Classical Mechanics/Introduction

Overview: What is classical mechanics

As a prelude to this course, let me describe what classical mechanics is about.

Classical mechanics is a part of physics that deals with the motion of point masses (very small things) and rigid bodies (large things that can rotate as a whole but cannot change their shape). This is very useful in practice, since many objects in real life can be approximately considered to be either point masses or rigid bodies in most situations.

Typical problems solved in classical mechanics are:

To find the trajectory of a stone thrown into the air with known initial velocity. (The stone is considered to be a point mass.)
To predict the motion of a spacecraft approaching some planet, if its initial position and velocity far from the planet are known. (The spacecraft is considered to be a point mass.)
To find out how many revolutions per minute a disc will be executing if we know the strength of the engine driving the rotation. (The disc is considered to be a rigid body rotating as a whole.)
To find how much energy and how much time is needed to accelerate a small object to a given speed. (Point mass.)
To find the frequency of oscillations in a system of point masses connected by springs.

Of course, one can consider also much more complicated problems than these. For example:

A light spinning top stands at an angle on the surface of a heavy cylinder that can roll along a horizontal plane without sliding. Determine the conditions that would allow the top to avoid falling off the cylinder. (Both the cylinder and the top are considered rigid bodies.)
A spacecraft launched from the Earth needs to reach the surface of Mars within a certain time. Predict the most appropriate time of year for this mission and determine the least amount of rocket fuel needed. (The spacecraft is considered a point mass moving in the gravitational field of the Sun, the Earth, and Mars.)

Mathematical methods used

Classical mechanics uses ordinary differential equations (ODEs) to describe the properties of bodies mathematically. Thus, coordinates, angles, etc. are numbers that depend on time, e.g. $x(t),y(t),z(t),\phi (t),\theta (t)$ , and satisfy certain (systems of) ODEs.

One can use several mathematical methods to solve such equations. In some cases, solutions can be found exactly, for instance: ${\ddot {x}}(t)+x(t)=0$ has the general solution $A\sin(t)+B\cos(t)$ . In other cases, solutions are found only in terms of integrals that one cannot evaluate in closed form. Sometimes, one can solve the equation approximately using some method such as perturbation theory. Finally, any ODE can be solved numerically (using a computer program) up to a certain precision.

Students of mechanics are expected to learn methods of solving certain standard differential equations that are exactly solvable, for instance: multidimensional harmonic oscillators, motion in 1-dimensional force field, motion in 3-dimensional central force field. Numerical methods for solving ODEs are important in practice but are usually not studied as part of classical mechanics because these methods are not specific to mechanics but are equally applicable to every differential equation. Numerical methods for solving various equations are best studied in a dedicated course that involves hands-on computer programming.

Newtonian mechanics

The first successful theory of classical mechanics is contained in Newton's three laws of mechanics that govern the motion of point masses:

There exist reference frames where a point mass not interacting with other bodies will move with constant speed in the same direction. (If this is not true in some reference frame, then that reference frame is not inertial. Further laws are formulated in inertial frames.)
A point mass interacting with other bodies moves with the acceleration ${\vec {a}}$ found from ${\vec {F}}=m{\vec {a}}$ , where ${\vec {F}}$ is the sum of all forces acting on the body, $m$ is the mass of the body, and ${\vec {a}}$ is the acceleration, i.e. the second derivative of the position vector ${\vec {r}}$ with respect to the time.
All forces are caused by other point masses, and whenever a point mass 1 exerts a force ${\vec {F}}$ on a point mass 2, the point mass 2 also exerts the force $-{\vec {F}}$ on the point mass 1.

The motion of all point masses is described by differential equations which can be solved directly as long as all relevant forces could be predicted or measured. I assume that you are already familiar with these laws and with typical situations where they apply (e.g. motion of bodies thrown at an angle near Earth) before you start studying theoretical mechanics.

In Newtonian mechanics, a rigid body is simply a collection of point masses connected by "rigid sticks." These sticks are "rigid" because they always produce exactly such forces as to keep constant distances between all points, regardless of any other forces or motions. Thus the motion of rigid bodies can be described without introducing any other special rules. One derives the concept of angular momentum, torque, etc., from Newton's laws without any additional postulates.

From Newtonian mechanics to "theoretical" mechanics

The necessity to consider point masses is certainly inconvenient if one needs to describe liquids and gases, so a special branch of mechanics with its own formalism was developed for that purpose, namely continuum mechanics (mechanics of continuous media). The formalism of continuum mechanics is generalized to field theory where the basic object is not a point mass but a field, i.e. some abstract "substance" that is present at all points in space and shows its influence at every point at once. (Examples are: gravitational field, electric field, and magnetic field.) Such substances may be described by a function of space and time, for example the vector field ${\vec {E}}(t,{\vec {r}})$ describes the electric field. The behavior of fields is usually governed by partial differential equations; for example, the electric field ${\vec {E}}(t,{\vec {x}})$ and the magnetic field ${\vec {B}}(t,{\vec {x}})$ satisfy Maxwell's equations.

As more and more complicated problems needed to be solved, various mathematical tools were developed to simplify and to generalize the mathematical description of mechanics. Finally, the Lagrangian and the Hamiltonian formulations of mechanics were discovered. These two formulations still remain the cornerstones of classical mechanics and field theory, as well as Einstein's theory of relativity, and thus indirectly of all modern theoretical physics. These formulations of mechanics are not based on the assumption of "forces" and are equally applicable to point masses, rigid bodies, fields, and continuous media. The main subject of theoretical mechanics (sometimes also called "analytical mechanics") is the study of these more refined and more general mathematical formulations of classical mechanics.

Overview: what is this "minimal standard course"

Much of theoretical physics is based on concepts from theoretical mechanics, such as the variation of the action, symmetry transformations, or the Hamiltonian. The goal of this course is to introduce the material that you absolutely need to learn if you would like to have a solid foundation for the study of theoretical physics.

In this text, I do not write the words "definition", "theorem", or "proof" like one does in mathematical texts. However, the same structure is present. I show new concepts in boldface within the sentence where they are defined.

Prerequisites

You can solve simple algebraic equations, such as $x-{\frac {2}{x}}-1=0$ .
You can solve simple differential equations such as ${\ddot {x}}(t)+2x(t)=3$ with initial conditions such as $x(0)=1,{\dot {x}}(0)=2$ .
You can manipulate three-dimensional vectors, compute vector sums, scalar products, vector (cross) products, projections on axes, angles between lines.
You are familiar with basic linear algebra (matrices, matrix multiplication, eigenvectors, diagonalization of matrices).
You can compute (or quickly look up) elementary integrals such as $\int \limits _{3}^{4}{\sqrt {5x^{2}+8}}~dx$ .
You are familiar with multivariate calculus, can compute partial derivatives, for example $\partial _{y}{\sqrt {x^{2}+y^{2}}}$ .
You are familiar with (European) school-level mechanics, including Newton's laws, forces in static configurations, motion in straight lines and in circles, and basic ideas about rotation of rigid bodies and torque.

Core material

The minimal standard course in theoretical mechanics consists of:

Initial study of the Lagrangian and the Hamiltonian formalisms.
- A general formulation of mechanics, applicable to all systems, can be achieved using a variation principle (also known as the action principle). All information about a particular mechanical system is encapsulated by its Lagrangian, which is a function $L(q_{i},{\dot {q}}_{i},t)$ . We derive the equations of motion in the general case, for an arbitrary mechanical system.
- You already know how simple mechanical problems can be solved by elementary methods using forces, torques, etc., and now you will see how the same problems are reformulated using Lagrangians. You will see that is rather straightforward to find the Lagrangian for any particular system. Then it becomes apparent that many problems are much easier to solve using these formulations.
- You need to gain some experience formulating various mechanical problems in terms of Lagrangians. Basically, you should be able to deal with any practical situation involving point masses, springs, sticks, ropes, frictionless rails, pendulums, balls rolling along cylinders rolling along planes, and so on. You need to learn how to best choose the coordinates or how to discover conservation laws in such systems. This comes only with experience solving practice problems.
- Rotation of rigid bodies presents extra difficulties because of the complicated geometry. You need to learn some methods for dealing with these situations efficiently (tensor of inertia, Euler angles, rotating frames of reference).
- The Hamiltonian formulation of mechanics is derived from the Lagrangian formulation. (For solving practical problems, the Hamiltonian formalism is less useful than the Lagrangian one, although it is somewhat more elegant mathematically. However, the Hamiltonian formalism is of such extraordinary importance in theoretical physics that it is usually studied early on, within the course of theoretical mechanics.)
- Special relativity is usually introduced at this point using the formalism of four-dimensional vectors. In the Lagrangian formulation, this is a perfectly ordinary mechanical theory with a somewhat unusual Lagrangian. You need to develop some physical intuition for the various relativistic effects and also become familiar with the four-dimensional description of the world as a spacetime. Without special relativity and the four-dimensional view of the world, the road to much of theoretical physics is essentially closed.
Study of the mathematical methods needed to actually solve practical problems.
- You need to learn how to solve various differential equations and systems of such equations. The typical equation is that of the harmonic oscillator with a driving force, ${\ddot {x}}+\omega ^{2}x=f(t)$ . Typically, you need to be able to find general solutions as well as particular solutions for given initial conditions. At this point we study only equations that can be solved exactly.
- You need to gain experience analyzing the behavior of solutions of various types of equations: linear systems, oscillations and resonance, and some simple kinds of solvable nonlinear equations such as the Kepler problem. These are very old problems that occur time and again in physics, so you should become very familiar with the qualitative features of their solutions.
- You need to learn some basic ideas of the calculus of variations. In particular, you need to understand the concept of a functional, and to learn to compute functional derivatives (i.e. "variations" of functionals). Otherwise you cannot really understand the use of the variational principle, which is very important in all of theoretical physics.

You usually need to solve several practice problems for each mathematical method, so that you can see how these methods work and gain experience.

Extra material

The above topics are considered standard because one cannot continue studying theoretical physics without having mastered them. There are also more advanced topics that build upon the standard ones and lead to other areas of physics:

Methods of perturbation theory, nonlinear oscillations, parametric resonance, adiabatic invariants (useful for celestial mechanics, chaos, and other things).
Use of quaternions instead of Euler angles to describe the position of rigid bodies (useful for numerical simulations of missile motion).
Elements of scattering theory: elastic/inelastic scattering; cross-sections in central potentials (preparation for particle physics).
A general definition of "symmetry" and the derivation of conservation laws (Noether's theorem). This is essentially a preparation for gauge field theory, which is the basis for particle physics.
Symmetries and oscillations in multi-dimensional systems (e.g. oscillations of molecules).
The action-angle variables and questions of integrability (preparation for chaos theory).
Canonical transformations, Hamilton-Jacobi equation -- tools to discover integrable systems. (Needed mostly for mathematical physics.)
Integral invariants, symplectic structures (tools to discover conservation laws and to explain the structure of the theory more deeply; useful in field theory).
Dealing with constrained degenerate systems (preparation for gauge theories).
Elements of continuous mechanics; continuity equations, stress-energy tensor, Euler equation. (Useful in hydrodynamics, electrodynamics of media, kinetics, etc.)

Some of these topics are usually included in a theoretical mechanics course at the lecturer's preference. History of theoretical mechanics is usually not studied; rather, students learn the contemporary, very much streamlined and simplified formulation of mechanics.

Suggested books to study

There is an extraordinarily large number of textbooks in theoretical mechanics, because it is a fairly old and well-studied subject. You need any textbook on classical mechanics that you can understand and that talks about "Lagrangians" early on. (Books that only talk about accelerations, forces, and torques may be quite advanced but they do not cover the subject of theoretical mechanics.)

H. Goldstein. Classical mechanics. - Has everything standard in it and quite a few advanced topics. An old classic.
L.N. Hand, J.D. Finch. Analytical mechanics (Cambridge, 1998). - A fresher, more didactic exposition of mechanics. Standard material.
V.I. Arnold. Mathematical methods of classical mechanics. -- Has almost nothing standard in it but is excellent for a more mathematically minded student.

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