An Examination of the Derivation of the Lagrange Equations of Motion

The Lagrange equations of motion are familiar to anyone who has worked in physics. However, their range of validity is rarely, if ever, a topic for discussion. Following on an earlier examination of the consequences for these equations if the mass is not assumed constant, this note will look carefully at the other assumptions made and consider any further consequences resulting. The form of the equations applicable in electromagnetism will also be reviewed in the light of these discussions. The Lagrange Equations of Motion. To follow the basic outline in Synge and Griffith [1], suppose (x, y, z) are the Cartesian coordinates of a typical particle of a system and suppose we have a holonomic system of n degrees of freedom described by generalised coordinates qi, i = 1, 2,...., n. Then, dx = ∑ ∂x ∂qi dqi n i=1 , ?̇? = ∑ ∂x ∂q1 q?̇? n i=1 with similar equations for both ?̇? and ?̇?. From the second equation, it is seen immediately that ∂?̇? ∂?̇?i = ∂x ∂qi . It is straightforward to show that the operators d dt and ∂ ∂qi commute. Then d dt ∂ ∂?̇?i ( 1 2 ?̇?) = d dt (?̇? ∂?̇? ∂?̇?i ) = ?̈? ∂?̇? ∂q?̇? + ?̇? d dt ( ∂?̇? ∂?̇?i ) = ?̈? ∂x ∂qi + ?̇? d dt ( ∂x ∂qi ) = ?̈? ∂x ∂qi + ?̇? ∂ ∂qi (?̇?) = ?̈? ∂x ∂qi + ∂ ∂qi ( 1 2 ?̇?) That is d dt ∂ ∂?̇?i ( 1 2 ?̇?) − ∂ ∂qi ( 1 2 ?̇?) = ?̈? ∂x ∂qi with similar equations for y and z. The next step is to multiply these equations by m, sum over all particles of the system and add the three resulting equations together to give d dt ∂ ∂qi {∑ 1 2 P m(?̇? + ?̇? + ?̇?)} − ∂ ∂qi {∑ 1 2 m(?̇? + ?̇? + ?̇?)