Physics - Dynamics Linear - Newtons Laws

Physics - Dynamics - Analytical Techniques

When we have a complex mechanical system and we want to understand how it would respond to different inputs, or if we want to simulate it in some way, then it would be useful if we had a general method to apply.

It would be really good if we could find a way to specify a complex mechanical system in a computer and then use the computer to simulate its behavior without the need for problem specific coding or other human intervention.

I don't think there is a single method which will produce a simple result in every case, so these pages aim to list some of the most promising approaches:

D'Alemberts Equation

Hamiltons Method

Lagrangian Method

Action Principle

Instead of expressing the dynamics equations in terms of differential equations we can use integral equations, this can help to simplify many types of problem.

In classical mechanics the Newtonian equations of motion are equivalent to minimizing the action over the set of all paths.

In Quantum Mechanics (QM) all paths have a probability but paths with a lower action have a higher probability.

The action of a particle (point element of matter) is determined by the Lagrangian L(x,v) which is a function of its position and its velocity.

momentum = ∫ F dt

energy(work) = ∫ F•dx

more here

Symmetry

Symmetry is an important topic for maths and physics.

Symmetry is important for many branches of mathematics including geometry (see this page) and group theory (see this page). Its importance can become apparent in unexpected places, for example, solving quintic equations.

We say that an object is symmetric, with respect to a given mathematical operation, if this operation does not change the object.

Nothers Theorem (discussed further on this page) says that, for every symmetry exhibited by a physical law, there is a corresponding observable quantity that is conserved. Virtually every theory, including relativity and quantum physics is based on symmetry principles.

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see also:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

Mathematical Methods for Physics & Engineering. Very comprehensive coverage of the mathematics required for undergraduate Physics.

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.

SuSE Linux 11.0. Operating system with a wide range of applications including Open Office. A good distribution for developers as it contains KDevelop. Java, Mono, etc. Can install itself as a dual-boot system with an existing Windows OS if required. For information about installing it see this page.

Can you help?

Please send me any improvements to here. I would appreciate ideas to make the pages more useful including error correction, ideas for new pages, improvements to wording. It helps if you quote the full URL of the page.

progam

I am working on a project which uses these principles, if you would like to help me with this you are welcome to join in, here:

http://sourceforge.net/projects/mjbworld/

This site may have errors. Don't use for critical systems.