Let be a path-independent function and be a path-dependent function; then
— 1.5.1 Varying a path
— Structure and Interpretation of Classical Mechanics
Assume that is a path-independent function, so that we can use Eq. 1.26:
Assume that is a path-independent function, so that any value of depends on the value of at that moment only, instead of depending on the whole path :
What kind of product is it here? Is it just a dot product? Probably not.
— Me@2019-10-12 03:42:01 PM
2019.10.13 Sunday (c) All rights reserved by ACHK
llamaz 1 hour ago [-]
I think the calculus of variations might be a better approach to introducing ODEs in first year.
You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of , you end up with an ODE (viz. the Euler-Lagrange equation).
This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system.
Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you’ve motivated ODEs and given the students a reason to solve ODEs with constant coefficients.
— Hacker News
2019.10.02 Wednesday ACHK
— Me@2017-11-09 01:26:59 PM
2019.10.01 Tuesday (c) All rights reserved by ACHK