Sea ice concentration: Difference between revisions

Revision as of 04:10, 14 January 2014

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.^[1] In addition to the energy, each of these tops involves three additional constants of motion that give rise to the integrability.

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^[2]^[3] is special symmetric top with a unique ratio of the moments of inertia satisfy the relation

$I_{1} = I_{2} = 2 I_{3}$ ,

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^[4] is defined by three principal axes, defined by the three orthogonal vectors ${\hat{e}}^{1}$ , ${\hat{e}}^{2}$ and ${\hat{e}}^{3}$ with corresponding moments of inertia $I_{1}$ , $I_{2}$ and $I_{3}$ . In a Hamiltonian formulation of classical tops, the conjugate dynamical variables are the components of the angular momentum vector $\vec{L}$ along the principal axes

$(l_{1}, l_{2}, l_{3}) = (\vec{L} \cdot {\hat{e}}^{1}, \vec{L} \cdot {\hat{e}}^{2}, \vec{L} \cdot {\hat{e}}^{3})$

and the z-components of the three principal axes,

$(n_{1}, n_{2}, n_{3}) = (\hat{z} \cdot {\hat{e}}^{1}, \hat{z} \cdot {\hat{e}}^{2}, \hat{z} \cdot {\hat{e}}^{3})$

The Poisson algebra of these variables is given by

${l_{a}, l_{b}} = ϵ_{a b c} l_{c}, {l_{a}, n_{b}} = ϵ_{a b c} n_{c}, {n_{a}, n_{b}} = 0$

If the position of the center of mass is given by ${\vec{R}}_{c m} = (a {\hat{e}}^{1} + b {\hat{e}}^{2} + c {\hat{e}}^{3})$ , then the Hamiltonian of a top is given by

$H = \frac{(l_{1})^{2}}{2 I_{1}} + \frac{(l_{2})^{2}}{2 I_{2}} + \frac{(l_{3})^{2}}{2 I_{3}} + m g (a n_{1} + b n_{2} + c n_{3}),$

The equations of motion are then determined by

${\dot{l}}_{a} = {H, l_{a}}, {\dot{n}}_{a} = {H, n_{a}}$

Euler Top

The Euler top is an untorqued top, with Hamiltonian

$H_{E} = \frac{(l_{1})^{2}}{2 I_{1}} + \frac{(l_{2})^{2}}{2 I_{2}} + \frac{(l_{3})^{2}}{2 I_{3}},$

The four constants of motion are the energy $H_{E}$ and the three components of angular momentum in the lab frame,

$(L_{1}, L_{2}, L_{3}) = l_{1} {\hat{e}}^{1} + l_{2} {\hat{e}}^{2} + l_{3} {\hat{e}}^{3} .$

Lagrange Top

The Lagrange top is a symmetric top with the center of mass along the symmetry axis at location, ${\vec{R}}_{c m} = h {\hat{e}}^{3}$ , with Hamiltonian

$H_{L} = \frac{(l_{1})^{2} + (l_{2})^{2}}{2 I} + \frac{(l_{3})^{2}}{2 I_{3}} + m g h n_{3} .$

The four constants of motion are the energy $H_{L}$ , the angular momentum component along the symmetry axis, $l_{3}$ , the angular momentum in the z-direction

$L_{z} = l_{1} n_{1} + l_{2} n_{2} + l_{3} n_{3},$

and the magnitude of the n-vector

$n^{2} = n_{1}^{2} + n_{2}^{2} + n_{3}^{2}$

Kovalevskaya Top

The Kovalevskaya top ^[2]^[3] is a symmetric top in which $I_{1} = I_{2} = 2 I_{3} = I$ and the center of mass lies in the plane perpendicular to the symmetry axis ${\vec{R}}_{c m} = h {\hat{e}}^{1}$ . The Hamiltonian is

$H_{K} = \frac{(l_{1})^{2} + (l_{2})^{2} + 2 (l_{3})^{2}}{2 I} + m g h n_{1} .$

The four constants of motion are the energy $H_{K}$ , the Kovalevskaya invariant

$K = ξ_{+} ξ_{-}$

where the variables $ξ_{\pm}$ are defined by

$ξ_{\pm} = (l_{1} \pm i l_{2})^{2} - 2 m g h I (n_{1} \pm i n_{2}),$

the angular momentum component in the z-direction,

$L_{z} = l_{1} n_{1} + l_{2} n_{2} + l_{3} n_{3},$

and the magnitude of the n-vector

$n^{2} = n_{1}^{2} + n_{2}^{2} + n_{3}^{2} .$

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

↑ Audin, M. Spinning Tops: A Course on Integrable Systems. New York: Cambridge University Press, 1996.
↑ ^2.0 ^2.1 S. Kovalevskaya, Acta Math. 12 177–232 (1889)
↑ ^3.0 ^3.1 A. M. Perelemov, Teoret. Mat. Fiz., Volume 131, Number 2, Pages 197–205 (2002)
↑ Herbert Goldstein Charles P. Poole , John L. Safko, Classical Mechanics, (3rd Edition), Addison-Wesley (2002)

[Audin-1] Audin, M. Spinning Tops: A Course on Integrable Systems. New York: Cambridge University Press, 1996.

[kovalevskaya-2] 2.0 ^2.1 S. Kovalevskaya, Acta Math. 12 177–232 (1889)

[Perelemov-3] 3.0 ^3.1 A. M. Perelemov, Teoret. Mat. Fiz., Volume 131, Number 2, Pages 197–205 (2002)

[Goldstein-4] Herbert Goldstein Charles P. Poole , John L. Safko, Classical Mechanics, (3rd Edition), Addison-Wesley (2002)

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
-Some misguided and over-zealous officers within the public schools’ warfare in opposition to violence have turned good college students into collateral harm. They've instituted zero-tolerance rules and procedures that fail to differentiate between, say, a scholar who comes to highschool armed due to gang-associated exercise and a Boy Scout who forgets before faculty to take his pocket knife out of his pocket. Unreasonable zero-tolerance requirements make it sure that faculty officials will not make their disciplinary decisions based mostly on sound judgment – and that some good kids will be punished for no good motive. Miles Rankin is one instance of zero-tolerance’s collateral injury.<br><br>Oceanside California Knife Laws. As all the time I am not a lawyer and these movies are strictly for informational Purposes only if you want authorized Recommendation Search out A Criminal Lawyer. As at all times read and keep a copy of all pertaining knife laws for yourself, apply stating them so that you sound assured and clever, you're your finest advocate. Stopping the method at the initial contact is better than wining a court docket case after plenty of authorized motion. I finished at Spartan Blades. They make high-end mounted blades with the intense needs of the militaryin thoughts. I stopped to seek out out moreabout their unique CQB device. Right here’s theinside story.<br><br>Window tint utilized properly with the proper rating can block as much as 50 % of warmth from getting into the room from sunlight, not to mention blocking 99 p.c of ultraviolet rays, rays that fade furnishings and carpets. Perhaps you may have a window that faces the street and also you prefer to hold the blinds open however are concerned about privateness. There may be window tinting that's geared to privateness more than solar blockage. The reflective quality will likely be totally different when privacy is the principle concern. You have your decide of [http://Www.todaylogin.com/blog/54434/cold-steel-recon-1-nutnfancy/ coloring] from bronze to copper to steel grey.<br><br>To expose the observe, take away the lathe trim from across the inside the header. Find the nails and use a nail punch to drive them via the trim. Use a utility knife to chop out the edges that are stuffed with caulk and paint; this will prevent peeling the paint. You can [http://www.thebestpocketknifereviews.com/cold-steel-recon-1-review/ Cold Steel Recon 1 Tactical Knife With G-10 Handle Tanto Point With Black Blade] now access the monitor easier with a small crescent wrench. In the following paragraphs I'll go over the qualities that I require from an EDC knife. In reading you will also discover my definition of an EDC knife, must you be confused about what should be required and anticipated of an Each Day Carry knife.<br><br>In fact, I wouldn't be an excellent lawyer with out giving a couple of caveats earlier than I start. First, remember that carrying any weapon, even one which’s authorized, could cause you numerous grief with law enforcement. Cops routinely write tickets and make arrests for things they incorrectly suppose is against the law. Being discovered “not responsible” will not make up for the time and aggravation of getting arrested and missing work - not to mention the price of hiring an legal professional. Also, this text solely covers it's illegal for any person to carry concealed, certain knives, legally described as "dirks" and "daggers," i.e., any fastened-blade knife or stabbing weapon.<br><br>Mounted blades don't fold. They've both a full tang, or a stick tang that makes them extremely stout and secure. Mounted blade knives are generally extra strong and bigger general, which adds to the protection of the knife along with the power. This energy makes them the knife of alternative for outside use. They virtually at all times come with a sheath to guard the blade and to permit carrying the knife in a protected manner. The truth is, they're often called "sheath knives ". Mounted blade knives do not need to be oiled like folding knives , and do not must be cleaned as frequently.<br><br>So without additional adieu, on to the knives. The following is an inventory of 5 knives that I think rank among the finest pocket knives when all points are thought-about. I ought to notice that my recommendations usually are not only based mostly on efficiency, but additionally consolation, value, build quality and my overall [http://www.foxnews.com/us/2013/09/18/student-suspended-for-10-days-for-accidentally-bringing-pocket-knife-to/ gerber tools] impression of the knife. Are there other nice knives on the market? Certain. Will you be happy with one in every of these? You wager. Spyderco Para-Navy 2 Now that you've an understanding of what goes into determining the very best pocket knife , here's a record of the top rated knives out there proper now.
+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]].
+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]]