Custom made Lagragian and Help Wanted

[jfy4]

Hi Everyone,

I'm interested in forming Lagrange's equations of motion using a Lagrangian I made up today. It looks like this:
$$\mathcal{L}(\dot{\psi},\psi)=\sqrt{\langle \dot{\psi}_b|C^{\dagger}_bC_a|\dot{\psi}_a\rangle}$$

where $$C^{\dagger}_b$$ is a creation operator for a basis $$b$$ etc... and a dot represents a full derivative with respect to some parametrization $$\tau$$.

Now there is no $$\psi$$ dependence so I believe I am only interested in this part:
$$-\frac{d}{d\tau}\frac{\partial\mathcal{L}}{\partial\dot{\psi}_a}=0$$

I have gotten this far:
$$-\frac{d}{d\tau}\frac{\partial\mathcal{L}}{\partial\dot{\psi}_a}=-\frac{d}{d\tau}\left(\frac{1}{2}\frac{1}{\mathcal{L}}\left(\int \partial_{\dot{\psi}_a}\dot{\psi}_{b}^{\ast}C^{\dagger}_bC_a\dot{\psi}_a d\tau + \int \dot{\psi}_{b}^{\ast}\partial_{\dot{\psi}_a}(C^{\dagger}_bC_a)\dot{\psi}_a d\tau + \int \dot{\psi}_{b}^{\ast}C^{\dagger}_bC_a\partial_{\dot{\psi}_a}\dot{\psi}_a d\tau\right)\right)=0.$$

The first partial turns to 0 i think, the second partial I'm not sure how to interpret, but I think it's zero too because of no $$\dot{\psi}_a$$ dependence and the last partial I believe is 1. Anyone got a comment on accuracy or how to clean it up any better?

Thanks in advance.

 

[TriTertButoxy]

It is not clear from your notation what $\mathcal{L}(\dot{\psi},\psi)=\sqrt{\langle \dot{\psi}_b|C^{\dagger}_bC_a|\dot{\psi}_a\rangle}$ means. Are we to interpret the Lagrangian as being the square-root of some quadratic form of your dynamical variables $\psi$? Typically, the bra and ket notations are reserved for quantum mechanics; not classical lagrangian mechanics.

 

[jfy4]

Yes, I had in mind the scalar product between those two vectors with the square root. After turning the first two partials to zero, I made this progress with the last integral. I integrated by parts:

$$\int\dot{\psi}_{b}^{\ast}C_{b}^{\dagger}C_a d\tau=\psi_{b}^{\ast}C_{b}^{\dagger}C_a-\int \psi_{b}^{\ast}\frac{d}{d\tau}(C_{b}^{\dagger}C_a) d\tau=\psi_{b}^{\ast}C_{b}^{\dagger}C_a-\int \psi_{b}^{\ast}\frac{dC_{b}^{\dagger}}{d\tau}C_a d\tau-\int \psi_{b}^{\ast}C_{b}^{\dagger}\frac{dC_a}{d\tau} d\tau$$

any other suggestions?