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## Homework Statement

So I'm deriving Lagrange's equations using Hamilton's principle which states that the motion of a dynamical system follows the path, consistent with any constraints, that minimise the time integral over the lagrangian [itex]L = T-U[/itex], where [itex]T[/itex] is the kinetic energy and [itex]U[/itex] is the potential energy.

We define the lagrangian as [itex]L = L(q_j,\dot{q}_j,t)[/itex]. Now I want to let [itex]q_j = q_j^{(0)}+\delta q_j[/itex], where [itex]\delta q_j[/itex] is the variation of [itex]q_j[/itex]. We also define

[tex]

S=\int_{t_1}^{t_2}L(q_j,\dot{q}_j,t)dt

[/tex]

So according to Hamilton's principle we would now like to minimise [itex]S[/itex]. At extremum we have [itex]\delta S = 0[/itex], i.e. the variation of S is zero.

**Now, my problem:**

My first experience with calculus of variations was to find Euler's equation. We considered then the functional

[tex]

J = \int_{x1}^{x2}f(y,y',x)dx

[/tex]

and our goal was to find the function [itex]y[/itex] that minimise S. To do this we let [itex]y(x,\alpha) = y(x)+\alpha\eta(x)[/itex], and set [itex]\frac{\partial J}{\partial\alpha}|_{\alpha=0}=0[/itex], where [itex]\eta[/itex] is some arbitrary function. This would give us an equation in [itex]y(x)[/itex] since we evaluate at [itex]\alpha=0[/itex].

So, returning to the derivation of Lagrange's equations. I set to find [itex]q_j[/itex] that minimise [itex]S[/itex] in similar fashion as we did for [itex]J[/itex]. But this time I do not have any [itex]\alpha[/itex] that I can put to zero. Should I instead take [itex]\delta S|_{\delta q_j = 0}=0[/itex]? For unless I have understood it wrong, it is actually [itex]q_j^{(0)}[/itex] that we want to find, right? I mean it seems strange to find [itex]q_j[/itex] to be some curve [itex]q_j^{(0)}[/itex] added by some arbitrary variations.

At the same time, I am not sure whether evaluating [itex]\delta q_j[/itex] at makes sense. What increase my doubts is that neither in my text book (Marion and Thornton) nor at any lecture has it been emphasized that we evaluate with [itex]\delta q_j = 0[/itex].

Anyhow, I am thankful for answers.