MST324 TMA04 Finished

Woohoo thats my last maths assignment away for this batch of courses. Of course with the switch to the October starts it wont be the last maths assignment of 2012.

So the focus of the last block of MST324 is the caclulus of variations. In my undergraduate lectures we had about 1 lecture on this fascinating subject. A typical problem that can be solved by this technique is what is the shape formed by a rope or cable under it's own weight when slung between two points. It has become a key technique in many branches of physics, especially quantum physics and the development of the path integral formalism by Feynman. Of course none of this heady stuff is covered here. However the applications to mechanical problems is of interest and it culminates in a new way of presenting mechanics based on the minimisation of the Energy known as Lagrangian mechanics. This provides a simpler way of obtaining the equations of motion for complicated systems instead of analysing the forces on a body. Any of my readers who have done MST209 will know just how confusing it can be to set up the forces even for the simplest of mechanical systems.

The essential part of the calculus of variations is to see what happens when small variations are made to a function under an Integral. Going back to the example of a cable hung between two points there are many possible functions which could pass between the two points and so one considers a collection of functions a functional rather than just one function.  By considering small variations in the Functional it is possible to obtain a set of differential equations for the Function which minimises the variation. These are called the Euler Lagrange Equations.

So to a brief overview of the TMA

The first question involved  caclculating the effect of small variations on a function. Seeing what happened if parts of the function were set to a constant. This involved using the Binomial theorem to expand a function under a square root sign and calculating the difference. Comparison with a generalised Taylor theorem then gives certain conditions. Whilst in principle straightforward this was a bit tricky algebraically (indeed that is a theme of this TMA). Still I think I got this one out

Questions 2 and 3 were concerned with obtaining the Euler lagrange equations for various functionals. Question 2 was again straightforward but applying the boundary conditions to the Functional led to quite a messy solution. Question 3 was concerned with showing that a transformation applied to a given Functional gave a simpler form of the  Euler Lagrange Equations for the original Functional. This was probably the easiest question of the TMA. Again I think I did justice to the questions.

Finally question 4 a question of applying Lagrangian Mechanics to the problem of a bead on a rotating parabolic Wire. There was a whopping 40 marks for this question

The first step is to obtain expressions for the Kinetic Energy and the Potential energy in terms of coordinates which are easier to express the problem in. The so called generalised coordinates.
This involved setting up a conversion between the postion of the particle expressed in Cartesian coordinates and the more convenient coordinates. The Lagrangian is the difference between the Kinetic energy and the Potential Function. The equations of motion are then derived for the generalised coordinates by applying the Euler Lagrange Equations. For this Lagrangian the equations of motion for one of the coordinates seemed quite complicated. The rest of the question then involved solving the equations of motion to obtain the angular frequency of oscillations of the bead for various conditions. This was algebraically quite involved using Taylor series and dropping terms of higher order. As there was no show that the solutions are of the form ... This made the question quite tricky to see if you were on the right track. However as the last part asked us to comment on the relation between the various angular frequencies and as my answer showed quite a simple relationship between them. I'm reasonably confident I got the algebra correct. The thing with complicated algebra is to just keep going and not to panic.

So overall I'm reasonably confident of a high mark for this one. Just as well as I need to make up for the last TMA.

I'm going to take a complete break from maths this week. The past few weeks have been a nightmare as I desperately tried to complete the TMA's . I will start revision in earnest after next Monday. For now I just want to chill.