PHY321 Classical Mechanics I, Michigan State University, Spring 2023
Contents
PHY321 Classical Mechanics I, Michigan State University, Spring 2023¶
This is the Jupyter-Book for the Classical Mechanics course at MSU
Learning outcomes¶
After the course you should:
be able to analyze forces that act on objects, apply Newton’s laws to determine the equations of motion, and solve these analytically and numerically,
Know about inertial frames and their relation to accelerating and rotating frames (non-inertial frames)
Know about forces, work, energy, angular momentum, linear momentum and conservation laws
Know about various types of motions, falling objects, objects moving in various fields
Know how to analyze energy diagrams and defining effective potential
Have knowledge about small oscillations, Harmonic oscillator potential and equations of motion
Have knowledge about transformation of variables that allow for analytical solutions, example two-body problems
Have knowledge about central forces and two-body problems, center-of-mass and relative coordinates as reference frame
Have knowledge about two-body scattering problems, classical scattering cross section
Have knowledge about Variational calculus and Lagrangian formalism
Know how to derive the equations of motion from the Lagrangian formalism with and without constraints (Lagrangian multipliers)
To solve many of these problems, we have through different projects and weekly exercises studied many systems numerically, from falling objects with and without friction/air resistance, small oscillations (harmonic oscillator), gravitational problems and other central force problems, rotations and the classical pendulum. To solve these systems, we have applied different algorithms for solving differential equations. These are
Euler-Cromer and Velocity-Verlet as energy conserving algorithms (time-independent forces)
Runge-Kutta family of algorithms for time-dependent forces We have also, in connection with for example the work-energy theorem studied methods for evaluating integrals. These are
Numerical integration using the Trapezoidal, midpoint and Simpson’s rule.
You should also have acquired skills in structuring a numerical project, as well as having developed a critical understanding of the pros and cons of the methods and an understanding of their limits and what can go wrong. Computing means solving scientific problems using computers. It covers numerical as well as symbolic computing. Computing is also about developing an understanding of the scientific process by enhancing algorithmic thinking when solving problems. Computing competence has always been a central part of the science and engineering education. In particular, some of the competences that are important in the development of your own understanding of computations, we would like to emphasize
derivation, verification, and implementation of algorithms
understanding what can go wrong with algorithms
overview of important, known algorithms for solving mechanics problems (To a extent large differential equations and integration)
understanding how algorithms are used to solve mathematical problems
Making science (your results) reproducible
algorithmic thinking for gaining deeper insights about scientific problems