Definition of order of a partial differential equation

[Kashmir]

How is the order of a partial differential equation defined?

This is said to be first order: $\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0$

And this second order :$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q_{i}}}\right)-\frac{\partial L}{\partial q_{i}}=0$

What's the proper definition?

Thank you

 

[mathman]

The dot over the q makes the second line second order.

 

[Kashmir]

The dot over the q makes the second line second order.

Thank you, could you please tell me the definition of second order for partial D.E ?

 

[Astronuc]

The order for partial D.E., like an ordinary D.E., refers to the highest order derivative in the D.E.

For example, a D.E. with $\partial^2{y}/\partial{x}^2$, would be 2nd order if no higher derivatives were present, and similarly with d²y/dx².

Unfortunately, I've gone blank about mixing time and position/space derivatives, and ordinary with partial.

There are many online tutorials concerning DEs, both ODE and PDE.
https://users.aber.ac.uk/ruw/teach/260/classification.php
https://tutorial.math.lamar.edu/classes/calciii/highorderpartialderivs.aspx

https://www.math.toronto.edu/jko/APM346_summary_1_2020.pdf
https://www.csc.kth.se/utbildning/kth/kurser/DN1213/numme06/utdelat/kap10.pdf