Exact solutions of classical central-force problems

<h2 id="general-problem">General problem</h2>
<p>Let 
  
    
      
        r
        =
        1
        
          /
        
        u
      
    
    {\displaystyle r=1/u}
  
. Then the <a href="/facts/Binet_equation/Y3jkM6rb">Binet equation</a> for 
  
    
      
        u
        (
        φ
        )
      
    
    {\displaystyle u(\varphi )}
  
 can be solved numerically for nearly any central force 
  
    
      
        F
        (
        1
        
          /
        
        u
        )
      
    
    {\displaystyle F(1/u)}
  
.  However, only a handful of forces result in formulae for 
  
    
      
        u
      
    
    {\displaystyle u}
  
 in terms of known functions.  The solution for 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  
 can be expressed as an integral over 
  
    
      
        u
      
    
    {\displaystyle u}

</p>

φ
        =
        
          φ
          
            0
          
        
        +
        
          
            L
            
              2
              m
            
          
        
        
          ∫
          
            u
          
        
        
          
            
              d
              u
            
            
              
                E
                
                  
                    t
                    o
                    t
                  
                
              
              −
              V
              (
              1
              
                /
              
              u
              )
              −
              
                
                  
                    
                      L
                      
                        2
                      
                    
                    
                      u
                      
                        2
                      
                    
                  
                  
                    2
                    m
                  
                
              
            
          
        
      
    
    {\displaystyle \varphi =\varphi _{0}+{\frac {L}{\sqrt {2m}}}\int ^{u}{\frac {du}{\sqrt {E_{\mathrm {tot} }-V(1/u)-{\frac {L^{2}u^{2}}{2m}}}}}}

<p>A central-force problem is said to be "integrable" if this integration can be solved in terms of known functions.
</p><p>If the force is a power law, i.e., if 
  
    
      
        F
        (
        r
        )
        =
        a
        
          r
          
            n
          
        
      
    
    {\displaystyle F(r)=ar^{n}}
  
, then 
  
    
      
        u
      
    
    {\displaystyle u}
  
 can be expressed in terms of <a href="/facts/Trigonometric_function/czej7ur0">circular functions</a> and/or <a href="/facts/Elliptic_function/mF03oI04">elliptic functions</a> if 
  
    
      
        n
      
    
    {\displaystyle n}
  
 equals 1, -2, -3 (circular functions) and -7, -5, -4, 0, 3, 5, -3/2, -5/2, -1/3, -5/3 and -7/3 (elliptic functions).<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>If the force is the sum of an inverse quadratic law and a linear term, i.e., if 
  
    
      
        F
        (
        r
        )
        =
        
          
            a
            
              r
              
                2
              
            
          
        
        +
        c
        r
      
    
    {\displaystyle F(r)={\frac {a}{r^{2}}}+cr}
  
, the problem also is solved explicitly in terms of <a href="/facts/Weierstrass%2527s_elliptic_functions/GHRXDknk">Weierstrass elliptic functions</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/E._T._Whittaker/b7wl3DMJ">Whittaker ET</a> (1937). <i><a href="/facts/Analytical_Dynamics_of_Particles_and_Rigid_Bodies/sn7za3vW">A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies</a></i> (4th ed.). New York: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-35883-5. {{cite book}}: ISBN / Date incompatibility (help)</li>
<li>Izzo,D. and Biscani, F. (2014). <i>Exact Solution to the constant radial acceleration problem</i>. Journal of Guidance Control and Dynamic.{{cite book}}:  CS1 maint: multiple names: authors list (link)</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Whittaker, pp. 80–95. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Izzo and Biscani <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Exact solutions of classical central-force problems open-in-new

Exact solutions of classical central-force problems