# Clarifications regarding the Maupertuis/Jacobi principle

I'm slightly confused regarding the Maupertuis' Principle. I have read the Wikipedia page but the confusion is even in that derivation. So, say we have a Lagrangian described by $$\textbf{q}=(q^1,...q^n)$$ of the form $$L = \frac{1}{2} \Big(\frac{d\textbf{q}}{dt}\Big)^2 - V(\textbf{q}).$$ Now we want to define a 'path' P, in the configuration space $$\textbf{q}(s)$$ parameterized by $$s$$, which is defined using the following equation $$ds^2:=\sum_{j=1}^n (dq^j)^2$$ with the condition $$\textbf{q}(s_f)=\textbf{q}_f$$ and $$\textbf{q}(0)=\textbf{q}_0$$. We now define the integral $$B[P]= \int_0^{s_f}\: ds\: \sqrt{2[V(\textbf{q}(s))-V(\textbf{q}_0)]}.$$

Now I have read that, (source: Erick Weinberg: Classical solutions in QFT Chapter 9 page 179),

'In classical mechanics Jacobi’s principle tells us that for a system described by a Lagrangian of the form mentioned earlier, a path from $$q_0$$ to $$q_f$$ that minimizes the above integral $$B[P]$$ gives a solution of the equations of motion whose time evolution is determined by $$\frac{1}{2}\Big(\frac{d\textbf{q}}{dt}\Big)^2 = V(\textbf{q}_0)-V(\textbf{q})$$'.

This is the exact statement that I want to prove.

If I prove this then I can verify his other claim that if the path that will minimise $$B$$ is $$\textbf{q}(s)$$ then this is the same path that minimises the action integral $$S=\int dt L$$ with $$\textbf{q}(t_0)=\textbf{q}_0$$ and $$\textbf{q}(t_f)=\textbf{q}_f$$. Because I can differentiate the 'energy conservation equation' which will be the Equations of Motion I am familiar with.

So to prove the statement I cannot use the energy conservation equation right? So then I use a similar approach for deriving the Euler Lagrange equations $$\delta B= \int_0^{s_f} ds \frac{-\sum_j \frac{\partial V}{\partial q^j} \delta q_j}{\sqrt{2[V(\textbf{q}(s))-V(\textbf{q}_0)]}}.$$ Since the $$\delta q_j$$'s are independent, this gives me $$\frac{\partial V}{\partial q_j}=0$$ which is incorrect. But I am trying to find where I am going wrong. I am mimicking the Euler-Lagrange equation, thinking of $$s$$ in some way as $$t$$ and $$L$$ as the term in the square root in the expression for B. There are no $$\frac{dq}{ds}$$ terms and $$V(\textbf{q}_0)$$ is a constant. To recap I want to derive the energy conservation equation that the path that minimises $$B[P]$$ should satisfy. Any leads? I suspect there may be something wrong in the way I've set up the question, like a wrong assumption or something.

The main point is that in Maupertuis' principle one restricts to virtual paths with constant and same fixed energy $$E$$ but with free endpoint times $$t_i$$ and $$t_f$$. OP's variations don't respect this. For more information, see e.g. Refs. 1 & 2.
2. L.D. Landau & E.M. Lifshitz, Mechanics, vol. 1, 1976; $$\S 44$$.