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Unread 30 Nov 2006, 17:04   #1
vampire_lestat
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maths

i have a question to all you physicists/mathamaticians:

How is hamiltonian mechanics simpler than lagrangian (in what way is 2 first order DE's simpler than one second order)?

What's the point of the hamilton-jacobi stuff, it seems to me that although the equation is simpler to solve than the other two, doing the transform to get it back into your orginal co-ordinates is far more complex.

and other thing, why bother with this generalised co-ordinate stuff anyway, how is it better than newtonian mechanics?
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Unread 30 Nov 2006, 18:11   #2
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Re: maths

The apple actually fell through time and space and into the 7th dimension where it morphed into Terry Pratchett. Newton actually felt a quantum shadow bounce of his nonce.
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Unread 30 Nov 2006, 18:51   #3
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Re: maths

Quote:
Originally Posted by vampire_lestat
How is hamiltonian mechanics simpler than lagrangian (in what way is 2 first order DE's simpler than one second order)?
What do you mean 'simpler'? The Lagrangian/Hamiltonian formulation is far more mathematically elegant than the Newtonian one, but when it comes to solving a specific problem, the simplest formulation to use is going to depend on what exactly youre doing. 'Simpler' is a pretty ambigous term - do you mean simpler in terms of solving problems, or simpler in a more theoretical sense (ie bringing out the fundamental structure).

Quote:
in what way is 2 first order DE's simpler than one second order
Where are you getting the second order equation from? The (Euler-)Lagrange equations are first order, as are Hamilton's.

Quote:
and other thing, why bother with this generalised co-ordinate stuff anyway, how is it better than newtonian mechanics?
Having a set of equations which are independent of any particular coordinate system lets you choose a set of coordinates which are the easiest to work with for a particular problem. With Newtonian mechanics youre somewhat tied to Cartesian coordinates and changing to another set can make things really messy. With the Lagrangian coordinate-independent formulation you can pick whatever set of coordinates will make the problem simplest to solve and not have to worry about transforming your equations.

I assume youve seen the horrifically tedious algebra that can be involved in a relatively simple coordinate change, such as tranforming the laplacian operator into spherical coordinates*. With generalised coordinates, you dont have to ever worry about doing that sort of thing - you just pick a set of coordinates that will make the problem easier to solve (eg by exploiting a symmetry present in the system) and the equations remain unchanged.


* I've seen this being done and it was significantly simpler than the method on that page but it was still extremely messy and best avoided.

Last edited by Nodrog; 30 Nov 2006 at 19:27.
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Unread 30 Nov 2006, 22:07   #4
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Re: maths

......

I need to do more maths study....or im going to get eaten alive at uni
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Unread 30 Nov 2006, 22:18   #5
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Re: maths

Quote:
Originally Posted by The General8
......

I need to do more maths study....or im going to get eaten alive at uni
Or you could pick a better course. Maybe one which disposes of mathematics altogether.
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Unread 30 Nov 2006, 22:25   #6
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Re: maths

I always like to confuse people with this simple equation:

1 + 1 = 10

But IT people might get that one more than some math people.
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Unread 1 Dec 2006, 00:08   #7
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Re: maths

Quote:
Originally Posted by lestat
in what way is 2 first order DE's simpler than one second order
Because first order DEs are easier than second order? I'm not quite sure what to say.

And as nod says, things which are independent, or easily atapted, to varying types of coordinate systems are fairly valuable because you cannot always assume that a situation is best modelled in Cartesian or spherical polar or whatever you decided at the start.

Quote:
Originally Posted by jt25man
I always like to confuse people with this simple equation:

1 + 1 = 10

But IT people might get that one more than some math people.
Yes, very good. I bet you pronounce it as "one plus one equals ten" or somesuch, as well.
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Unread 1 Dec 2006, 05:53   #8
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Re: maths

Quote:
Originally Posted by MrL_JaKiri
Yes, very good. I bet you pronounce it as "one plus one equals ten" or somesuch, as well.
That equation doesn't really work out when you say it out loud, it confuses people when you say "one plus one equals one zero", but for posting in written form it works just fine.
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Unread 1 Dec 2006, 06:04   #9
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Re: maths

Quote:
Originally Posted by jt25man
I always like to confuse people with this simple equation:

1 + 1 = 10

But IT people might get that one more than some math people.
if you're smug about base 'jokes', you'll just love "why is halloween the same as christmas?" and "how many people can read hex if only you and dead people can read hex?"
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Unread 1 Dec 2006, 18:32   #10
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Re: maths

What are the answers?
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Unread 1 Dec 2006, 19:21   #11
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Re: maths

halloween is the same as christmas because OCT 31 = DEC 25

the dead hex one is 57006 ($DEAD+1)
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Unread 1 Dec 2006, 19:54   #12
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Re: maths

I don't get it.
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Unread 1 Dec 2006, 21:25   #13
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Re: maths

Quote:
Originally Posted by Cannon_Fodder
I don't get it.
OCT means octal and is the base-8 number system which goes from 0 to 7.
Thus 25 in 'our' (DEC = decimal) number system equals 31 in the octal system.

Similar HEX means hexadecimal, which goes from 0 to 15 (0, 1,..., 9, A, B, C, D, E, F), which, without going into details, means 'DEAD' = (13*16^3 + 14*16^2 + 10*16^1 + 13*16^0) + 'you' = 57006.
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Unread 1 Dec 2006, 21:50   #14
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Re: maths

I see.
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