How Can Continuous Group Theory Be Applied to Solve a Differential Equation?

[eljose]

Let be a differential equation :

$$y^{(n)}=F(t,y,\dot y ,\ddot y , \dddot y,..., y^{n-1})$$

then if we propose a Lagrangian so its euler-Lagrange equation gives:

$$\sum_{k=0}^{n}(-1)^{n}\frac{d^{2}}{dt^{2}}(\frac{\partial ^{n} L}{\partial \ y^{n} })=0$$

The differential equation can be derived from a variational principle...then my question is how could we applycontinous group theory to solve this differential equation thanks,or for example if i know that the differential equation has as a particular solution y(t)=exp(rt) where r can be a real or complex parameter.

 

[fresh_42]

The answer is called Lie theory, esp. Noether's theorem.
See e.g. https://www.amazon.com/dp/0387909699/?tag=pfamazon01-20