- #1
jonz13
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This is from a past paper (from a lecturer I don't particularly understand)
a) {4 marks} Find the Euler-Lagrange equations governing extrema of I subject to J=\text{constant}, whereI=\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})
andJ=\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.
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}
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).
Homework Statement
a) {4 marks} Find the Euler-Lagrange equations governing extrema of I subject to J=\text{constant}, whereI=\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})
andJ=\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.
Homework 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}
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).