# Theoretical physics in transportation

## Recent changes

• Nov 12, 2017 Lectures restructured. Jupyter notebooks and Hamilton equations added.
• Oct 15, 2017 Content of lectures 1--3 updated.
• Oct 8, 2017 New version of the pages created.

## Lectures

Here you find the short description of each lectures, usually complemented by relevant links.

## Old pages

In the past the course was taught in Mathematica and Maple. From that era, you might find the following links useful: I kept also some Maple-related material.

## Mathematical minimum

The course requires some mathematical preliminaries. In the following file you will find basic exercises which you have to be familiar with, otherwise you will not be able to follow the lectures.

## Syllabus

### Newtonian mechanics

• Newton's laws of motion. Law of force as differential equation of second order.
• Solution of simple equations of motion (uniform/accelerated motion, harmonic oscillator). Role of initial conditions.
• Momentum, angular momentum, kinetic energy, potential energy, force as the gradient of the potential.
• Index notation, Cartesian coordinates, summation convention.

### Lagrangian mechanics

• Generalized coordinates, polar coordinates, spherical coordinates.
• Motivation for Lagrange's formalism: coordinate independence. Example: mathematical pendulum.
• Coordinate transformations.
• Definition of generalized force, kinetic energy in generalized coordinates, potential energy in generalized coordinates
• Lagrange's equations and Lagrange's equations of the second kind.
• Applications: harmonic oscillator, mathematical pendulum, motion in central field.
• Generalized momentum, cyclic coordinates and conservation of momentum
• Hamilton's principle and derivation of Lagrange's equations

### Hamiltonian mechanics

• Legendre transformation of function of $n$ variables.
• Definition of generalized momentum (again) and application of Legendre transform to Lagrange's equations
• Motivation for Hamilton's formalism: first order/second order equations, replacement of generalized velocities by generalized momenta
• Derivation of Hamilton's equations. Physical meaning of the Hamiltonian (energy).
• Phase space and its geometrical interpretation. Finding phase trajectories for mathematical pendulum without explicit solution.
• Canonical transformations. Four types of generating functions. Hamiltonian as a generator of infinitesimal time translations.
• Generalized momentum as gradient of the action. Hamilton-Jacobi equation.
• Action-angle coordinates and their geometrical interpretation (torus)

### Dynamical systems

• Definition of dynamical system. Relation of dynamical systems to Hamiltonian formalism (Hamilton's equations are special case of dynamical systems)
• Critical (fixed) points of dynamical systems. Qualitative difference between two different fixed points oh mathematical pendulum.
• Linearization of dynamical system in the neighbourhood of critical point.
• Stability analysis of linear(ized) systems. Classification of critical points (stable/unstable nodes, stable/unstable foci, centers). Hyperbolic points. Hartman's theorem.
• Numerical solution of dynamical systems in Maple/Mathematica
• Attractors, limit cycles, strange attractors, fractals, Lyapunov exponents

### Two dimensional flow

• Definition of 2D flow. Stream function and potential function.
• Cauchy-Riemann conditions and complex potential. Laplace equation.
• Laurent series and solution for the flow past the cylinder. Joukowski transformation.
• Vortex panel method. Flow past arbitrary profile.
• Qualitative discussion of turbulent flow. Kolmogorov theory of fully developed turbulence, self-similarity, intermittency.
• Lift force and drag force, Joukowski theorem, qualitative discussion of turbulent boundary layer.

### Traffic models

• Phenomenological description of traffic flow. Continuity equation and relation between density and traffic flow.
• Macroscopic traffic models and diffusion equation. Numerical solution in Mathematica/Maple.
• Microscopic models.
• Intelligent driver model (IDM) I. Behaviour of driver on free road, numerical implementation.
• IDM II. Interaction of driver with other vehicles. Calculation of the acceleration and numerical implementation.
• MOBIL -- model of overtaking. Conditions for the overtake, advantage and politeness factors. Numerical implementation.
• Analysis of stability of IDM model. Formation of traffic jams, soliton-like solutions.