A Brief Review of Helmholtz Conditions
  It is well known that the equations of motion obtained from Newtons second law of motion can be obtained from a Lagrangian via the Euler-Lagrangian formulation if and only if the equations of motion satisfy the Helmholtz conditions. In this pedagogical article we give a simple proof of the above statement and show its application to simple mechanical and dissipative systems. 
Kushagra Nigam and Kinjal Banerjee
math-ph math.MP