No forces…
And so begins my few posts on analytical dynamics.
I like analytical dynamics, people will think me weird for this. But it is so much more pleasant than newtonian mechanics. Who needs forces when you have energies!
Firstly we need to consider generalised coordinates. These are coordinates which when you solve the equations of motion the solutions are valid in any coordinate set.
So if you consider a simple 1D swinging pendulum the generalised coordinate (q) will be the angle between the pendulum and the line from the pivot to ground (
).
But why would we want to do this other than for kicks?
Well the answer is quite simple to solve the newtonian equations of motion you need to solve N times second order differentials, where N is the number of objects in the system. This is quite a lot. And we can reduce this down to n times first order differential equations, using Hamilton-Jacobi mechanics, although I will only discuss this shortly in later updates because I don’t really like it.
Next time… The Legrangian formalisation, possibly a derivation or how to use it, depending how I’m feeling…
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