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Classical Mechanics (MP350) MP350 Classical Mechanics Jon-Ivar Skullerud with modifications by Brian Dolan December 11, 2020 1 Contents 1 Introduction 5 1.1 Physics is where the action is . .5 1.2 Overview . .6 2 Lagrangian mechanics 8 2.1 From Newton II to the Lagrangian . .8 2.2 The principle of least action . .8 2.2.1 Hamilton's principle . 11 2.3 The Euler{Lagrange equations . 12 2.4 Generalised coordinates . 15 2.4.1 The shortest path between two points (optional) . 20 2.4.2 Polar and spherical coordinates . 21 2.5 Lagrange multipliers [Optional] ........................ 24 2.5.1 Constraints . 24 2.6 Canonical momenta and conservation laws . 29 2.6.1 Angular momentum . 30 2.7 Energy conservation: the hamiltonian . 32 2.7.1 When is H conserved? . 33 2.7.2 The Energy and H .......................... 34 2.8 Lagrangian mechanics | summary sheet . 37 3 Hamiltonian dynamics 39 3.1 Hamilton's equations of motion . 40 3.2 Cyclic coordinates and effective potential . 42 3.3 Hamilton's equations from a variational principle . 44 3.4 Phase space [Optional] ............................ 45 3.5 Liouville's theorem [Optional] ......................... 48 2 3.6 Poisson brackets . 50 3.6.1 Properties of Poisson brackets . 50 3.6.2 Poisson brackets and conservation laws . 52 3.6.3 The Jacobi identity and Poisson's theorem . 53 3.7 Noethers theorem . 54 3.8 Hamiltonian dynamics | summary sheet . 57 4 Central forces 59 4.1 One-body reduction, reduced mass . 59 4.2 Angular momentum and Kepler's second law . 61 4.3 Effective potential and classification of orbits . 64 4.4 Integrating the energy equation . 64 4.5 The inverse square force, Kepler's first law . 66 4.5.1 The shapes of the orbits . 68 4.6 More on conic sections . 69 4.6.1 Ellipse . 70 4.6.2 Parabola . 72 4.6.3 Hyperbola . 72 4.7 Kepler's third law . 74 4.8 Kepler's equations . 76 4.9 Runge-Lenz vector . 77 4.10 Central forces | summary sheet . 78 5 Rotational motion 80 5.1 How many degrees of freedom do we have? . 81 5.1.1 Relative motion as rotation . 82 5.2 Rotated coordinate systems and rotation matrices . 82 5.2.1 Active and passive transformations . 84 5.2.2 Elementary rotation matrices . 84 5.2.3 General properties of rotation matrices . 84 5.2.4 The rotation group [optional] ..................... 87 5.3 Euler angles . 88 5.3.1 Rotation matrix for Euler angles . 89 5.3.2 Euler angles and angular velocity . 90 5.4 The inertia tensor . 91 3 5.4.1 Rotational kinetic energy . 91 5.4.2 What is a tensor? Scalars, vectors and tensors. 96 5.4.3 Angular momentum and the inertia tensor . 98 5.5 Principal axes of inertia . 98 5.5.1 Rotations and the inertia tensor . 98 5.5.2 Comments . 101 5.6 Equations of motion . 102 5.6.1 The symmetric heavy top . 102 5.6.2 Euler's equations for rigid bodies . 104 5.6.3 Stability of rigid-body rotations . 105 4 Chapter 1 Introduction 1.1 Physics is where the action is In these lectures we shall develop a very powerful (and beautiful) way of formulating Newtonian mechanics. The basic idea is to derive Newton's equations from a variational principle, meaning that for a given dynamical system we look for a function of the dynamical variables and velocities such that the time evolution of the system is obtained by minimising this function. The function is called the action for the system. This gives an extremely concise and elegant way of describing the dynamics: for systems with many degrees of freedom and/or many particles we do not need to write down a mess of complicated coupled differential equations to define the dynamics | we just write down a single function, the action. In principle we can write all the laws of physics on the back of a postage stamp. In order to solve the dynamics though we need to pick it apart, and that requires deriving dynamical equations from the action and solving them, which can still be quite complicated. But the simplicity and elegance of the variational formulation often points to a choice of variables that makes the solution easier. Moreover the action principle is the springboard to new physics. The methods introduced in this course can easily be extended to both special and general relativity and were instrumental in the development of quantum mechanics at the beginning of the 20th century. Indeed today the action is fundamental to our current understanding of the Standard Model of particle physics and relativistic quantum field theory, it is the principal tool used to study matter at the most fundamental level. We shall not sail into such exotic waters in this course though, we shall leave that to later modules. The focus here will remain on Newtonian mechanics, but there is a shift in emphasis, from Newtonian forces and acceleration to the more general and abstract formulations that were developed in the late 18th and the 19th century, associated with names like Euler, Lagrange, Hamilton and Jacobi. Therefore, this course is not more of the stuff you have already studied in modules like MP110, MP112 and MP205, but instead represents a completely new way of looking at mechanics, and one which forms the foundation of nearly all modern mathematical physics. The focus in this course is on methods and formulations rather than on answers or numbers. In part, this is because the key to solving complicated problems is very often 5 to formulate them properly and to select appropriate methods. However, there are other reasons for this shift in focus: • Often, we are not that interested in numerical solutions, but more in the qualitative features of a system, and we can find out a lot about this without doing any numerical calculations. • We will see that wildly different physical systems can look identical from a math- ematical point of view, so solving one can immediately give us the solution to the other. Starting with numerical calculations can obscure this. • Symmetries will play an extremely important role, and we will learn to identify and exploit symmetries to simplify and understand mechanical systems. Putting in numbers at the start will often hide the symmetries. The Lagrange{Hamilton formalism and the symmetry principles which we will become acquainted with here, are used all throughout modern physics: • quantum mechanics; • statistical mechanics; • condensed matter theory (quantum statistical mechanics) • classical field theory (electromagnetism, general relativity) • particle physics (quantum field theory and symmetry groups) • chaos theory • etc 1.2 Overview The module will cover the following topics: • The principle of least action (Hamilton's principle), the lagrangian and the Euler{ Lagrange equations. • Generalised coordinates (how to formulate a mechanical problem in the most sen- sible way given symmetries and constraints). • Canonical momenta and conservation laws; energy conservation. • Hamilton's equations of motion. • Poisson brackets. • Central force motion, angular momentum conservation. • Planetary motion, Kepler's laws. 6 • Rotations and rotation matrices. • Inertia tensor, principal axes of inertia. • Euler's equations of (rotational) motion. Learning outcomes At the end of this course, you should be able to: • formulate the basic principles of the Lagrange{Hamilton formalism; • use these principles to derive equations of motion for dynamical systems; • explain the relation between symmetries and conservation laws; • apply conservation laws to analyse the motion of dynamical systems; and • describe the mathematical properties of rotations and systems with rotational symmetry. 7 Chapter 2 Lagrangian mechanics 2.1 From Newton II to the Lagrangian In the coming sections we will introduce both the notion of a Lagrangian as well as the principle of least action. This will be an equivalent, but much more powerful, formulation of Newtonian mechanics than what can be achieved starting from Newton's second law. However, to introduce this new way of thinking, we will in this section give a short argument why the Lagrangian is a \natural" object to study. Consider now a single particle at position x in a potential V (x; t). The kinetic energy 1 2 of this particle is T = 2 mx_ . The equation of motion for this particle is @ mx¨ = − V (x; t): (2.1) @x What we ultimately seek, is a way to generate this equation of motion from a simpler d @ object. Playing around with this equation we note that we can write mx¨ = dt @x_ T . We may thus rewrite (2.1) as d @ @ T (_x) = − V (x; t): dt @x_ @x Note that since T does not depend on x and V does not depend onx _ we can rewrite the equation further as d @ @ − (T − V ) = 0: (2.2) dt @x_ @x This funny looking equation will be the starting point for this course. The difference d @ @ L = T − V we will call the Lagrangian, and the differential operator dt @x_ − @x will be obtained from the principle of least action. We will find that (2.2) is more general than meets the eye. Especially, it will look the same irrespective of the coordinate system that we are working in. The same thing can not be said for Newton II, which becomes much more complicated when the coordinate system is not the Cartesian one. 2.2 The principle of least action The starting point for the reformulation of classical mechanics is the principle of least action, which may be somewhat flippantly paraphrased as \The world is lazy", or in 8 the more flowery words of Pierre Louis Maupertuis (1744), Nature is thrifty in all its actions: The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena.
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