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| {{About|the two-body problem in classical mechanics|the career management problem of working couples|two-body problem (career)}}
| | There seems to be a belief that petite is synonymous with elfin, which petite women are tiny graceful creatures whom float delicately on a sea of petals and have waists the size of a typical womans neck. Not true. While various such women do exist, there are only because various (if not more) whom are petite plus, average size plus above (inside width) yet less than average height. Why do you care?<br><br>Number 6. Visit the venue before the wedding date, if possible. May it be within the easy backyard of the groom or the cliff of an island, it is very suggested to visit the spot where the ceremony is held. This might enable we greatly for it you'll understand what to expect and tackle. Simply observe the step, assuming we will shoot at the mountains.<br><br>However, the National Institutes of Health does not really recommend a weight reduction waist to height ratio objective for people with abdominal weight yet whom have BMIs inside the general and overweight range -- unless these individuals have two or even more risk factors for cardiovascular condition, or only the want to lose several fat.<br><br>Waist Circumference Measurement - A wise technique of measuring the amount of abdominal fat is by measuring the person's all-natural waist line. By getting the waist circumference, it is actually simpler to check when a person is at risk of getting a heart disease or any other health difficulties. Women with a waist line of 35 inches or more whilst 40 inches or more for men are taken as piece of the high risk category.<br><br>To discover out more, a analysis team searched healthcare literature for studies that looked at stroke risk plus body mass index with a minimum of four years of follow up.<br><br>I won't hesitate suggesting that it helped me a lot. I tried following his regulations for a month plus couldn't believe the results. It really worked for me. Later I found millions of individuals are absolutely associated with the [http://safedietplansforwomen.com/waist-to-height-ratio waist to height ratio] program. The benefit of joining Dr. Charles's fitness system is the fact that we can remain fit all time. I mean you don't have to spend hours working out in gym plus fitness centers. Once you join the program, you are given a list of food items that enable to burn body fat promptly. I am sure we will receive amazing results following joining the system.<br><br>So what exactly is the value of the BMI? On an individual level, completely none. It is a statistical tool that only holds a small value plus which is for utilize across a large population, so which the heavily muscled athletes with BMIs of 28+ are averaged out by the skinny fat individuals that have the aforementioned BMIs of 18-25 with body fat percentages of 25%. If weight is what you're worried about, I have the most perfect solution for you. Cut off the leg. We only lost a wise 1/5 of your weight. Are you pleased? Many of the time, fat reduction tries result in a smaller adaptation of the same body, twenty pounds lighter, however nonetheless lookin like a pear.<br><br>I was born in Utah United States; my parents likes to cook food, so I enrolled at culinary school at The Arts of Institutes main inside baking plus pastry. I employed inside among the ideal hotels in the city, nevertheless later decided to establish my own business. We already have more than 15 people in the company and it's growing so far. I enjoyed using my team and focus on improving more.I love to invest my time with my family or doing certain outdoor activities like hiking. |
| [[Image:orbit5.gif|thumb|right|400px|Two bodies with similar mass orbiting around a common [[Barycentric coordinates (astronomy)|barycenter]] with elliptic orbits.]]
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| In [[classical mechanics]], the '''two-body problem''' is to determine the motion of two point particles that interact only with each other. Common examples include a [[satellite]] orbiting a [[planet]], a [[planet]] orbiting a [[star]], two [[star]]s orbiting each other (a [[binary star]]), and a classical [[electron]] orbiting an [[atomic nucleus]] (although to solve this system correctly a quantum mechanical approach must be used).
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| The two-body problem can be re-formulated as two '''one-body problems''', a trivial one and one that involves solving for the motion of one particle in an external [[potential]]. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. By contrast, the [[three-body problem]] (and, more generally, the [[n-body problem|''n''-body problem]] for ''n'' ≥ 3) cannot be solved in terms of first integrals, except in special cases.
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| [[Image:orbit2.gif|thumb|right|200px|Two bodies with a slight difference in [[mass]] orbiting around a common [[Barycentric coordinates (astronomy)|barycenter]]. The sizes, and this particular type of orbit are similar to the [[Pluto]]-[[Charon (moon)|Charon]] system and also to Earth-Moon system in which the center of mass is inside the bigger body instead.]]
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| ==Reduction to two independent, one-body problems==
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| [[File:Two-body Jacobi coordinates.JPG|thumb|300px|Jacobi coordinates for two-body problem; Jacobi coordinates are <math>\boldsymbol{R}=\frac {m_1}{M} \boldsymbol{x}_1 + \frac {m_2}{M} \boldsymbol{x}_2 </math> and <math>\boldsymbol{r} = \boldsymbol{x}_1 - \boldsymbol{x}_2 </math> with <math>M = m_1+m_2 \ </math>.<ref name=Betounes>{{cite book |title=Differential Equations |author=David Betounes |url=http://books.google.com/?id=oNvFAzQXBhsC&pg=PA58 |isbn=0-387-95140-7 |page=58; Figure 2.15 |year=2001 |publisher=Springer}}</ref>]]
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| Let '''x'''<sub>1</sub> and '''x'''<sub>2</sub> be the positions of the two bodies, and ''m''<sub>1</sub> and ''m''<sub>2</sub> be their masses. The goal is to determine the trajectories '''x'''<sub>1</sub>(''t'') and '''x'''<sub>2</sub>(''t'') for all times ''t'', given the initial positions '''x'''<sub>1</sub>(''t'' = 0) and '''x'''<sub>2</sub>(''t'' = 0) and the initial velocities '''v'''<sub>1</sub>(''t'' = 0) and '''v'''<sub>2</sub>(''t'' = 0).
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| When applied to the two masses, [[Newton's laws of motion#Newton's second law|Newton's second law]] states that
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| :<math>
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| \mathbf{F}_{12}(\mathbf{x}_{1},\mathbf{x}_{2}) = m_{1} \ddot{\mathbf{x}}_{1} \quad \quad \quad (\mathrm{Equation} \ 1)
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| </math>
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| :<math>
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| \mathbf{F}_{21}(\mathbf{x}_{1},\mathbf{x}_{2}) = m_{2} \ddot{\mathbf{x}}_{2} \quad \quad \quad (\mathrm{Equation} \ 2)
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| </math>
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| where '''F'''<sub>12</sub> is the force on mass 1 due to its interactions with mass 2, and '''F'''<sub>21</sub> is the force on mass 2 due to its interactions with mass 1.
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| Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. ''Adding'' equations (1) and (2) results in an equation describing the [[center of mass]] ([[Barycentric coordinates (astronomy)|barycenter]]) motion. By contrast, ''subtracting'' equation (2) from equation (1) results in an equation that describes how the vector '''r''' = '''x'''<sub>1</sub> − '''x'''<sub>2</sub> between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories '''x'''<sub>1</sub>(''t'') and '''x'''<sub>2</sub>(''t'').
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| ===Center of mass motion (1st one-body problem)===
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| Addition of the force equations (1) and (2) yields
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| :<math>
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| m_{1}\ddot{\mathbf{x}}_1 + m_2 \ddot{\mathbf{x}}_2 = (m_1 + m_2)\ddot{\mathbf{R}} = \mathbf{F}_{12} + \mathbf{F}_{21} = 0
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| </math>
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| where we have used [[Newton's laws of motion|Newton's third law]] '''F'''<sub>12</sub> = −'''F'''<sub>21</sub> and where
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| :<math>
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| \ddot{\mathbf{R}} \equiv \frac{m_{1}\ddot{\mathbf{x}}_{1} + m_{2}\ddot{\mathbf{x}}_{2}}{m_{1} + m_{2}}
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| </math>
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| :<math>
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| \mathbf{R}
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| </math> is the position of the [[center of mass]] ([[Barycentric coordinates (astronomy)|barycenter]]) of the system.
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| The resulting equation:
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| :<math>
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| \ddot{\mathbf{R}} = 0
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| </math> | |
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| shows that the velocity '''V''' = ''d'''''R'''/''dt'' of the center of mass is constant, from which follows that the total momentum ''m''<sub>1</sub> '''v'''<sub>1</sub> + ''m''<sub>2</sub> '''v'''<sub>2</sub> is also constant ([[conservation of momentum]]). Hence, the position '''R''' (''t'') of the center of mass can be determined at all times from the initial positions and velocities.
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| ==Two-body motion is planar==
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| The motion of two bodies with respect to each other always lies in a plane (in the [[center of mass frame]]). Defining the [[linear momentum]] '''p''' and the [[angular momentum]] '''L''' by the equations
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| :<math>
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| \mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times \mu \frac{d\mathbf{r}}{dt}
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| </math> | |
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| the rate of change of the angular momentum '''L''' equals the net [[torque]] '''N''' | |
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| :<math>
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| \mathbf{N} = \frac{d\mathbf{L}}{dt} = \dot{\mathbf{r}} \times \mu\dot{\mathbf{r}} + \mathbf{r} \times \mu\ddot{\mathbf{r}} \ ,
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| </math>
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| and using the property of the [[vector cross product]] that '''v''' × '''w''' = '''0''' for any vectors '''''v''''' and '''''w''''' pointing in the same direction,
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| :<math>
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| \mathbf{N} \ = \ \frac{d\mathbf{L}}{dt} = \mathbf{r} \times \mathbf{F} \ ,
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| </math>
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| with '''F''' = μ ''d'' <sup>2</sup>'''r''' / ''dt'' <sup>2</sup>. | |
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| Introducing the assumption (true of most physical forces, as they obey [[Newton's laws of motion|Newton's strong third law of motion]]) that the force between two particles acts along the line between their positions, it follows that '''r''' × '''F''' = '''0''' and the [[conservation of angular momentum|angular momentum vector '''L''' is constant]] (conserved). Therefore, the displacement vector '''r''' and its velocity '''v''' are always in the plane [[perpendicular]] to the constant vector '''L'''.
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| == Laws of Conservation of Energy for each of two bodies for arbitrary potentials ==
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| In system of the center of mass for arbitrary potentials
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| :<math>~U_{12} = U(\mathbf{r}_1 - \mathbf{r}_2) </math>
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| :<math>~U_{21} = U(\mathbf{r}_2 - \mathbf{r}_1) </math>
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| the value of [http://vv-voronkov.spb.ru/EN/two-lawe.html energies] of bodies not change:
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| :<math>~E_1 = m_1 \frac{v_1^2}{2} + \frac{m_2} {m_1+m_2} U_{12} = Const_1(t) </math>
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| :<math>~E_2 = m_2 \frac{v_2^2}{2} + \frac{m_1} {m_1+m_2} U_{21} = Const_2(t) </math>
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| ==Central forces==
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| {{main|Classical central-force problem}}
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| For many physical problems, the force '''F'''('''r''') is a [[central force]], i.e., it is of the form
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| :<math>\mathbf{F}(\mathbf{r}) = F(r)\hat{\mathbf{r}}</math>
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| where r = |'''r'''| and '''r̂''' = '''r'''/r is the corresponding [[unit vector]]. We now have:
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| :<math>
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| \mu \ddot{\mathbf{r}} = {F}(r) \hat{\mathbf{r}} \ ,
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| </math> | |
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| where ''F(r)'' is negative in the case of an attractive force.
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| ==Work==
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| The total work done in a given time interval by the forces exerted by two bodies on each other is the same as the work done by one force applied to the total relative displacement.
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| ==See also==
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| * [[Kepler orbit]]
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| * [[Energy drift]]
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| * [[Equation of the center]]
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| * [[Euler's three-body problem]]
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| * [[Gravitational two-body problem]]
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| * [[Kepler problem]]
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| * [[n-body problem|''n''-body problem]]
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| * [[Virial theorem]]
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| * [[Two-body problem (career)]]
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| ==References==
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| {{reflist|1}}
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| ==Bibliography==
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| * {{cite book | author = [[Lev Landau|Landau LD]], [[Evgeny Lifshitz|Lifshitz EM]] | year = 1976 | title = Mechanics | edition = 3rd. | publisher = Pergamon Press | location = New York | isbn = 0-08-029141-4}}
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| * {{cite book | author = [[Herbert Goldstein|Goldstein H]] | year = 1980 | title = [[Classical Mechanics (textbook)|Classical Mechanics]] | edition = 2nd. | publisher = Addison-Wesley | location = New York | isbn = 0-201-02918-9}}
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| ==External links==
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| * [http://scienceworld.wolfram.com/physics/Two-BodyProblem.html Two-body problem] at [[ScienceWorld|Eric Weisstein's World of Physics]]
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| {{DEFAULTSORT:Two-Body Problem}}
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| [[Category:Concepts in physics]]
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| [[Category:Orbits]]
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| [[Category:Classical mechanics]]
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There seems to be a belief that petite is synonymous with elfin, which petite women are tiny graceful creatures whom float delicately on a sea of petals and have waists the size of a typical womans neck. Not true. While various such women do exist, there are only because various (if not more) whom are petite plus, average size plus above (inside width) yet less than average height. Why do you care?
Number 6. Visit the venue before the wedding date, if possible. May it be within the easy backyard of the groom or the cliff of an island, it is very suggested to visit the spot where the ceremony is held. This might enable we greatly for it you'll understand what to expect and tackle. Simply observe the step, assuming we will shoot at the mountains.
However, the National Institutes of Health does not really recommend a weight reduction waist to height ratio objective for people with abdominal weight yet whom have BMIs inside the general and overweight range -- unless these individuals have two or even more risk factors for cardiovascular condition, or only the want to lose several fat.
Waist Circumference Measurement - A wise technique of measuring the amount of abdominal fat is by measuring the person's all-natural waist line. By getting the waist circumference, it is actually simpler to check when a person is at risk of getting a heart disease or any other health difficulties. Women with a waist line of 35 inches or more whilst 40 inches or more for men are taken as piece of the high risk category.
To discover out more, a analysis team searched healthcare literature for studies that looked at stroke risk plus body mass index with a minimum of four years of follow up.
I won't hesitate suggesting that it helped me a lot. I tried following his regulations for a month plus couldn't believe the results. It really worked for me. Later I found millions of individuals are absolutely associated with the waist to height ratio program. The benefit of joining Dr. Charles's fitness system is the fact that we can remain fit all time. I mean you don't have to spend hours working out in gym plus fitness centers. Once you join the program, you are given a list of food items that enable to burn body fat promptly. I am sure we will receive amazing results following joining the system.
So what exactly is the value of the BMI? On an individual level, completely none. It is a statistical tool that only holds a small value plus which is for utilize across a large population, so which the heavily muscled athletes with BMIs of 28+ are averaged out by the skinny fat individuals that have the aforementioned BMIs of 18-25 with body fat percentages of 25%. If weight is what you're worried about, I have the most perfect solution for you. Cut off the leg. We only lost a wise 1/5 of your weight. Are you pleased? Many of the time, fat reduction tries result in a smaller adaptation of the same body, twenty pounds lighter, however nonetheless lookin like a pear.
I was born in Utah United States; my parents likes to cook food, so I enrolled at culinary school at The Arts of Institutes main inside baking plus pastry. I employed inside among the ideal hotels in the city, nevertheless later decided to establish my own business. We already have more than 15 people in the company and it's growing so far. I enjoyed using my team and focus on improving more.I love to invest my time with my family or doing certain outdoor activities like hiking.