After thinking on it again, it suddenly came to me, I was about to post my mistake but it seems you beat me to it! :smile:
I was so fixated on the path taken by the particle as a physical trajectory that I missed/forgot the fact that the action integral is over time: viewed over space and time...
Sorry, I'm being slightly loose in my notation for brevity: ##r^2=x_i x_i##, so yes ##V## is a function of position. But I'm not ignoring it: finding a stationary point of the action can indeed produce a solution which results in the particle moving in a way corresponding to what a...
Newton's laws do indeed follow from solving the E-L equations, but what I'm trying (unfortunately not very well!) to say is that when we set up the E-L equations there is also another solution that we typically ignore. Let me take the concrete example you gave...
I'm going to make this...
Yes, but Newton's Law is empirical, or else a consequence of solving the Euler-Lagrange equations for the system. But in many cases the E-L equations would also admit a 'motionless solution'.
I'm not saying there can't be dynamic systems. I'm saying that in many cases there is an apparent 'motionless solution' to the Euler-Lagrange equations, which is unphysical and I'm asking what principal (other than empiricism) we use to discard this solution.
Consider a point particle in a...
I realize to many people this might seem a silly or at least pointlessly philosophical question, but I was wondering if there's a deeper answer I've missed...
After a lengthy break in physics I've been playing around with some classical mechanics to try and probe my understanding (and memory!)...
Ughh, my bad. I read this when I was very tired and tried to answer the question too quicky (something I really shouldn't do!).
I saw NCStarGar's use of r in the centripetal and gravitational force equations and assumed he was making the assumption that M>>m and ignoring centre of mass...
No! The law is MP^2 = 4\pi^2a^3/G (see here). In which case there is no contradiction (hardly surprising given the above form of Kepler's Law is derived from Newton's law of gravity).
For clarity I'm going to refer to your 1kg object as a box in what follows, but obviously it can be anything you want.
The problem is that the gravitational potential energy belongs to the earth-box system not to the box itself. It is a failure to take this into account that leads you to...
Sorry for slow reply, been rather busy.
The limit only appears undefined because you are applying it term by term as you have written it. You need to group all the coefficients of 1/z^2 together (and similarly 1/z^3, etc. etc.). Then L'Hopital's rule or judicious Taylor expansion of sin and...
Keep going! You've differentiated twice, so you need to do another eleven. If you keep regrouping like terms together it shouldn't end up too difficult. Also don't forget the factorial multiplying the final answer...
Sorry for replying so late.
Glad to here it works. Although I can't say I know of any physical justification for the approximation you've used!
If you ever finish the game send me a PM, since as you may have guessed I'm a bit of a sailing buff!
Yes, but it's not just the function you need to differentiate. You should have this in your textbook already, but if not look here:
http://en.wikipedia.org/wiki/Residue_(complex_analysis)#Limit_formula_for_higher_order_poles
Yes the denominator of your fraction is zero there, so you need to test for a pole at z=0.
No it is a real pole. As I said in my previous post, if the limit of the function multiplied by z as z\rightarrow 0 was zero then it would be removable.
You have multiplied by z^{15}. That's not...