Theoretical Mechanics - what is where in video-lectures ==================================================== Lecture 1 0:00:00 Organizational information 0:08:50 What is it good for (motivation for the subject) 0:21:15 Constraints (introduction) 0:51:10 Holonomic and non-holonomic constraints (brief account) 0:55:10 3n-dimensional formalism (bar vectors) 1:12:30 Virtual displacements - concept 1:21:15 Mathematical box: 1-st order Taylor expansion via gradient ............................................................................................................ Lecture 2 0:02:25 Configuration space 0:04:00 Dimension of a space, number of degrees of freedom of the system 0:08:15 Plane and spherical pendulum 0:20:15 Virtual displacements - how to find them 1. 0:30:45 D'Alembert-Lagrange principle - derivation 0:51:50 The principle of virtual work 1:02:30 Chapter 0 - operations with vectors 1 (indices, Kronecker, Levi-Civita, ...) ............................................................................................................ Lecture 3 0:02:20 Lagrange's equations - beginning of the derivation (from the D'Al-Lagr. principle) 0:04:00 Parameterization of the configuration space 0:15:50 Generalized coordinates 0:17:05 Virtual displacements - how to find them 2. 0:31:45 Mathematical box: Non-degeneracy of the scalar product 0:44:10 Generalized forces 0:54:10 Generalized velocities 0:55:10 Kinetic energy - structure 1:15:30 Concrete examples (calculation of kinetic energy) ............................................................................................................ Lecture 4 0:17:00 Lagrange's equations for general forces 0:17:30 Rewriting it for potential forces 0:33:30 Lagrange function (Lagrangian) 0:35:20 Lagrange's equations (final form) 0:46:35 Example - plane pendulum 1:02:30 Chapter 0 - operations with vectors 2 (Davis-Cup, curl, grad, div, ...) ............................................................................................................ Lecture 5 0:00:00 Advantages of Lagrange's equations (in addition to constraints) 0:02:50 General covariance (arbitrary coordinates) 0:07:20 Example - free motion in a plane 0:24:45 The need for general covariance in physics 0:26:15 Conservation laws 0:28:20 Cyclic coordinates and conservation laws due to them 0:44:20 Generalized (= canonical) momenta 0:47:20 What are conservation laws good for? 0:57:10 Conservation of energy due to cyclic t (isolated system) ............................................................................................................ Lecture 6 0:05:25 Generalized potential energy 0:21:50 How it looks like for the Lorentz force 0:45:50 About order on the blackboard and the concept of good teaching 0:47:55 Interaction of degrees of freedom 0:50:50 On rubber stamp for Lagrange's equations 1:13:25 Ansatz - a method of solving very complicated differential equations ............................................................................................................ Lecture 7 0:00:20 Chapter 0 - (ordinary) potential energy 0:35:55 The principle of extremal action - introduction 0:46:20 Mathematical box: Calculus of variations (beginning) 0:47:25 Example (history) - brachistochrone 1:04:30 Shortest path connecting two points 1:07:55 Catenary 1:15:00 Functionals and functions regarded as grinders ............................................................................................................ Lecture 8 0:00:00 Mathematical box: Variational calculus (completion) 0:19:20 Euler's equation (from the calculus of variations) 0:20:50 Euler's equations for several variables 0:29:05 Ending of the mathematical box (on calculus of variations) 0:31:20 Dictionary on Euler - Lagrange stuff 0:37:00 Formulation of the principle of extremal action 0:45:10 Action and Noether's Theorem (idea) 0:56:30 Action and higher physics (quantum theory and relativity) 1:19:15 Action and freedom in the Lagrangian 1:25:55 Order reduction in variational equations ............................................................................................................ Lecture 9 0:00:00 Still to the last lecture - standard notations on action stuff 0:05:15 Hamilton's equations - introduction 0:16:00 Mathematical box: Legendre transformation in general 0:41:00 Using the results of the mathematical box 0:49:35 The resulting form of Hamilton's equations 0:53:05 Example - plane pendulum 1:03:00 Trick - Hamiltonian from Lagrangian using inverse matrix 1:12:00 Example - spherical pendulum 1:25:40 An example of a bad ansatz ............................................................................................................ Lecture 10 0:03:30 Phase flow - consequence of the 1st order of Hamilton's equations 0:13:40 Phase space 0:17:10 Liouville's theorem 0:24:00 Momenta versus velocities - discussion 0:32:15 Poisson brackets 0:48:48 Hamiltonian variables and quantum mechanics 1:06:10 Digression - Legendre transformation in thermodynamics 1:20:25 Phase portrait for harmonic oscillator 1 ............................................................................................................ Lecture 11 0:03:30 Canonical and "mechanical" momentum (still a Hamiltonian topic) 0:14:15 Phase portrait for harmonic oscillator 2 0:27:30 Phase portrait for a plane pendulum 0:33:40 Transition to dimensionless variables 1:15:15 Exact solution of a plane pendulum (not just small oscillations) ............................................................................................................ Lecture 12 0:00:00 Scaling - via equation of motion, free fall 0:12:35 Scaling - via equation of motion, a planet around the Sun 0:22:50 Scaling - via Lagrangian, free fall 0:27:00 Scaling - via Lagrangian, a planet around the Sun 0:30:50 Scaling - including parameters 0:42:42 Using "in practical world" 0:48:45 Summary of the whole week's ideas 0:50:20 Two-body problem - introduction (from next week already :-) 0:53:50 Simplification of U due to homogeneity of space 1:02:25 Simplification of U due to isotropy of space 1:15:40 Transition to center of mass and relative vector 1:22:40 Total and reduced mass, decoupling of degrees of freedom ............................................................................................................ Lecture 13 0:06:20 Problem of a single body in the central field 0:12:10 Central field - what does it mean 0:20:50 Consequences of the centrality of the field on motion in it - conservation of angular momentum 0:24:05 Consequence of conservation of angular momentum - (a.o.) motion in a plane 0:31:00 Planar Lagrangian and relevant conservation laws 0:38:30 Conservation laws as equations (instead of the Lagrange's) 0:44:20 Effective potential energy 0:46:30 Separation of variables and solutions in quadratures 0:54:30 Exclusion of time and equation of trajectory (another quadrature) 1:00:55 Monotonicity of the function phi (t) 1:04:40 Constant areal (sectorial) velocity (Kepler's 2-nd law) 1:11:40 Restrictions on the values of the variable r 1:14:45 Summary of knowledge about the solution for general U(r) 1:21:35 Kepler's problem - introduction - U(r) proportional to 1/r (force 1/r^2 - Newton and Coulomb) ............................................................................................................ Lecture 14 0:01:10 Kepler's problem - analysis of effective potential energy 0:12:20 Corresponding quadrature for the trajectory (r(phi) dependence) 0:22:00 The resulting trajectory equation and what it (supposedly) describes (conic sections) 0:22:00 The role of total energy 0:39:10 Content of the result in original polar coordinates 0:45:55 Content of the result in Cartesian coordinates 0:58:10 Kepler's first two laws 1:01:50 Calculation of semi-major axis and orbital period and Kepler's third law 1:12:10 Comparison with a similar result obtained cheaper (by scaling) ............................................................................................................ Lecture 15 0:00:30 Small oscillations - introduction 0:04:00 Small amount of energy - implications 0:09:45 Expansion of potential energy 0:18:00 Expansion of kinetic energy 0:25:00 Resulting approximate T,U and, consequently, L 0:30:20 Resulting equations of motion in original coordinates 0:33:40 New coordinates (linear combinations of the original ones) 1:02:30 Modes in the new coordinates 1:07:35 Modes rewritten back into the original coordinates 1:11:30 Modes in original coordinates as a group of prisoners 1:14:45 Modes - ansatz directly in the original coordinates ............................................................................................................ Lecture 16 0:15:00 What does plugging the ansatz into the equations of motion give 0:21:20 Secular equation 0:29:50 Algorithm for solving problems for small oscillations 0:35:20 Example - two balls on three springs - modes 0:57:10 Example - three balls on four springs - modes = ? 1:06:30 Example - spherical pendulum 1:10:20 Example - string ............................................................................................................ Lecture 17 0:00:05 Non-inertial frames - short review of the last lecture 0:02:05 Basis rotating together with non-inertial frame 0:08:40 Mathematical box: Differential equation for rotating vector 0:26:30 Equation for rotating basis 0:28:35 Computation of dotted and double-dotted r 0:36:20 Insertion into Newton's equation of motion 0:38:20 Resulting equation - 4 "additional" forces (Coriolis, Euler, Inert, Cetrifug) 0:44:00 Specification for the frame (,,laboratory") fixed in Bratislava 0:55:40 Resulting equation for Bratislava 0:59:00 Perturbation method - idea and application to free fall in Bratislava ............................................................................................................ Lecture 18 0:02:50 Calculation of the kinetic energy of a rigid body 0:23:40 The first occurrence of the inertia tensor 0:28:15 Calculation of the angular momentum of a rigid body 0:34:30 The second occurrence of the inertia tensor 0:35:30 Comparison with translational motion (there is also a - masked - tensor) 0:38:30 Mathematical box - what is a tensor (in our case - bilinear form and linear operator) 0:58:00 Changing the components (matrix) of the tensor when changing the basis 1:08:25 Laboratory and body frames (and so laboratory and body tensor components) 1:16:55 The canonical form of the inertia tensor ("principal" axis of the body) 1:20:50 Moment of inertia ................................................ ................................................ ........ Lecture 19 0:03:30 Possible canonical forms of the inertia tensor 0:04:00 An example of a "canonical form" for which there is no body :-( 0:12:00 Classification of tops 0:17:10 Typical bodies for individual types of tops 0:26:00 Motion of a free spherical top - rotates uniformly around a fixed axis 0:34:05 Euler's equations for the rotational motion of a rigid body - introduction 0:38:10 Euler's angles 0:43:30 Derivation of (the structure of) Euler's kinematic equations 0:57:00 Derivation of Euler's dynamic equations 1:11:40 Brief analysis of the entire system of Euler's equations 1:13:55 Visual motivation (experiment) for certain top motion 1:17:00 Free symmetrical top - Euler's equations 1:23:10 Free symmetrical top - ansatz .......................................................................................................... Lecture 20 0:01:10 Continuum mechanics - common stuff (for both hydrodynamics and elasticity) 0:03:10 Volumetric force as the volume integral of its density 0:09:55 Surface forces and the need for a stress tensor 0:16:20 Cauchy formula (for stress tensor) 0:18:05 Why stress tensor is needed (comparison with inertia tensor) 0:24:25 Stress (traction) vector 0:26:50 Total surface force as surface integral 0:29:40 Mathematical box - mnemonics for Gauss' theorem 0:35:30 Total force (volume plus surface) as volume integral alone 0:38:25 Momentum balance and the general equation of motion of the continuum 0:51:00 Angular momentum balance and stress tensor symmetry 1:12:45 Brief summary 1:15:10 Euler's and Lagrange's approach in describing the continuum 1:20:30 Velocity field (in the Euler approach) .......................................................................................................... Lecture 21 0:03:20 Hydrodynamics - introduction 0:05:00 Acceleration calculation (in Euler kinematics) 0:16:35 Expression of volumetric force (we take gravitational force) 0:16:55 Stress tensor expression for an ideal fluid 0:23:00 Dry water 0:26:10 Euler's equation for the motion of an ideal (= non-viscous) fluid 0:31:30 Continuity equation 0:46:50 Bernoulli's equation 1 (B = constant on the streamline) 1:04:30 Vorticity and vorticity-free flow 1:06:40 Bernoulli's equation 2 (B = constant in bulk) 1:07:20 A few words about vorticity flow 1:15:40 Flow rate through a small hole (Torricelli formula) 1:18:45 Hydrostatics and water pressure in the aquarium .......................................................................................................... Lecture 22 0:01:00 Archimedes' law 0:07:30 Thought experiment as a tool of knowledge 0:09:15 Continuity equation for an incompressible fluid 0:11:35 Laminar flow 0:14:00 Level in a spinning bucket (via ansatz in the Euler equation) 0:37:50 Viscosity - search for necessary correction in the stress tensor 0:39:50 Phenomenological approach in physics 0:41:20 Illustrative model - a board dragged on the surface 0:53:45 The resulting viscous term in the stress tensor 1:00:00 The resulting (= Navier-Stokes) equation 1:01:00 An alarm clock created from a vacuum 1:02:20 Boundary conditions - Navier-Stokes and Euler 1:07:25 A note on the second (bulk) viscosity 1:08:45 Navier-Stokes for an incompressible fluid 1:09:50 Idealized examples and science 1:12:20 Flow in a vertical toilet pipe .......................................................................................................... Lecture 23 0:00:40 Flow in a vertical toilet pipe - continued 0:04:40 Dirichlet problem for v(r) 0:22:15 Summary of the result 0:25:00 Poiseuille's formula - total flow proportional to R^4 0:32:10 R^4 and angina pectoris 0:37:00 Elastic continuum - Lagrangian approach, displacement field 0:43:30 Tuning fork yesterday and today 0:44:30 What is deformation? 0:49:10 Strain tensor (from displacement field) 1:07:15 Strain tensor for translations and rotations 1:12:45 Trace of the deformation tensor = divergence of displacement field 1:18:35 Mathematical box - determinant of matrix I + a bit = 1 + trace of the bit 1:24:00 Volume expansion = the trace (and the divergence) ................................................ ................................................ ........ Lecture 24 0:04:35 Calculation of the strain tensor for (unrealistic) uniform stretching 0:14:50 Poisson effect 0:16:55 Calculation of the strain tensor for simple shear 0:23:30 Hooke's law (tensorial version) 0:36:10 Tensor Cijkl - number of components, symmetry 0:45:55 Homogeneous and isotropic medium 0:49:40 Mathematical box: Homogeneous and isotropic tensors 1:03:50 Assembling Cijkl from delta and epsilon (Lego style) 1:15:20 Homogeneous and isotropic continuum - Lamé coefficients 1:16:40 Hooke's law for homogeneous and isotropic continuum 1:19:45 Equation of motion of elasticity - acceleration and divergence of sigma ................................................ ................................................ ........ Lecture 25 0:05:10 Equation of motion of elastic continuum - result (Lamé equation) 0:08:35 Ansatz in the Lamé equation: plane linearly polarized wave 0:22:50 Necessary calculations of all terms for this ansatz 0:33:00 Analysing the equation created by substituting ansatz - longitudinal and transverse waves 0:45:35 Volume expansion for longitudinal and transverse waves (calculation and intuition) 0:48:30 Velocity of longitudinal and transverse waves (calculation and intuition) 0:54:50 Longitudinal, transverse and "general" waves 0:57:00 Sound waves in an ideal fluid 0:59:00 Linearization of equations for an ideal fluid (Euler and continuity equation) 1:12:20 A road to wave equations 1:16:30 Barotropic case - p = p(rho) 1:19:20 Wave equations for the barotropic case 1:21:45 Where does barotropicity come from: isotherm and adiabata (fast and slow heat exchange)