Helmholtz theorem (classical mechanics)
The Helmholtz theorem of classical mechanics reads as follows:
Let 
  
    
      
        H
        (
        x
        ,
        p
        ;
        V
        )
        =
        K
        (
        p
        )
        +
        φ
        (
        x
        ;
        V
        )
      
    
    {\displaystyle H(x,p;V)=K(p)+\varphi (x;V)}
  
 be the Hamiltonian of a one-dimensional system, where 
  
    
      
        K
        =
        
          
            
              p
              
                2
              
            
            
              2
              m
            
          
        
      
    
    {\displaystyle K={\frac {p^{2}}{2m}}}
  
 is the kinetic energy and  
  
    
      
        φ
        (
        x
        ;
        V
        )
      
    
    {\displaystyle \varphi (x;V)}
  
 is a "U-shaped" potential energy profile which depends on a parameter 
  
    
      
        V
      
    
    {\displaystyle V}
  
. Let 
  
    
      
        
          
            ⟨
            ⋅
            ⟩
          
          
            t
          
        
      
    
    {\displaystyle \left\langle \cdot \right\rangle _{t}}
  
 denote the time average. Let 

  
    
      
        E
        =
        K
        +
        φ
        ,
      
    
    {\displaystyle E=K+\varphi ,}
  

  
    
      
        T
        =
        2
        
          
            ⟨
            K
            ⟩
          
          
            t
          
        
        ,
      
    
    {\displaystyle T=2\left\langle K\right\rangle _{t},}
  

  
    
      
        P
        =
        
          
            ⟨
            
              −
              
                
                  
                    ∂
                    φ
                  
                  
                    ∂
                    V
                  
                
              
            
            ⟩
          
          
            t
          
        
        ,
      
    
    {\displaystyle P=\left\langle -{\frac {\partial \varphi }{\partial V}}\right\rangle _{t},}
  

  
    
      
        S
        (
        E
        ,
        V
        )
        =
        log
        ⁡
        ∮
        
          
            2
            m
            
              (
              
                E
                −
                φ
                
                  (
                  
                    x
                    ,
                    V
                  
                  )
                
              
              )
            
          
        
        
        d
        x
        . {\displaystyle S(E,V)=\log \oint {\sqrt {2m\left(E-\varphi \left(x,V\right)\right)}}\,dx.} Then 

  
    
      
        d
        S
        =
        
          
            
              d
              E
              +
              P
              d
              V
            
            T
          
        
        .