t help very much when, in general, physical systems aren’t easily describable by them. So instead, we introduce some abstraction and think of generalized coordinates, each of which we call \(q_{i}\). Each one of these coordinates could be cartesian \((x,y,z)\), cylindrical \((\rho ,\theta ,z)\), spherical \((\rho ,\theta ,\phi)\), or even other ways of placing a particle in space, such as a distance traveled along a constrained path, or an angle defined by the movement of a particle, or anything that suits us. Each generalized coordinate can also change in time, yielding a generalized velocity, and these are denoted using the dot notation for derivatives as \(\dot{q}_{i}\). By a simple extension, we can speak of other generalized quantities, such as generalized momentum \(p_{i} = \frac{\partial L}{\partial \dot{q}_{i}}\). This particular quantity will come in handy in later on.

A Lagrangian is defined as the difference of the kinetic and potential energies of the system in question, and it will have the following form: $$ L = \frac{1}{2}m\mathbf{\dot{r}}^{2}-U(\mathbf{r}).$$ The function U corresponds to the potential energy, and in general it will depend on the particle’s position. The first term is the familiar kinetic energy term \(\frac{1}{2}m\mathbf{v}^{2}\) from kindergarten, with the velocity expressed in terms of r.

Notice that the Lagrangian depends on the generalized coordinates and velocities, since each r is defined in those terms. L being a vector equation, we can separate the different \(q_{i}\)’s and their velocities into what are known as the Euler-Lagrange equations, which can be conveniently abbreviated as $$ \frac{\partial L}{\partial q_{i}} = \frac{\mathrm{d} }{\mathrm{d} t}\frac{\partial L}{\partial \dot{q_{i}}}. $$
So the rough strategy for solving problems in the Lagrangian formulation of Classical Mechanics is this: write L explicitly in terms of your generalized coordinates and velocities, making sure to include the specific (mathematical) description of U as well if it’s available. Then, plug the resulting expression for L into the Euler-Lagrange equation for as many coordinates as you have, perform the corresponding derivatives, and finally integrate whatever is left to solve each equation (i.e., get rid of whatever time derivatives are left over after you plug things in). If you’re successful, you will end up with as many equations of motion as you had generalized coordinates, and those equations of motion combine to give you the overall motion of the particle.

There are many nuances and neat things to go into with regards to the Lagrangian approach, but I’ll leave them aside for now. As a summary, notice that you end up with a second-order differential equation for each coordinate involved, and also notice that the setup for solving a problem begins with considerations of energy rather than forces. One important consequence of this is that any forces of constraint that may act on the object are naturally incorporated into the equations, without the necessity of force diagrams.

Hamiltonian Mechanics


A second approach to generalizing Newton’s laws is that of Hamiltonian Mechanics, which is a similar abstraction to that of Lagrange. First, we remember the concept of a generalized momentum, and also consider that for many systems the Lagrangian doesn’t depend explicitly on time $$\frac{\partial L}{\partial t} = 0.$$ When this is the case, we can define another quantity, the Hamiltonian, as $$H = \sum_{i=1}^{n}p_{i}\dot{q}_{i} – L.$$In addition, if the relationship between generalized and cartesian coordinates doesn’t depend on time either, the Hamiltonian is equal to the sum of the kinetic and potential energies of the system: $$H = T + U .$$In the same way that we used L in the Lagrangian formulation to derive our equations of motion, the Hamiltonian formulation relies on operating on H. In this case, the Hamiltonian is plugged into the equations $$\dot{q}_{i} = \frac{\partial H}{\partial p_{i}}\:  ,\:  \dot{p}_{i} = -\frac{\partial H}{\partial q_{i}}$$ to yield the equations of motion. This may seem like more work than the Lagrangian approach, but notice that we are left with sets of first-degree differential equations instead of second-degree ones. Even though we have twice as many equations as the Lagrangian approach, they are usually trivial to solve, and they include all the advantages of the Euler equations as far as dealing with constraints.

Hamiltonian mechanics in particular is important for later use in Quantum Mechanics and, as was the case with the Lagrangian formulation, there’s many details and asides that I left out for the sake of brevity. Both formulations are usually overkill for problems about blocks sliding down incline planes, but they become indispensable when working with more complicated systems. As I re-study mechanics for my PhD, I will try to post more on the details of each of these approaches, as well as practical examples of their use.

This post was the result of summarizing some of my notes on the subject, but obviously you would do well to look up these topics online. The Wikipedia entries are quite good, and then there are several recommendable textbooks. First, the Wiki links:
Newtonian Mechanics
Lagrangian Mechanics
Hamiltonian Mechanics
Books that I’m familiar with include the following:
1) Taylor, John R., Classical Mechanics. Great for undergraduate level students and laymen with calculus background. Informal, conversational style.
2) Goldstein, H. Classical Mechanics. Standard graduate text for students who want to get serious and be grown-ups. Very long-winded at times, but solid.
3) Landau & Lifshitz, Mechanics. Standard graduate text for masochists and people who like their textbooks lean and mean.

No comments:

Post a Comment