Homework Statement
A mountain is modeled by z=f(x,y) which is a known function.
a) What are the differential equations for x(t) and y(t) that minimize the distance between 2 points.
b) If
z=f(x,y)=(\sin^2 2\pi x)(\sin^2 2\pi y)
Solve the equations.
Homework Equations
The...
I read in hand and finch (analytical mechanics) that if you assume you have a lagrangian:
L=(\phi,\nabla\phi,x,y,z)
Then what does the euler lagrange equation look like in vector notation. I know that if you have a function with more than 1 independent variable then the euler-lagrange...
actually I have a typo in the original integral it should be \left(\frac{n[y]}{n_0}\right)^2 and I figured out what the problem was. To get the expression we had to take a square root. So there should be a +/- sign. If we use the + sign we don't get a solution because RHS < 0 and LHS > 0, but we...
I know mathematic gives an analytical expression in terms of \alpha but the equation seems to have no solution. (the integral is negative and the left hand side is positive)
Homework Statement
This is from hand and finch. We proved in the previous problem that (using euler lagrange equation):
x=\int_0^y\frac{dy}{\sqrt{\left(\frac{n[y]}{n_0}\right)-1}}
where n_0 is the refractive index at y=0 and x=0. The ray enters horizontally.
As an actual computation...
Homework Statement
So if +x points downward and +y points rightwards then the functional that needs to be minimized is:
\sqrt{2g}T[y]=\int_{x_0}^{x_1}\frac{dx}{\sqrt{x}}\sqrt{1+\left(\frac{dy}{dx}\right)^2}
Homework Equations
I think we just have to use the Euler lagrange...
I think that I had a problem a year or two back where I think we solved this problem. However, I think I must be mistaken because the equation of motion cannot be analytically solved.
I don't have the solution.
I still am thinking about what would happen when the ladder leaves the wall (I...
Oh yea sorry ... I missed that part. We do know the initial angle. My question was what is the physical condition that determines the angle that the ladder flys off given the initial angle. (that is, what mathematical constraint do we need in conjunction with the equation of motion to determine...
Homework Statement
A ladder of length L is leaning against a frictionless wall. The floor is frictionless too. It starts to fall. At what angle will it leave the wall.
Homework Equations
The Attempt at a Solution
I can derive the equation of motion...
Hey guys,
So I have to make a presentation on this topic. Does anyone of you know of any recent applications of this phenomenon or helpful introductory paper/article? I'm doing my own independent research too but thought that you guys might know of a very helpful resource/idea that I can look...
Hi,
I am an undergrad majoring in Physics. I want to be a theoretical physicist. What language would help me more (in any way)? German or Chinese?
Thanks,
Homework Statement
This is from baby rudin:
If E\subset X and if f is a function defined on X, the restriction of f to E is the function g whose domain of definition is E, such that g(p)=f(p) for p\in E. Define f and g on R^2 by: f(0,0)=g(0,0)=0, f(x,y)=\frac{xy^2}{x^2+y^4}...
Firstly note that for any 2x and y, we want the rectangle to have two of it sides running over the base and height of the triangle and one of its corners touching the hypotenuse to maximize the area.
With this observation, we get a relation between 2x and y. (note that I'm putting P over A...
Could you explain the problem a little more. How does the triangle relate to the rectangle?
I'm guessing this is a problem in lagrange multipliers, where we need to maximize A = 2xy, with a certain restraint on the variables that needs to yet describe.
Yea using calculus, we can can show that \alpha^\frac{1}{p} is the minimum point of this function in the positive domain and since this function is positive for x>0, \forall n\,(\,x_n\geq\alpha^\frac{1}{p}\,) and the proof is complete.
I was just wondering whether we can do it without calculus...