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In classical mechanics, the precession of a top under the influence of gravity is not in general, an integrable problem. There are however three famous cases that are integrable, the Euler, the Lagrange and the [[Kovalevskaya Top|Kovalevskaya top]].<ref name = "Audin">Audin, M. Spinning Tops: A Course on Integrable Systems. New York: Cambridge University Press, 1996.</ref> In addition to the energy, each of these tops involves three additional constants of motion that give rise to the [[Integrable system#Hamiltonian systems and Liouville integrability|integrability]].
Whether you may have a helper or not, having a second ladder is an asset when hanging ceiling drywall. By positioning each ladder so that every finish of the drywall sheet can be accessed shortly, the job will go a lot sooner. In case you can grasp them, drywall stilts are even a greater plan. These adjustable stilts will make you the proper peak to work on the drywall with out the ladder. Another various to the ladder could be a very good bench made sturdy enough to work from, lengthy enough to achieve the complete length of a bit of drywall and tall enough to access the ceiling.<br><br>First, you need some clothes. Mountain males put on leather. Not black polished leather like you would find at a Harley Davidson convention, no it is advisable to get your fingers on a large buckskin. You may go to your native Mountain Man Supply [http://en.wikipedia.org/wiki/Pocket_knife gerber knive] Outlet, but one of the simplest ways to get your hands on a buckskin is to go out and kill your individual deer. Preferably out of season to point out that you just don’t care about local legal guidelines. You are a [http://Web.Seoulmi.Hs.kr/wiki/index.php/How_To_Disassemble_Kershaw_Blur mountain] man. You are the legislation. Next skin the deer, and treatment the leather-based.<br><br>Well, that depends – when you don’t care about knives, simply go pick up a cheap Opinel knife online at Ragweeds Forge or a Rough Rider or Steel Warrior trapper or stockman from Smoky Mountain Knifeworks or from Ebay. They will do the job acceptably and be quite a bit better than any gas station knife. There's [http://www.thebestpocketknifereviews.com/kershaw-blur-review/ Kershaw Ken Onion Blur Folding Knife With Speed Safe] also the option of a really small fixed-blade knife that rides in a pocket, but most people want a folding pocket knife (or are pressured because of state or local regulation to hold a folding knife). Should you love a quality merchandise, learn on –<br><br>Middle Cross - Heart Cross knife liners has been within the business for over 25 years with their wonderful designs. Significantly these knives have also withstood the take a look at of time and sturdiness for the purchasers. Their experts are getting somebody who has not solely created the products [http://en.wikipedia.org/wiki/Knife knife sets] time and time again. Glenn Waters - Nearly from 15 years as a knife maker Glenn Waters has the knowledge of the method to being an clever knife maker solely. For his or her excellent products units them apart from the remainder of the knife trade.<br><br>Twenty-pound paper is the standard weight for many enterprise wants. In the beginning, utilizing prime quality linen or different specialty paper is just not vital. Nonetheless, it's in all probability a good idea to maintain a small supply of nice paper on hand for instances when you'll want to make written communications with high-stage individuals, corresponding to prospective purchasers, or consultants you want to thank for help given. Individuals develop first impressions of you through your written communications If you are keeping it simple, that's okay. A nicely-developed letterhead on 20-pound printer paper can look very skilled.<br><br>Police, then again, have acknowledged that the evidence indicates otherwise; Rachel Pittman, they are saying, acted alone when she brutally murdered Amanda Doss and her children. A detective on the case has said that they don't expect to make any extra arrests. It does not seem that Rachel herself has accused anybody else of being involved, or indicated to police that she was in any means coerced to kill the Doss household. Until another arrest is made, or until the details come out in trial, rumours will proceed to hover over this case.<br><br>The gut hook blade is a novel type of blade used in each pocket knives and stuck blade knives These blades have a sharp hook to the highest aspect of the blade close to the tip. This characteristic is designed for skinning animals. The hook is dug into the pores and skin and is pulled like a zipper for the preliminary reduce. It's designed to chop into the animal pores and skin without hitting the muscle or inner organs. Selecting the Knife There isn't a one specific kind or model of knife that is best for whittling. Mostly, it's a matter of comfort.<br><br>Mikey Yurco had a desk arrange and it’s all the time fun to stopand talk with Mickey. He’s a veryinnovative knife maker who brings his martial arts expertise to hisdesigns. He’s not afraid to experimentand produces a wide variety of blades and sharply pointed objects. We had a pleasant dialogue of knife carry modesfor self-defense and the need to have the ability to entry a knife with eitherhand. It’s a tough task to develop ahigh diploma of dexterity in both hands. As Mickey says “I’d give my right arm to be ambidextrous.” SCRANTON - A 12-year-old boy is charged with bringing a pocket knife onto faculty property on Thursday, Scranton police mentioned.
 
The Euler top describes a free top without any particular symmetry, moving in the absence of any external torque.  The Lagrange top is a symmetric top, in which the center of gravity lies on the symmetry axis.  The [[Kovalevskaya Top|Kovalevskaya top]]<ref name ="kovalevskaya">S. Kovalevskaya, Acta Math. 12 177–232 (1889)</ref><ref name = "Perelemov">A. M. Perelemov, Teoret. Mat. Fiz.,  Volume 131,  Number 2, Pages 197–205 (2002)</ref> is special symmetric top with a unique ratio of the moments of inertia satisfy the relation
 
<math> I_1=I_2= 2 I_3</math>,
 
and in which the center of gravity is located in the plane perpendicular to the symmetry axis.
 
==Hamiltonian Formulation of Classical tops==
 
A classical top<ref name="Goldstein">Herbert Goldstein
Charles P. Poole , John L. Safko, Classical Mechanics, (3rd Edition), Addison-Wesley (2002)</ref> is defined by three principal axes, defined by the three orthogonal vectors <math> \hat{\mathbf{e}}^1</math>, <math> \hat {\mathbf{e}}^2</math> and  <math> \hat{\mathbf{e}}^3</math> with corresponding moments of inertia <math>I_1</math>, <math> I_2</math> and <math>I_3</math>.   In a Hamiltonian formulation of classical tops, the conjugate dynamical variables are the components of the angular momentum vector <math>\vec{L}</math> along the principal axes
 
<math>(l_1, l_2, l_3)= (\vec{L}\cdot \hat {\mathbf{e}}^1,\vec{L}\cdot \hat {\mathbf{e}}^2,\vec{L}\cdot \hat {\mathbf{e}}^3) </math>
 
and the z-components of the three principal axes,
 
<math>(n_1, n_2, n_3)= (\mathbf{\hat{z}}\cdot \hat {\mathbf{e}}^1,\mathbf{\hat{z}}\cdot \hat {\mathbf{e}}^2,\mathbf{\hat{z}}\cdot \hat {\mathbf{e}}^3) </math>
 
The Poisson algebra of these variables is given by
 
<math>
\{ l_a,l_b\} = \epsilon_{abc}l_c, \  \{l_a, n_b\} = \epsilon_{abc}n_c, \ \{n_a, n_b\} = 0
</math>
 
If the position of the center of mass is given by <math> \vec{R}_{cm} = (a \mathbf{\hat e}^1 + b \mathbf{\hat e}^2 + c\mathbf{\hat e}^3)</math>, then the Hamiltonian of a top is given by
 
<math>
H = \frac{(l_1)^2}{2I_1}+\frac{(l_2)^2}{2I_2}+\frac{(l_3)^2}{2I_3}+ mg (a n_1 + bn_2 + cn_3),
</math>
 
The equations of motion are then determined by
 
<math>
\dot{l}_a = \{ H, l_a\}, \dot{n}_a = \{H, n_a\}
</math>
 
==Euler Top==
 
The Euler top is an untorqued top, with Hamiltonian
 
<math>
H_E = \frac{(l_1)^2}{2I_1}+\frac{(l_2)^2}{2I_2}+\frac{(l_3)^2}{2I_3},
</math>
 
The four constants of motion are the energy <math>H_E</math> and  
the three components of angular momentum  in the lab frame,
 
<math>
(L_1,L_2,L_3) = l_1 \mathbf{\hat e}^1 +l_2\mathbf{\hat e}^2+ l_3 \mathbf{\hat e}^3.
</math>
 
==Lagrange Top==
 
The Lagrange top is a symmetric top with the center of mass along the symmetry axis at location, <math> \vec{R}_{cm} = h\mathbf{\hat e}^3</math>, with Hamiltonian
 
<math>
H_L= \frac{(l_1)^2+(l_2)^2}{2I}+\frac{(l_3)^2}{2I_3}+ mgh n_3.
</math>
 
The four constants of motion are the energy <math> H_L</math>, the angular momentum component along the symmetry axis, <math> l_3</math>, the angular momentum in the z-direction
 
<math>
L_z = l_1n_1+l_2n_2+l_3n_3,
</math>
 
and the magnitude of the n-vector
 
<math>
n^2 = n_1^2 + n_2^2 + n_3^2
</math>
 
==Kovalevskaya Top==
 
The Kovalevskaya top <ref name = "kovalevskaya"/><ref name = "Perelemov"/> is a symmetric top in which <math> I_1=I_2= 2I_3=I</math> and the center of mass lies in the plane perpendicular to the symmetry axis <math>\vec R_{cm} = h \mathbf{\hat e}^1</math>.   The Hamiltonian is
 
<math>
H_K= \frac{(l_1)^2+(l_2)^2+ 2 (l_3)^2}{2I}+ mgh n_1.
</math>
 
The four constants of motion are the energy <math> H_K</math>,  the Kovalevskaya invariant
 
<math>
K = \xi_+ \xi_-
</math>
 
where the variables <math>\xi_{\pm}</math> are defined by
 
<math>
\xi_{\pm} = (l_1\pm i l_2 )^2- 2 mgh I(n_1\pm i n_2),
</math>
 
the angular momentum component in the z-direction,
 
<math>
L_z = l_1n_1+l_2n_2+l_3n_3,
</math>
 
and the magnitude of the n-vector
 
<math>
n^2 = n_1^2 + n_2^2 + n_3^2.
</math>
 
==References==
{{reflist}}
 
[[Category:Hamiltonian mechanics]]

Revision as of 19:50, 7 February 2014

Whether you may have a helper or not, having a second ladder is an asset when hanging ceiling drywall. By positioning each ladder so that every finish of the drywall sheet can be accessed shortly, the job will go a lot sooner. In case you can grasp them, drywall stilts are even a greater plan. These adjustable stilts will make you the proper peak to work on the drywall with out the ladder. Another various to the ladder could be a very good bench made sturdy enough to work from, lengthy enough to achieve the complete length of a bit of drywall and tall enough to access the ceiling.

First, you need some clothes. Mountain males put on leather. Not black polished leather like you would find at a Harley Davidson convention, no it is advisable to get your fingers on a large buckskin. You may go to your native Mountain Man Supply gerber knive Outlet, but one of the simplest ways to get your hands on a buckskin is to go out and kill your individual deer. Preferably out of season to point out that you just don’t care about local legal guidelines. You are a mountain man. You are the legislation. Next skin the deer, and treatment the leather-based.

Well, that depends – when you don’t care about knives, simply go pick up a cheap Opinel knife online at Ragweeds Forge or a Rough Rider or Steel Warrior trapper or stockman from Smoky Mountain Knifeworks or from Ebay. They will do the job acceptably and be quite a bit better than any gas station knife. There's Kershaw Ken Onion Blur Folding Knife With Speed Safe also the option of a really small fixed-blade knife that rides in a pocket, but most people want a folding pocket knife (or are pressured because of state or local regulation to hold a folding knife). Should you love a quality merchandise, learn on –

Middle Cross - Heart Cross knife liners has been within the business for over 25 years with their wonderful designs. Significantly these knives have also withstood the take a look at of time and sturdiness for the purchasers. Their experts are getting somebody who has not solely created the products knife sets time and time again. Glenn Waters - Nearly from 15 years as a knife maker Glenn Waters has the knowledge of the method to being an clever knife maker solely. For his or her excellent products units them apart from the remainder of the knife trade.

Twenty-pound paper is the standard weight for many enterprise wants. In the beginning, utilizing prime quality linen or different specialty paper is just not vital. Nonetheless, it's in all probability a good idea to maintain a small supply of nice paper on hand for instances when you'll want to make written communications with high-stage individuals, corresponding to prospective purchasers, or consultants you want to thank for help given. Individuals develop first impressions of you through your written communications If you are keeping it simple, that's okay. A nicely-developed letterhead on 20-pound printer paper can look very skilled.

Police, then again, have acknowledged that the evidence indicates otherwise; Rachel Pittman, they are saying, acted alone when she brutally murdered Amanda Doss and her children. A detective on the case has said that they don't expect to make any extra arrests. It does not seem that Rachel herself has accused anybody else of being involved, or indicated to police that she was in any means coerced to kill the Doss household. Until another arrest is made, or until the details come out in trial, rumours will proceed to hover over this case.

The gut hook blade is a novel type of blade used in each pocket knives and stuck blade knives These blades have a sharp hook to the highest aspect of the blade close to the tip. This characteristic is designed for skinning animals. The hook is dug into the pores and skin and is pulled like a zipper for the preliminary reduce. It's designed to chop into the animal pores and skin without hitting the muscle or inner organs. Selecting the Knife There isn't a one specific kind or model of knife that is best for whittling. Mostly, it's a matter of comfort.

Mikey Yurco had a desk arrange and it’s all the time fun to stopand talk with Mickey. He’s a veryinnovative knife maker who brings his martial arts expertise to hisdesigns. He’s not afraid to experimentand produces a wide variety of blades and sharply pointed objects. We had a pleasant dialogue of knife carry modesfor self-defense and the need to have the ability to entry a knife with eitherhand. It’s a tough task to develop ahigh diploma of dexterity in both hands. As Mickey says “I’d give my right arm to be ambidextrous.” SCRANTON - A 12-year-old boy is charged with bringing a pocket knife onto faculty property on Thursday, Scranton police mentioned.

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