In Lagrangian mechanics we have the Euler-Lagrange equations, which are defined as
$$\frac{d}{dt}\Bigg(\frac{\partial L}{\partial \dot{q}_j}\Bigg) – \frac{\partial L}{\partial q_j} = 0,\quad j = 1, \ldots, n.$$
In Hamiltonian mechanics we have that
$$\dot{q}_j = \frac{\partial H}{\partial p_j}, \quad \dot{p}_j = -\frac{\partial H}{\partial q_j}, \quad j = 1,\ldots n,$$
where
$$p_i = \frac{\partial L}{\partial \dot{q}_i}.$$
What has been confusing me is that the coordinates $p,\,\dot{p},\,q,\,\dot{q}$ are all actually paths, meaning they are functions of time. Thus the Lagrangian $L$ and the Hamiltonian $H$ are actually functionals, and I would suspect that we need to use a variational derivative instead of an ordinary one above.
However despite this some textbooks on classical mechanics treat $p,\,\dot{p},\,q,\,\dot{q}$ as coordinates and not as paths, so they take derivatives and partial derivatives and not variational derivatives.
I am getting confused between these two viewpoints. How is it justified to take derivatives as if $p,\,\dot{p},\,q,\,\dot{q}$ were coordinates and not functions? When should one use the variational derivative instead?
Best Answer
The Lagrangian is a function, not a functional. The action is a functional, and is defined as
$$S[q; t_0,t_1] = \int_{t_0}^{t_1} L\big(q(t), \dot q(t), t\big) \mathrm dt$$ The partial derivatives which appear in the Euler-Lagrange equations are slot derivatives. One might write $$\big(\partial_1 L\big) (a,b,c) = \lim_{\epsilon\rightarrow 0} \frac{L(a+\epsilon,b,c)-L(a,b,c)}{\epsilon}$$ $$\big(\partial_2 L\big) (a,b,c) = \lim_{\epsilon\rightarrow 0} \frac{L(a,b+\epsilon,c)-L(a,b,c)}{\epsilon}$$ $$\big(\partial_3 L\big) (a,b,c) = \lim_{\epsilon\rightarrow 0} \frac{L(a,b,c+\epsilon)-L(a,b,c)}{\epsilon}$$
Demanding that the action be stationary for arbitrary smooth perturbations $\eta$ which vanish at $t_0$ and $t_1$ is to demand that $$\frac{d}{d\epsilon} S[q+\epsilon\eta;t_0,t_1] \bigg|_{\epsilon=0} $$ $$= \int_{t_0}^{t_1} \left[ \big(\partial_1L\big)(q(t), \dot q(t),t) -\frac{d}{dt} \big(\partial_2 L\big)(q(t),\dot q(t), t)\right]\eta(t)\ \mathrm dt = 0$$ which implies that the integrand must vanish - hence the EL equations.
From a terminology standpoint, if we can write $$\frac{d}{d\epsilon}S[q+\epsilon \eta;t_0,t_1] = \int_{t_0}^{t_1}\bigg(\ldots\bigg) \eta(t) \mathrm dt $$ then we call $\big(\ldots\big)$ the variational (or functional) derivative of $S$, and typically write it as $\delta S/\delta q$ or something similar.
The confusion arises when we write $$\big(\partial_1L\big)(q(t),\dot q(t), t) \equiv \frac{\partial L}{\partial q} $$ This is confusing, but as the number of "slots" of $L$ increases, there is really no viable alternative that isn't notationally horrific. You just have to understand what's being talked about.