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{{About|the two-body problem in classical mechanics|the career management problem of working couples|two-body problem (career)}} | |||
[[Image:orbit5.gif|thumb|right|400px|Two bodies with similar mass orbiting around a common [[Barycentric coordinates (astronomy)|barycenter]] with elliptic orbits.]] | |||
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). | |||
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. | |||
[[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.]] | |||
==Reduction to two independent, one-body problems== | |||
[[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>]] | |||
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). | |||
When applied to the two masses, [[Newton's laws of motion#Newton's second law|Newton's second law]] states that | |||
:<math> | |||
\mathbf{F}_{12}(\mathbf{x}_{1},\mathbf{x}_{2}) = m_{1} \ddot{\mathbf{x}}_{1} \quad \quad \quad (\mathrm{Equation} \ 1) | |||
</math> | |||
:<math> | |||
\mathbf{F}_{21}(\mathbf{x}_{1},\mathbf{x}_{2}) = m_{2} \ddot{\mathbf{x}}_{2} \quad \quad \quad (\mathrm{Equation} \ 2) | |||
</math> | |||
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. | |||
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''). | |||
===Center of mass motion (1st one-body problem)=== | |||
Addition of the force equations (1) and (2) yields | |||
:<math> | |||
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 | |||
</math> | |||
where we have used [[Newton's laws of motion|Newton's third law]] '''F'''<sub>12</sub> = −'''F'''<sub>21</sub> and where | |||
:<math> | |||
\ddot{\mathbf{R}} \equiv \frac{m_{1}\ddot{\mathbf{x}}_{1} + m_{2}\ddot{\mathbf{x}}_{2}}{m_{1} + m_{2}} | |||
</math> | |||
:<math> | |||
\mathbf{R} | |||
</math> is the position of the [[center of mass]] ([[Barycentric coordinates (astronomy)|barycenter]]) of the system. | |||
The resulting equation: | |||
:<math> | |||
\ddot{\mathbf{R}} = 0 | |||
</math> | |||
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. | |||
==Two-body motion is planar== | |||
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 | |||
:<math> | |||
\mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times \mu \frac{d\mathbf{r}}{dt} | |||
</math> | |||
the rate of change of the angular momentum '''L''' equals the net [[torque]] '''N''' | |||
:<math> | |||
\mathbf{N} = \frac{d\mathbf{L}}{dt} = \dot{\mathbf{r}} \times \mu\dot{\mathbf{r}} + \mathbf{r} \times \mu\ddot{\mathbf{r}} \ , | |||
</math> | |||
and using the property of the [[vector cross product]] that '''v''' × '''w''' = '''0''' for any vectors '''''v''''' and '''''w''''' pointing in the same direction, | |||
:<math> | |||
\mathbf{N} \ = \ \frac{d\mathbf{L}}{dt} = \mathbf{r} \times \mathbf{F} \ , | |||
</math> | |||
with '''F''' = μ ''d'' <sup>2</sup>'''r''' / ''dt'' <sup>2</sup>. | |||
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'''. | |||
== Laws of Conservation of Energy for each of two bodies for arbitrary potentials == | |||
In system of the center of mass for arbitrary potentials | |||
:<math>~U_{12} = U(\mathbf{r}_1 - \mathbf{r}_2) </math> | |||
:<math>~U_{21} = U(\mathbf{r}_2 - \mathbf{r}_1) </math> | |||
the value of [http://vv-voronkov.spb.ru/EN/two-lawe.html energies] of bodies not change: | |||
:<math>~E_1 = m_1 \frac{v_1^2}{2} + \frac{m_2} {m_1+m_2} U_{12} = Const_1(t) </math> | |||
:<math>~E_2 = m_2 \frac{v_2^2}{2} + \frac{m_1} {m_1+m_2} U_{21} = Const_2(t) </math> | |||
==Central forces== | |||
{{main|Classical central-force problem}} | |||
For many physical problems, the force '''F'''('''r''') is a [[central force]], i.e., it is of the form | |||
:<math>\mathbf{F}(\mathbf{r}) = F(r)\hat{\mathbf{r}}</math> | |||
where r = |'''r'''| and '''r̂''' = '''r'''/r is the corresponding [[unit vector]]. We now have: | |||
:<math> | |||
\mu \ddot{\mathbf{r}} = {F}(r) \hat{\mathbf{r}} \ , | |||
</math> | |||
where ''F(r)'' is negative in the case of an attractive force. | |||
==Work== | |||
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. | |||
==See also== | |||
* [[Kepler orbit]] | |||
* [[Energy drift]] | |||
* [[Equation of the center]] | |||
* [[Euler's three-body problem]] | |||
* [[Gravitational two-body problem]] | |||
* [[Kepler problem]] | |||
* [[n-body problem|''n''-body problem]] | |||
* [[Virial theorem]] | |||
* [[Two-body problem (career)]] | |||
==References== | |||
{{reflist|1}} | |||
==Bibliography== | |||
* {{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}} | |||
* {{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}} | |||
==External links== | |||
* [http://scienceworld.wolfram.com/physics/Two-BodyProblem.html Two-body problem] at [[ScienceWorld|Eric Weisstein's World of Physics]] | |||
{{DEFAULTSORT:Two-Body Problem}} | |||
[[Category:Concepts in physics]] | |||
[[Category:Orbits]] | |||
[[Category:Classical mechanics]] |
Revision as of 02:55, 27 December 2013
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.
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 stars 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).
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 for n ≥ 3) cannot be solved in terms of first integrals, except in special cases.
Reduction to two independent, one-body problems
Let x1 and x2 be the positions of the two bodies, and m1 and m2 be their masses. The goal is to determine the trajectories x1(t) and x2(t) for all times t, given the initial positions x1(t = 0) and x2(t = 0) and the initial velocities v1(t = 0) and v2(t = 0).
When applied to the two masses, Newton's second law states that
where F12 is the force on mass 1 due to its interactions with mass 2, and F21 is the force on mass 2 due to its interactions with mass 1.
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 (barycenter) motion. By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector r = x1 − x2 between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories x1(t) and x2(t).
Center of mass motion (1st one-body problem)
Addition of the force equations (1) and (2) yields
where we have used Newton's third law F12 = −F21 and where
- is the position of the center of mass (barycenter) of the system.
The resulting equation:
shows that the velocity V = dR/dt of the center of mass is constant, from which follows that the total momentum m1 v1 + m2 v2 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.
Two-body motion is planar
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
the rate of change of the angular momentum L equals the net torque N
and using the property of the vector cross product that v × w = 0 for any vectors v and w pointing in the same direction,
with F = μ d 2r / dt 2.
Introducing the assumption (true of most physical forces, as they obey 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 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.
Laws of Conservation of Energy for each of two bodies for arbitrary potentials
In system of the center of mass for arbitrary potentials
the value of energies of bodies not change:
Central forces
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. For many physical problems, the force F(r) is a central force, i.e., it is of the form
where r = |r| and r̂ = r/r is the corresponding unit vector. We now have:
where F(r) is negative in the case of an attractive force.
Work
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.
See also
- Kepler orbit
- Energy drift
- Equation of the center
- Euler's three-body problem
- Gravitational two-body problem
- Kepler problem
- n-body problem
- Virial theorem
- Two-body problem (career)
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.
Bibliography
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534