Introduction-Classical and Relativistic Mechanics-Lecture Handout, Exercises of Classical and Relativistic Mechanics

Download Introduction-Classical and Relativistic Mechanics-Lecture Handout and more Exercises Classical and Relativistic Mechanics in PDF only on Docsity! 1 Introduction There are two approaches to classical mechanics: • The Lagrangian approach: treats position and velocity as fundamental and describes how they change in time if you are given the Lagrangian, a function of position and velocity. • The Hamiltonian approach: treats position and momentum as fundamental and describes how they change in time if you are given the Hamiltonian, a function of position and momentum. What is it? It is the energy (e.g. kinetic energy + potential energy). The Hamiltonian approach, which is what we are going to describe, leads to a lot of interesting mathematics. It turns out that giving the position and momentum of a particle specifies a point in a space called the ‘phase space’ or ‘state space’ of the particle. Mathematically this is often a ‘cotangent bundle’ — an important concept from the theory of manifolds. A cotangent bundle is an example of a ‘symplectic manifold’, which in turn is an example of a more general thing called a ‘Poisson manifold’. These are the things we want to understand. But we will work our way up to these abstractions starting from the basics! In our historical overview, I led up through the history of physics to Newton’s great Principia and his three laws. Now we will actually start doing some physics à la Newton. We are going to start out by thinking about a classical particle moving in n-dimensional space. Think n = 3 if you like — but other dimensions are interesting too! 2 A classical particle in Rn In classical mechanics a particle traces out a path in some space, say Rn: q:R→ Rn Here R stands for ‘time’ and Rn stands for ‘space’: the particle’s position in space is a function of time. We say that q(t) is the position of the particle at time t. We define the velocity v(t) = q̇(t) = dq(t) dt and acceleration a(t) = q̈(t) = dv(t) dt . Any particle has a mass m > 0, and Newton’s 2nd law says F = ma, i.e. F (t) = mq̈(t) for all t ∈ R, where F :R→ Rn is called the force. We will assume that q, F and indeed all functions we discuss are smooth or C∞ — meaning they have infinitely many continuous derivatives. Then, with luck, we can solve this second order differential equation, namely Newton’s second law: q̈(t) = F (t) m 1 docsity.com for q(t) if we know F (t), m, q(t0), and also its time derivative q̇(t0) for some t0 (often called “time zero”). Examples: 1. A free particle. If F (t) = 0, we have a free particle. Then q̈(t) = 0 so q̇(t) is constant, say v ∈ Rn, and q(t) = q(0) + tv = q(0) + tq̇(0). We recover Newton’s first law: a free particle moves along a line in Rn at constant velocity. 2. A particle near the Earth’s surface feeling only the force of gravity. This force is approximately independent of time and position: F (t) = (0, 0,−mg). Here we are in R3 and g is the downwards acceleration due to gravity - approximately 9.8 meters/second2. Homework 1: Solve Newton’s second law F = ma for q(t) ∈ R3 for this F (t) - find q(t) in terms of the initial position q(0) and q̇(0). Hint: the path it traces out is a parabola! 3. The simple harmonic oscillator. Here n = 1: picture of mass m on spring at equlibrium and picture of stretched spring with height difference q(t) caption: The equilibrium position has q = 0. caption: The position q(t) is measured relative to the equilibrium position. The force is approximately given by Hooke’s Law: F (t) = − kq(t), where k > 0 is the spring constant. There is a famous fact that physicists often quote, which is that ‘to first order everything is linear’. This is tautological, of course, but it really says that most functions are differentiable and we can approximate by the first derivative. So, it’s not so absurd to think that approximately, at least, the force will depend in a linear way on position in many situations. That’s why the harmonic oscillator is so important. Homework 2: Solve Newton’s second law for q(t) ∈ R (when the force is given by Hooke’s law) in terms of m, k, q(0), q̇(0) and find the period P of the oscillation and the frequency ω, where ω = 2π P Hint: It oscillates! 3 Momentum and Energy To go further, let’s study two ways to integrate the force F :R→ Rn: ∫ t2 t1 F (t)dt ∈ Rn ∫ t2 t1 F (t) · q̇(t)dt ∈ R 2 docsity.com