I was thinking that I would do a series of posts about classical mechanics from a mathematician’s standpoint. This post will be an overview of the Newtonian formalism of the subject, with some important examples so the usual physics jargon can be introduced. Future posts will cover the Hamiltonian and Lagrangian formalisms with all of the geometry involved.
Classical mechanics is the study of particles moving continuously through Euclidean space. In classical mechanics, there are three independent concepts, or “fundamental units”, that we have to worry about; namely mass, length, and time. In Newtonian mechanics, mass is the easiest one to model; it is simply a positive number, , an intrinsic value assigned to our particle. (Although starting in quantum field theory and special relativity, mass takes on a much deeper meaning.) Physically, mass is a measure of a particle’s inertia, its resistance to change. Our concept of length is modeled by Euclidean space, which we will take to be an affine space modeled on a real -dimensional vector space. (In physics, we obviously would assume that , but there is no need for that assumption here.) Of course, to have a coherent concept of length, we must equip our space with a translation-invariant metric: the Euclidean metric, for example. Finally, time will be modeled by an affine space based on a 1-dimesional real vector space, again with a translation invariant metric. Also, we impose an orientation on , which corresponds to being able to tell when you are moving forward or backward in time.
Now that we have a description of the spaces we will be working in, we can start talking about particles. We can describe a particle by a continuous path that assigns to a time the location of the particle at that time. Clearly, not all paths describe physical situations…we need a way of identifying which paths are “real.” To do this, we need to be given one more piece of information: a potential. A potential is just a (usually smooth) function , and it represents a type of energy. The fundamental postulate of Newtonian mechanics is that the physical paths must satisfy . Here, . It is important to note that this is really equations, one for each coordinate. This formula goes by many names: it is called Newton’s Second Law, but it is also known as the equations of motion or the Euler-Lagrange equations for the system.
Let denote the set of solutions to the equations of motion . Generally, we assume is of some form that elements of exist for all time. We don’t want our particles leaving our universe in finite time! We know from our experience with ordinary differential equations that a unique solution to the equations of motion exists for any initial position in and any velocity (tangent) vector at that point. Therefore, we get a bijection from to as follows: Fix a point ; we can map to . We can now assign a smooth manifold structure to by requiring that this map be a diffeomorphism. It turns out that this smooth structure does not depend on the choice of . This structure is very important in practice; it means we can use calculus to analyze our state of solutions!
Let’s compute some examples. In these examples, we are going to fix an initial time . This makes everything notationally easier, and we don’t lose any information.
Example 1: The free particle. If is identically 0 (or in fact, just constant), then we can quickly compute the solution space . This illustrates what is called Newton’s First Law: an object travelling at a constant velocity will continue to do so unless acted upon by an outside force.
Example 2: The harmonic oscillator: For this example, let , and suppose there is a point in that we can label 0 so the potential has the form for some positive . Then the equation of motion becomes . We know that the solution space of this equation is . (A quick note: many of you probably expected to see a linear combination of and as the general form of the solution, but by using the angle addition formulas, we can express this as a single cosine.) In classical mechanics, this equation describes the motion of things such as a stretched spring or a pendulum. But this example’s greatest importance comes as a model in quantum mechanics and quantum field theory.
I would like to mention one more example now, namely when and . This potential yields the famed inverse square laws of gravity and electrostatics. The solutions to this system are conic sections, but I would like to delay the derivation of this fact until the Lagrangian formulation.
I have yet to mention anything about energy or momentum. Momentum of a particle is defined as . Note that if the potential is constant, then momentum is independent of time. Indeed, . This is the law of Conservation of momentum. Kinetic energy, or energy of motion, is defined by the formula . The total energy of the particle is defined to be . Note that this gives the law of Conservation of energy: . Throughout all of this, we assumed that the potential depended only on position. Of course, we could also allow an explicit time dependence in the potential. In this case, energy is not conserved. What we have seen in both of the cases of energy and momentum is that conservation of a quantity is implied by some symmetry. These are instances of something called Noether’s Theorem, which I’ll talk about when we get to Lagrangian mechanics.
That’s it for Newtonian mechanics. It’s rather dry…we’re only solving a second order ODE, and we have no intuition as to where it came from or why it should work. To get more intuition into the subject, we will need to explore the geometry of space more thoroughly, which will lead to the development of the Hamiltonian formalism. That’s it for now. See you soon.