IMAGE CREDIT: NASA/JPL
SOURCE FOR CONTENT: Classical Dynamics of Particles and Systems. Thornton and Marion. 5th Edition.
Consider a functional
where . By the method used in a previous section of the aforementioned text, we may write
Additionally, we will find it useful to define
Further we may also define an integral functional by way of integrating Eq.(1) over the interval , and introducing a variational parameter we have
Two necessary conditions that are used to derive the Euler-Lagrange equation include
Let us take the derivative of with respect to yielding
Carrying out the derivative operator on the right-hand-side of Eq.(7) we get
From Eqs.(2) and (3) we see that
Thus Eq.(8) becomes…
Consider the second term under the summation. We may make use of integration by parts to obtain the following
By the necessary condition (Eq.(5)), it follows that
Additionally, in Eq.(13) above, we have also made use of the condition that . Since for any , then the terms in the brackets must vanish, yielding the Euler-Lagrange Equation for several dependent variables.
Update: the next posts will be those discussed on my Facebook page. Namely, I intend to continue with my Research Series and my series in Tensor Calculus and General Relativity with various ancillary posts in my Astrophysics Series.