# Landau/Lifshitz action as a function of coordinates [duplicate]

In Landau/Lifshitz' "Mechanics", $$\S43$$, 3ed, the authors consider the action of a mechanical system as a function of its final time $$t$$ and its final position $$q$$. They consider paths originating at some point $$q^{(1)}$$ at a time $$t_1$$, and terminating at the same point $$q$$ after different times $$t$$. The total derivative of the action is then

$$\frac{dS}{dt}=\frac{\partial S}{\partial t}+\sum_i\frac{\partial S}{\partial q_i}\dot{q}_i\tag{p.139}.$$

The authors then claim that $$\frac{\partial S}{\partial q_i}$$ can be replaced by $$p_i$$. I don't understand this step. L+L derived this formula for $$p_i$$ by considering paths starting from the same point $$q^{(1)}$$, but passing through different endpoints after a common time interval. Why should $$p_i=\frac{\partial S}{\partial q_i}\tag{43.3}$$ continue to hold when the action is varied in a different way?

## 1 Answer

As defined in your question $$p(q,t) = \frac{\partial S(q,t)}{\partial q}$$ is the momentum at time $$t$$ of the classical (least action) path which passes through the points $$(q_1,t_1)$$ and $$(q,t)$$. Changing the end point $$(q,t)$$ will generally change the corresponding path but $$\frac{\partial S}{\partial q}$$ will always be the momentum at time $$t$$ for whatever path that happens to be.

That being said, in your total derivative equation the action is being evaluated on a classical path uniquely specified by certain end points, $$q(t) \equiv q_{(q_1,t_1,q_2,t_2)}(t)$$, to make it a function of only time $$S(q(t), t)$$. So by definition the classical path passing through the points $$(q_1,t_1)$$ and $$(q(t), t)$$ will be this same path for all $$t$$. This means that for all times $$t$$, $$p(t) = \frac{\partial S(q(t),t)}{\partial q}$$ is the momentum of the classical path at time $$t$$.