Analytical Dynamics I
- Sciences
- 300 level
- 3 credit units
- 145 pages
- 14 units
This course, Analytical Dynamics, is designed to teach students how mathematics can be applied to solve problems in contemporary science, technology, and engineering. It covers the basics of analytical dynamics, exposing students to the skills needed to achieve proficiency in this area of applied mathematics. The course explores concepts such as constraints, Lagrange's equations, simple harmonic motion, and Hamiltonian theory, preparing students for advanced studies and practical applications in various fields.
About this course
- Difficulty
- Intermediate
- Study hours
- 208 hours
- Maths
- Advanced
- Content
- Theoretical, problem solving
- Practical work
- No
- MTH211: Calculus
- PHY202: Classical Mechanics
- Assignments
- Tutor marked assignments
- Final examination
What you'll read
The real module and unit structure of MTH315, taken from the course material NOUN publishes.
One paragraph, so you can see how it reads
MTH315 · UNIT 2 HOLONOMIC AND NON-HOLONOMIC CONSTRAINTS
Consequently, the displacement described in equation (30) above is called virtual displacement to distinguish it from an actual displacement of the system occurring in a time interval [t, t + dt] during which the forces and constraints may be changing.
What you should be able to do
- Apply Lagrange's and Hamilton's equations to solve dynamics problems.
- Differentiate between holonomic and non-holonomic constraints.
- Analyze simple harmonic motion and related systems.
- Apply Newton's laws to analyze motion in various contexts.
- Calculate work, power, and energy in mechanical systems.
- Apply calculus of variations to solve problems in mechanics.
What it prepares you for
- Mechanical Engineer
- Aerospace Engineer
- Robotics Engineer
- Physics Researcher
- Applied Mathematician
- Aerospace
- Robotics
- Automotive
- Manufacturing
- Research and Development
- MATLAB
- Mathematica
Where it gets hard
The units students slow down on, and what makes each one heavy.
- Module 2:
Unit 1: Lagrange's Equation
Lagrange's equations require a strong understanding of calculus and differential equations, making it difficult for students without a solid mathematical foundation.
- Module 7:
Unit 3: The Hamilton-Jacobi Equation
The Hamilton-Jacobi equation involves advanced concepts of partial differential equations and canonical transformations, requiring significant analytical skills.
A suggested way through it
13 weeks, about 52 hours in total. Yours will differ.
- Week 1Module 1:
Unit 1: Degree of Freedom · 2 hours
Define degree of freedom and relate it to discrete and continuous systems.. Calculate the total kinetic energy of a system of particles.. Apply the conservation theorem for linear momentum to solve problems..
Unit 2: Constraints · 2 hours
Define constraints and differentiate between holonomic and non-holonomic constraints.. Explain D'Alambert's principle and its applications.. Solve problems involving virtual work and virtual displacement..
- Week 2Module 2:
Unit 1: Lagrange's Equation · 4 hours
Apply Lagrange's equations to solve dynamical problems.. Determine the Lagrange function for particles moving in a conservative force field.. Derive Lagrange's equations for holonomic constraints..
- Week 3Module 3:
Unit 1: Impulsive Motion · 4 hours
Define impulsive motion and identify equations of motion for impulsive forces.. Explain the concept of a conservative force field.. Solve problems involving the impact of two forces..
- Week 4Module 4:
Unit 1: Simple Harmonic Motion · 2 hours
Define simple harmonic motion (SHM) and identify the forces causing it.. Explain the suspension of a particle by an elastic string.. Define conical pendulum and solve related problems..
Unit 2: Collation of Smooth Spheres · 2 hours
Define and explain the collision of smooth spheres.. Apply the laws for the impact of spheres in direct and indirect impacts.. Solve problems involving the collision of smooth spheres..
- Week 5Module 5:
Unit 1: Newton's Law of Motion · 4 hours
State Newton's laws of motion and apply them to solve problems.. Define force and solve problems involving forces acting on a particle.. Apply Newton's laws to describe the motion of a particle in space..
- Week 6Module 5:
Unit 2: Work, Power and Energy · 3 hours
Define work, power, and energy and solve related problems.. Apply the principle of linear momentum to solve problems.. Apply the principle of angular momentum to solve problems..
Unit 3: Rectilinear Motion · 1 hour
Define rectilinear motion and solve problems involving uniform force fields.. Explain uniformly accelerated motion and solve related problems.. Define weight and acceleration due to gravity and apply them in calculations..
- Week 7Module 6:
Unit 1: Reduction of Coplanar Forces Acting on a Rigid Body to a Force and a Couple · 4 hours
Reduce coplanar forces acting on a rigid body to a single force and a single couple.. Calculate the center of mass of simple bodies.. Analyze the motion of the center of mass..
- Week 8Module 6:
Unit 2: Moment of a Force · 4 hours
Define the moment of a force and solve related problems.. Explain the concept of couples and their properties.. Apply the conditions for equilibrium of a particle to solve problems..
- Week 9Module 7:
Unit 1: The Hamiltonian · 4 hours
Define the Hamiltonian and state Hamilton's equations.. Explain ignorable or cyclic coordinates and their significance.. Define phase space and state Liouville's theorem..
- Week 10Module 7:
Unit 2: The Calculus of Variation · 4 hours
Define the calculus of variation and state Hamilton's principle.. Explain canonical or contact transformations.. Determine the condition for a transformation to be canonical..
- Week 11Module 7:
Unit 3: The Hamilton-Jacobi Equation · 4 hours
State the Hamilton-Jacobi equation and solve it for simple systems.. Analyze cases where the Hamiltonian is independent of time.. Define phase integrals, action, and angle variables..
- Week 12Review
Review: Modules 1-4 · 4 hours
Review all modules. Work on assignments.
- Week 13Review
Review: Modules 5-7 · 4 hours
Review all modules. Work on assignments.
Preparing for the exam
- Thoroughly review all worked examples in the study units.
- Practice solving problems from the TMAs and self-assessment exercises.
- Create concept maps linking Lagrange's equations, Hamilton's equations, and conservation laws.
- Focus on understanding the applications of each principle rather than just memorizing formulas.
- Allocate sufficient time to practice solving problems involving constraints and impulsive motion.
- Review past exam papers to familiarize yourself with the question formats and difficulty level.
Questions students ask about this course
What is MTH315 about?
This course, Analytical Dynamics, is designed to teach students how mathematics can be applied to solve problems in contemporary science, technology, and engineering. It covers the basics of analytical dynamics, exposing students to the skills needed to achieve proficiency in this area of applied mathematics. The course explores concepts such as constraints, Lagrange's equations, simple harmonic motion, and Hamiltonian theory, preparing students for advanced studies and practical applications in various fields.
How many units does MTH315 have?
MTH315, Analytical Dynamics I, has 14 units across 7 modules, over 145 pages of course material. You can read it one unit at a time.
How many credit units is MTH315?
MTH315 carries 3 credit units, at 300 level in Sciences.
Is MTH315 hard?
MTH315 is rated intermediate level, with advanced mathematical content. It is mostly theoretical and problem solving work.
How long does MTH315 take to study?
About 208 hours of study, spread across its 14 units.
How is MTH315 assessed?
MTH315 is assessed by assignments, tutor marked assignments and final examination.
What do I need before starting MTH315?
MTH211: Calculus PHY202: Classical Mechanics
What can I do with MTH315?
Mechanical Engineer, Aerospace Engineer, Robotics Engineer, Physics Researcher and Applied Mathematician.