M2A2: Dynamics I (Classical Mechanics)

Jeroen Lamb

Department of Mathematics, Imperial College, London SW7 2BZ, UK

DISCLAIMER: This note is written to help you prepare for the exam, and only serves as such. You cannot derive any right from this note. In particular, I explicitly reserve the right to ask questions in the exam concerning material discussed in the lectures which is not mentioned in the short description below.

The exam will test you on:

• Theory (ideas presented in the lectures)
• Application of the theory to examples (along the line of problems and coursework)

The problems and coursework provides you with an idea of the kind of questions you may expect on the exam, as do the M2A2 exams of the previous years. However, I will also test your knowledge of the theory that has been discussed in the lectures. To help you preparing for this part, I will summarize the syllabus (as treated in the lectures) below, and provide some``mock questions'', indicative of the level of understanding I expect you to have of the course-notes.

Chapter 1: Variational Calculus

The Euler-Lagrange equation:

• in one independent variable, with fixed and free boundary conditions
  
• with several independent variables, with fixed and free boundary conditions
  
• with several dependent variables, with fixed and free boundary conditions (see also coursework # 1)
  
• for integral of function depending on a function y and the first and second derivatives y' and y''.
• example: brachistochrone, with fixed begin and end points, with variable end point, and varaible end point on a prescribed curve

Mock questions: derive the E-L equations in one of the above situations, reproduce the results for one of the brachistrochrone problems.

Constrained optimalization: Lagrange multipliers

• Lagrange function and Lagrange multipliers for constraint variational problems
• Geometric derivation of Lagrange multipliers for constraint problem to 2d surface in R³
• Examples: geodesics in R² (line), on the 2-sphere S² (great circle), isoperimetric problem, derivation of Helmholtz equation

Mock question: derive that optimizing a Lagrange functional leads to solving a constrained variational problem (for a given example), calculate geodesics on a plane, sphere, cylinder, etc.

Chapter 2: From Newton to Lagrange and Hamilton

Newton's equations: forces

• Three Newton's laws.
• Forces, lamellar forces and potentials, Newton's law of gravitation, Lorentz law of electromagnetic force
• configurations and paths, rate of change of a quantity along a path in configuration space, conserved quantities
• work and kinetic energy: rate of work done on a system, along physical paths
• Examples: work done by lamellar forces, work done by electromagnetic force
• Degrees of freedom and constraints: holonomic constraints

Mock question: derive one of the identities concerning work done on a path (as mentioned above), define what is meant by a holonomic constraint

Generalized coordinates: d'Alembert's hypothesis and Lagrange's equations

• transformation to generalized coordinates (also time-dependent, as used in some of the problems/coursework), transformation of kinetic energy (when quadratic in velocities).
• equations of motion in new coordinates: work done on system in terms of generalized forces, d'Alembert's equations of motion (relating the generalized force to derivatives of the kinetic energy)
• d'Alembert's hypothesis: constrained forces do no work on the system
• d'Alembert's principle
• generalized potential: Lagrange's equations of motion when a generalized force can be derived from a generalized potential function
• Examples of Lagrangian systems: free particle, particle in lamellar field (in cartesean and spherical coordinates), charged particle in an electromagnetic field
• action and Hamiltons' principle: definition of action, Hamilton's principle of extremal action leads to Lagrange's equations of motion

Mock questions: derive Langrange's equation from d'Alemberts equations, find generalized potential for some given force, show that d'Alemberts and Lagrange's equations of motion have the same form in different coordinate systems.

Hamiltonian systems:

• Hamiltonian and total energy
• Conditions under which H=T+V
• Conservation of H, conservation of total energy
• Canonical momentum and the p-q description
• Transformation from Lagrangian to Hamiltonian mechanics via a Legendre transformation
• Derivation of Hamilton's equations of motion, from Lagrange's equations of motion
• examples: lamellar, spherical/polar coordinates, charged particle in electromagnetic field (as example of system with velocity-dependent potential)
• Phase space
• Poisson brackets: time-derivative in terms of Poisson brackets, Hamilton's equations in terms of Poisson brackets

Mock questions: define Poisson brackets, and demonstrate their applications; demonstrate transformation from Lagrangian to Hamiltonian via Legendre transformation

Noether's theorem: conserved quantities and symmetry:

• Conserved quantities generate symmetry transformation; and examples (definition of a symmetry transformation as a transformation that maps a solution of the equation to another solution)
• Symmetry and conservation of momenta: translation symmetry versus linear momentum and rotational symetry versus angular momentum;
• the notion of ``integrability'' of a mechanical system

Mock question: demonstrate the relationship between symmetry and conserved quantities in an application.

Normal modes:

• (Liapunov) stability of an equilibrium solution (check the stability of an equilibrium by checking positive definiteness of the quadratic part of the potential)
• Normal modes: what are they? (families of periodic solutions near a stable equilibrium; one family per degree of freedom)
• Calculation of normal mode in quadratic approximation: examples
• True normal modes: what happens to normal mode solutions if we don't have a quadratic system?

Mock question: calculate (approximate) and discuss normal mode solutions around a stable equilibrium in some application (including: show that an equilibrium is stable ...)

Rigid body dynamics:

• Center of mass coordinates; decoupling of total motion of a rigid body in motion of the center of mass and local motion (rotations around the center of mass)
• Inertia tensor and relation between angular momentum and angular velocity; prinicipal axes and principal moments of inertia
• Equations of motion in body coordinates (Euler's equations of motion; total external torque); precession for an axisymmetric body.
• Equations of motion in fixed coordinates (Eulerian angles: do not learn the expression of in terms of the angles by heart, but be comfortable in using them in an example...)
• Free motion of an axisymmetric rigid body in a fixed coordinate frame (``frisbee'' example: two wobbles per revolution)
• Motion of an axisymmetric top (rotating axisymmetric body feeling a gravitational force); Lagrangian and derivation of the equations of motion; sleeping top; precession and nutation (for as far as discussed in the lectures)

Mock question: derive relation between angular velocity and angular momentum vector; derive relation between time-derivative of angular moment vector and total external torque; etc .