This is from a past paper (from a lecturer I don't particularly understand)(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

a) {4 marks} Find the Euler-Lagrange equations governing extrema of [itex] I [/itex] subject to [itex] J=\text{constant} [/itex], where[tex]I=\int_{t_1}^{t_2}\text{d}t \frac{1}{2}(x\dot{y}-y\dot{x})=\int f(t,x,y,\dot{x},\dot{y})[/tex]

and[tex]J=\int_{t_1}^{t_2}\text{d}t (\dot{x}^2+\dot{y}^2)=\int g(t,x,y,\dot{x},\dot{y})[/tex]

b) {8 marks} show that for the problem in part a) the extremal curves satisfy [itex](x-\alpha)\dot{x}+(y-\beta)\dot{y}=0[/itex] where [itex]\alpha[/itex] and [itex]\beta[/itex] are constants.

2. Relevant equations

From an earlier part of the question I have two Euler-Lagrange equations (one differentiating w.r.t. [itex]y[/itex] aswell)[tex]\frac{\partial (f-\lambda g)}{\partial x}-\frac{\mathrm{d} }{\mathrm{d} t}\frac{\partial (f-\lambda g)}{\partial \dot{x}}=0[/tex]

and I think I can write, due to no dependence on [itex]t[/itex] (another one with [itex]y[/itex] again)[tex](f-\lambda g) - \dot{x}\frac{\partial (f-\lambda g)}{\partial \dot{x}}=\mathrm{constant}[/tex]

3. The attempt at a solution

For part a) I'm not particularly sure what I am being asked for, or if the equation above is the answer. for part b) I have tried subbing into the equations above and can get out linear equations for [itex]x(t) \text{ and } y(t)[/itex] and get a few dead ends, I'm not really sure what approach to use (a definite answer to part a) would probably help).

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Calculus of variations: Euler-Lagrange

**Physics Forums | Science Articles, Homework Help, Discussion**