# Calculus of variations: Euler-Lagrange

1. Apr 25, 2012

### jonz13

This is from a past paper (from a lecturer I don't particularly understand)
1. The problem statement, all variables and given/known data
a) {4 marks} Find the Euler-Lagrange equations governing extrema of $I$ subject to $J=\text{constant}$, where$$I=\int_{t_1}^{t_2}\text{d}t \frac{1}{2}(x\dot{y}-y\dot{x})=\int f(t,x,y,\dot{x},\dot{y})$$
and$$J=\int_{t_1}^{t_2}\text{d}t (\dot{x}^2+\dot{y}^2)=\int g(t,x,y,\dot{x},\dot{y})$$
b) {8 marks} show that for the problem in part a) the extremal curves satisfy $(x-\alpha)\dot{x}+(y-\beta)\dot{y}=0$ where $\alpha$ and $\beta$ are constants.
2. Relevant equations
From an earlier part of the question I have two Euler-Lagrange equations (one differentiating w.r.t. $y$ aswell)$$\frac{\partial (f-\lambda g)}{\partial x}-\frac{\mathrm{d} }{\mathrm{d} t}\frac{\partial (f-\lambda g)}{\partial \dot{x}}=0$$
and I think I can write, due to no dependence on $t$ (another one with $y$ again)$$(f-\lambda g) - \dot{x}\frac{\partial (f-\lambda g)}{\partial \dot{x}}=\mathrm{constant}$$
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 $x(t) \text{ and } y(t)$ and get a few dead ends, I'm not really sure what approach to use (a definite answer to part a) would probably help).

2. Apr 26, 2012

### vela

Staff Emeritus
For part (a), you want to take the specific f and g you've been given and substitute them into the general Euler-Lagrange equation you cited as a relevant equation.