Lagrangian Dynamics

PHYS20402 Lagrangian Dynamics 10 Semester 2 Undergraduate Level 2 Sean Freeman Unit coordinator Department of Physics & Astronomy 1 FHEQ level (Framework for Higher Education Qualifications) ' Middle part of Bachelors ' European Credit Transfer & Accumulation System Rating : 5.0 Lagrangian Dynamics Lagrangian Dynamics To introduce the Lagrangian and Hamiltonian formulations of classical mechanics. To develop the knowledge and skills required to solve a variety of dynamical problems involving more than one degree of freedom. On completion successful students will be able to:1. Choose an appropriate set of generalised coordinates to describe a dynamical system and obtain its Lagrangian in terms of those coordinates and the associated 'velocities'. Derive and solve the corresponding equations of motion. Treat small oscillations as an eigenvalue problem.2. Apply a variational principle to solve simple problems involving constraints.3. Appreciate symmetries and how they manifest themselves in terms of constants of the motion.4. Obtain generalised momenta and thus the Hamiltonian of a dynamical system. Derive and solve the equations of motion in Hamiltonian form.   1.  IntroductionReview of Newtonian mechanics:  internal forces, external forces, forces of constraint.  Rotational problems and polar coordinates. Conservation laws and conservative systems. Partial derivatives. 2.  Lagrangian DynamicsThe energy method plus other conservation laws. The Lagrangian and Lagrange’s equation. Small oscillations and normal modes. 3.  Calculus of VariationsFunctional minimization. The Euler-Lagrange equations. Constrained variation. Hamilton’s principle of least action. Lagrangian dynamics. 4.  The Hamiltonian FormalismLegendre transformations. Generalized momenta, the Hamiltonian and Hamilton's equations. Phase space.  Liouville’s theorem 5.  Symmetries and Conservation LawsGenerators of transformations. Poisson brackets. Symmetries of the Lagrangian produce constants of motion. Noether’s theorem. 6.  Normal Modes from MatricesNormal modes from symmetries. Review of mathematics of matrices:  eigenvalues and eigenvectors. Diagonalizing a matrix using its eigenvectors. Small oscillations as eigenvalue problems. 7.  Special TopicsLagrangian for charged particle moving in electric and magnetic fields. Continuous systems: the Lagrangian Density. <h4 class="ui header" style="-webkit-text-stroke-width:0px;border-style:none;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);font-family:"Segoe UI", Lato, "Helvetica Neue", Arial, Helvetica, sans-serif;font-size:1.1rem !important;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;letter-spacing:normal;line-height:1.28571em;margin:-0.142857em 0px 1rem;orphans:2;padding:0px;scrollbar-color:auto;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">CUIP Teaching Methods</h4><div class="ui grid" style="-webkit-box-align:stretch;-webkit-box-direction:normal;-webkit-box-orient:horizontal;-webkit-text-stroke-width:0px;align-items:stretch;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);display:flex;flex-flow:wrap;font-family:"Segoe UI", Lato, "Helvetica Neue", Arial, Helvetica, sans-serif;font-size:14px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;margin:-1rem;orphans:2;padding:0px;scrollbar-color:auto;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;"><div class="five wide column" style="box-sizing:inherit;display:inline-block;padding:1rem;position:relative;vertical-align:top;width:333.854px;"><figure class="table" style="width:305.854px;"><table class="ui table" style="background-color:rgb(255, 255, 255);border-collapse:separate;border-radius:0.285714rem;border-spacing:0px;border:1px solid rgba(34, 36, 38, 0.15);box-shadow:none;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);font-size:1em;margin:0px;scrollbar-color:auto;text-align:left;vertical-align:middle;" id="teaching_table"><thead class="mobile hidden" style="box-shadow:none;box-sizing:inherit;text-align:inherit;vertical-align:inherit;"><tr style="box-sizing:inherit;"><th style="background-color:rgb(249, 250, 251);border-bottom:1px solid rgba(34, 36, 38, 0.1);border-left-style:none;border-radius:0.285714rem 0px 0px;border-right-color:!important;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);cursor:auto;padding:0.928571em 0.785714em;text-align:inherit;text-transform:none;transition:background 0.1s, color 0.1s;vertical-align:inherit;">Activity</th><th class="right aligned" style="background-color:rgb(249, 250, 251);border-bottom:1px solid rgba(34, 36, 38, 0.1);border-left-style:none;border-radius:0px 0.285714rem 0px 0px;box-sizing:inherit;color:rgba(0, 0, 0, 0.87);cursor:auto;padding:0.928571em 0.785714em;text-align:right;text-transform:none;transition:background 0.1s, color 0.1s;vertical-align:inherit;">Hours</th></tr></thead><tbody style="box-sizing:inherit;counter-reset:numdivcounter 0;text-align:inherit;vertical-align:inherit;"><tr style="box-sizing:inherit;"><td style="border-right-color:!important;border-top-style:none;box-sizing:inherit;padding:0.785714em;text-align:inherit;transition:background 0.1s, color 0.1s;">Lectures</td><td class="right aligned" style="border-top-style:none;box-sizing:inherit;padding:0.785714em;text-align:right;transition:background 0.1s, color 0.1s;">22</td></tr></tbody></table></figure></div></div> 1 Written exam 100% Model answers will be issued within one week of issuing each example sheet. Informal Q&A sessions will be organised to allow students to clarify any questions on the lecture material or on the model answers. PHYS10101 Dynamics Pre-Requisite Compulsory PHYS10302 Vibrations & Waves Pre-Requisite Compulsory PHYS10372 Mathematics 2 Pre-Requisite Compulsory <div class="ewa-rteLine" style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:"Aptos Narrow";font-size:16px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:start;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:pre-wrap;widows:2;word-spacing:0px;">Pre-requisites:</div><ul><li>Vibrations & Waves PHYS10302;</li><li>Mathematics 2 PHYS10372 or MATH24420 PDEs and vector calculus</li></ul>Anti-requisites:<ul><li>PHYS20401 (25/26 only)</li></ul> N Modern Classical Mechanics T. M. Helliwell,V. V. Sahakian, Cambridge University Press<o:p></o:p>Other Textbooks at a similar level to this course:<o:p></o:p>Classical Mechanics Kibble, T. W. B., Berkshire, F. H.<o:p></o:p>Classical Mechanics Taylor, John R., Sausalito<o:p></o:p>Textbooks that cover same material, but go further and are more formal, theoretical or mathematical (or all three!); <o:p></o:p>The Variational Principles of Mechanics Cornelius Lanczos<o:p></o:p>Classical Mechanics  Herbert Goldstein, John L. Safko,  Charles P. Poole<o:p></o:p>Mechanics L.D. Landau and E.M. Lifshitz <o:p></o:p>Supplementary Reading:<o:p></o:p>Classical Mechanics : the Theoretical Minimum Leonard Susskind and George Hrabovsky<o:p></o:p>The Lazy Universe: an Introduction to the Principle of Least Action Jennifer Coopersmith<o:p></o:p> Assessment written exam 1.5 Lectures 22 0 76.5