Wikipedia:Reference desk/Archives/Mathematics/2018 August 8

= August 8 =

Higher order derivatives
(refactored from Talk:Lagrangian mechanics and Talk:Lagrange multiplier)

Hi I've been trying to find an answer to my question but I can't find it anywhere so I'd thought I'd ask here...

Given a functional $$L(q,\dot{q},t)$$ and a constraint of the form $$g(q,t)=C$$ one can form a Lagrangian of the form $$L(q,\dot{q},t)+\lambda(g(q,t)-C)$$ and get the Euler-Lagrange equation from varying the corresponding action. i.e. $$ 0=\delta S\rightarrow\frac{\partial L}{\partial q}-\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)=\lambda\frac{\partial g}{\partial q} $$

But what if the constraint equation depended on higher order derivatives, i.e. $$g(q,\dot{q},\cdots,t)=C$$?

Can I still form a Lagrangian of the form $$L(q,\dot{q},\cdots,t)+\lambda(g(q,\dot{q},\cdots,t)-C)$$ and get the Euler-Lagrange equation from varying the corresponding action. i.e. $$ 0=\delta S\rightarrow\frac{\partial L}{\partial q}-\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)+\cdots=\lambda\left(\frac{\partial g}{\partial q}-\frac{d}{dt}\left(\frac{\partial g}{\partial\dot{q}}\right)+\cdots\right) $$

— Preceding unsigned comment added by 2604:3D09:A47F:F630:8C32:405D:8927:B5CD (talk)


 * Systems whose constraints depend on velocities and higher order derivatives are generally nonholonomic systems where the state of the system depends on the path history. These are generally harder to solve than holonomic systems. But the techniques of variational calculus and Lagrange multipliers should be able to be used here, too. -- 17:41, 8 August 2018 (UTC)

Ok so they can be harder to solve but theoretically possible using the same outline as the Lagrange multipliers technique for holonomic systems? Thanks for your help. One more question, what exactly makes it harder to solve? Could you give an example please? — Preceding unsigned comment added by 2604:3D09:A47F:F630:5148:5614:E8C6:8D1B (talk)
 * Please sign your posts with  . See Signatures. Richard-of-Earth (talk) 19:21, 8 August 2018 (UTC)
 * Also, please, take into account that $$\lambda=\lambda(t)$$ is a function, not a number. Ruslik_ Zero 20:53, 8 August 2018 (UTC)

Hi I heard about that but never seen it in an example. Would i be right to guess that it is a function of time explicitly when the Lagrangian is a function of time explicitly? Can you give an example cause I never seen $$\lambda$$ to be a function of time....thanks — Preceding unsigned comment added by 2604:3d09:a47f:f630:5148:5614:e8c6:8d1b (talk) 22:14, 8 August 2018 (UTC)