page_title: Two-body problem
section_title: Reduction to two independent, one-body problems
hierarchical_section_title: Two-body problem / Reduction to two independent, one-body problems
caption_attribution_description: English: Two-body Jacobi coordinates
context_page_description: In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored.
  The most prominent case of the classical two-body problem is the gravitational case, arising in astronomy for predicting the orbits of objects such as satellites, planets, and stars. A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions.
  A simpler "one body" model, the "central-force problem", treats one object as the immobile source of a force acting on other. One then seeks to predict the motion of the single remaining mobile object. Such an approximation can give useful results when one object is much more massive than the other.
  However, the one-body approximation is usually unnecessary except as a stepping stone.
context_section_description: The complete two-body problem can be solved by re-formulating it 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. 
  Let x₁ and x₂ be the vector 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. The two dots on top of the x position vectors denote their second derivative with respect to time, or their acceleration vectors.
  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).
caption: Jacobi coordinates for two-body problem; Jacobi coordinates are 
    
      
        
          
            R
          
          =
          
            
              
                m
                
                  1
                
              
              M
            
          
          
            
              x
            
            
              1
            
          
          +
          
            
              
                m
                
                  2
                
              
              M
            
          
          
            
              x
            
            
              2
            
          
        
      
      {\displaystyle {\boldsymbol {R}}={\frac {m_{1}}{M}}{\boldsymbol {x}}_{1}+{\frac {m_{2}}{M}}{\boldsymbol {x}}_{2}}
    
   and 
    
      
        
          
            r
          
          =
          
            
              x
            
            
              1
            
          
          −
          
            
              x
            
            
              2
            
          
        
      
      {\displaystyle {\boldsymbol {r}}={\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{2}}
    
   with 
    
      
        
          M
          =
          
            m
            
              1
            
          
          +
          
            m
            
              2
            
          
           
        
      
      {\displaystyle M=m_{1}+m_{2}\ }
    
  .[1]
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