Let $L(q_i,\dot{q_i},t)$ be the(a?) Lagrangian of a physical system. Assume that the gen. coordinates $q_i$ transform under a certain Group G as $q_i\rightarrow q_i'=f_i(q_j,\theta_k)$ where $f_i$ are functions depending upon $q_i$'s and parameters $\theta_k$'s satisfying appropriate conditions (invertibility etc.)

Does the Lagrangian transform as $L\rightarrow L'=L(q_i',\dot{q_i}',t)$ or as $L\rightarrow L'=L(q_i(q_i'),\dot{q_i}(q_i',\dot{q_i}'),t)$ or in some other way?

What does it mean when we say that $L$ is invariant under transformations of group G? Do we mean that the numerical value of $L$ is unchanged at any t? Or the form of $L'(q_i',\dot{q_i}',t)$ is same as that of $L(q_i,\dot{q_i},t)$? Or something else?

For example, let's take the Lagrangian density of complex scalar field (I seem to be more comfortable with field theory formulation) - $$\mathcal{L}=\left(\partial_{\mu} \Phi\right)^{\dagger} \partial^{\mu} \Phi-\left(\Phi^{\dagger} \Phi\right)$$ under group G=$U(1)$, $$\begin{aligned} \Phi \rightarrow \Phi^{\prime} &=e^{-i \alpha} \Phi \\ \Phi^{\dagger} \rightarrow \Phi^{\prime \dagger} &=e^{i \alpha} \Phi^{\dagger} \end{aligned}$$ What would be the expression of transformed $\mathcal{L\rightarrow L}'$=?

Again, what does "invariance of $\mathcal{L}$ under $U(1)$" mean, both physically and mathematically?


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