What is Calculus of variations: Definition and 154 Discussions
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.
Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.
I have this question,
Express the length of a given curve r = r(\theta) in radial co-ordinates. Using the Variational principle derive the shortest path between two points is a line.
Ive drawn a picture with two angles (measured from the x-axis) \theta_1 and \theta_2 so that r(\theta_1) =...
I have another difficult question regarding calculus of variations.
A particle travels in the (x,y) plane has a speed u(y) that depends on the distance of the particle from the x-axis. The direction of travel subtends an angle \theta with the x-axis that can be controlled to give the minimum...
I am facing a difficult integral here for calculus of variations. The question reads:
Find the extremum to the integral:
I[y(x)] = \int_{Q}^{P} (dy/dx)^2(1+dy/dx)^2 dx
where P = (0,0) and Q = (1,2)
Hi all,
I seeking some advice about the calculus of variations.
I am an undergraduate and i am enrolled in a topic of the above mentioned. After successfully completing the requirments for the topic, 3 weeks after commencement i am feeling way out of my depth. I understand that the calculus...