现在时间是:
当前位置:首 页 >> Basic Informations>> 图像处理>> 文章列表

Euler–Lagrange equation

作者:   发布时间:2013-01-28 10:52:10   浏览次数:994

 

Euler–Lagrange equation-(only for study)

From Wikipedia, the free encyclopedia
  (Redirected from Euler-Lagrange equation)
 

In calculus of variations, the Euler–Lagrange equationEuler's equation,[1] or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph Louis Lagrange in the 1750s.

Because a differentiable functional is stationary at its local maxima and minima, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing (or maximizing) it. This is analogous to Fermat's theorem in calculus, stating that where a differentiable function attains its local extrema, itsderivative is zero.

In Lagrangian mechanics, because of Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler–Lagrange equation for theaction of the system. In classical mechanics, it is equivalent to Newton's laws of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations.

Contents

  [hide

[edit]History

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.

Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.[2]

[edit]Statement

The Euler–Lagrange equation is an equation satisfied by a function, q, of a real argument, t, which is a stationary point of the functional

displaystyle S(q) = int_a^b L(t,q(t),q'(t)), mathrm{d}t

where:

  • q is the function to be found:
    begin{align}
    q colon [a, b] subset mathbb{R} & to     X 
    t & mapsto x = q(t)
    end{align}
such that q is differentiable, q(a) = xa, and q(b) = xb;
  • q′ is the derivative of q:
    begin{align}
    q' colon [a, b] & to     T_{q(t)}X 
    t & mapsto v = q'(t)
    end{align}
TX being the tangent bundle of X (the space of possible values of derivatives of functions with values in X);

The Euler–Lagrange equation, then, is given by

L_x(t,q(t),q'(t))-frac{mathrm{d}}{mathrm{d}t}L_v(t,q(t),q'(t)) = 0.

where Lx and Lv denote the partial derivatives of L with respect to the second and third arguments, respectively.

If the dimension of the space X is greater than 1, this is a system of differential equations, one for each component:

frac{partial L(t,q(t),q'(t))}{partial x_i}-frac{mathrm{d}}{mathrm{d}t}frac{partial L(t,q(t),q'(t))}{partial v_i} = 0
quad text{for } i = 1, dots, n.

[edit]Examples

A standard example is finding the real-valued function on the interval [ab], such that f(a) = c and f(b) = d, the length of whose graph is as short as possible. The length of the graph of f is:

 ell (f) = int_{a}^{b} sqrt{1+(f'(x))^2},mathrm{d}x,

the integrand function being L(xyy′) = 1 + y′ ² evaluated at (xyy′) = (xf(x), f′(x)).

The partial derivatives of L are:

frac{partial L(x, y, y')}{partial y'} = frac{y'}{sqrt{1 + y'^2}} quad text{and} quad
frac{partial L(x, y, y')}{partial y} = 0.

By substituting these into the Euler–Lagrange equation, we obtain


begin{align}
frac{mathrm{d}}{mathrm{d}x} frac{f'(x)}{sqrt{1 + (f'(x))^2}} &= 0 
frac{f'(x)}{sqrt{1 + (f'(x))^2}} &= C = text{constant} 
Rightarrow f'(x)&= frac{C}{sqrt{1-C^2}} := A 
Rightarrow f(x) &= Ax + B
end{align}

that is, the function must have constant first derivative, and thus its graph is a straight line.

[edit]Classical mechanics

[edit]Basic method

To find the equations of motions for a given system, one only has to follow these steps:

  • From the kinetic energy T, and the potential energy V, compute the Lagrangian L = T - V.
  • Compute frac{partial L}{partial q}.
  • Compute frac{partial L}{partial dot{q}} and from it, frac{d}{dt}frac{partial L}{partial dot{q}}. It is important that dot{q} be treated as a complete variable in its own right, and not as a derivative.
  • Equate frac{partial L}{partial q} = frac{d}{dt}frac{partial L}{partial dot{q}}. This is the Euler–Lagrange equation.
  • Solve the differential equation obtained in the preceding step. At this point, dot{q} is treated "normally". Note that the above might be a system of equations and not simply one equation.

[edit]Particle in a conservative force field

The motion of a single particle in a conservative force field (for example, the gravitational force) can be determined by requiring the action to be stationary, by Hamilton's principle. The action for this system is

S = int_{t_0}^{t_1} L(t, mathbf{x}(t), mathbf{dot{x}}(t)),mathrm{d}t

where x(t) is the position of the particle at time t. The dot above is Newton's notation for the time derivative: thus (t) is the particle velocity, v(t). In the equation above, L is theLagrangian (the kinetic energy minus the potential energy):

L(t, mathbf{x}, mathbf{v}) = frac{1}{2}m sum_{i=1} ^{3} v_i^2 - U(mathbf{x}),

where:

  • m is the mass of the particle (assumed to be constant in classical physics);
  • vi is the i-th component of the vector v in a Cartesian coordinate system (the same notation will be used for other vectors);
  • U is the potential of the conservative force.

In this case, the Lagrangian does not vary with its first argument t. (By Noether's theorem, such symmetries of the system correspond to conservation laws. In particular, the invariance of the Lagrangian with respect to time implies the conservation of energy.)

By partial differentiation of the above Lagrangian, we find:

frac{partial L(t,mathbf{x},mathbf{v})}{partial x_i} = -frac{partial U(mathbf{x})}{partial x_i} = F_i (mathbf{x})quad text{and} quad
frac{partial L(t,mathbf{x},mathbf{v})}{partial v_i} = m v_i = p_i,

where the force is F = −U (the negative gradient of the potential, by definition of conservative force), and p is the momentum. By substituting these into the Euler–Lagrange equation, we obtain a system of second-order differential equations for the coordinates on the particle's trajectory,

F_i(mathbf{x}(t)) = frac{mathrm d}{mathrm{d}t} m dot{x}_i(t) = m ddot{x}_i(t),

which can be solved on the interval [t0t1], given the boundary values xi(t0) and xi(t1). In vector notation, this system reads

mathbf{F}(mathbf{x}(t)) =  mmathbf{ddot x}(t)

or, using the momentum,

 mathbf{F} = frac {mathrm{d}mathbf{p}} {mathrm{d}t}

which is Newton's second law.

[edit]Variations for several functions, several variables, and higher derivatives

[edit]Single function of single variable with higher derivatives

The stationary values of the functional


I[f] = int_{x_0}^{x_1} mathcal{L}(x, f, f', f'', dots, f^{(n)})~mathrm{d}x ~;~~
f' := cfrac{mathrm{d}f}{mathrm{d}x}, ~f'' := cfrac{mathrm{d}^2f}{mathrm{d}x^2}, ~
f^{(n)} := cfrac{mathrm{d}^nf}{mathrm{d}x^n}

can be obtained from the Euler–Lagrange equation[3]


cfrac{partial mathcal{L}}{partial f} - cfrac{mathrm{d}}{mathrm{d} x}left(cfrac{partial mathcal{L}}{partial f'}right) + cfrac{mathrm{d}^2}{mathrm{d} x^2}left(cfrac{partial mathcal{L}}{partial f''}right) - dots +
(-1)^n cfrac{mathrm{d}^n}{mathrm{d} x^n}left(cfrac{partial mathcal{L}}{partial f^{(n)}}right)  = 0

under fixed boundary conditions for the function itself as well as for the first n-1 derivatives (i.e. for all f^{(i)}, i in {0, ..., n-1}). The endpoint values of the highest derivative f^{(n)} remain flexible.

[edit]Several functions of one variable

If the problem involves finding several functions (f_1, f_2, dots, f_n) of a single independent variable (x) that define an extremum of the functional


I[f_1,f_2, dots, f_n] = int_{x_0}^{x_1} mathcal{L}(x, f_1, f_2, dots, f_n, f_1', f_2', dots, f_n')~mathrm{d}x
~;~~ f_i' := cfrac{mathrm{d}f_i}{mathrm{d}x}

then the corresponding Euler–Lagrange equations are[4]


begin{align}
cfrac{partial mathcal{L}}{partial f_i} - cfrac{d}{dx}left(cfrac{partial mathcal{L}}{partial f_i'}right) = 0
end{align}

[edit]Single function of several variables

A multi-dimensional generalization comes from considering a function on n variables. If Ω is some surface, then


I[f] = int_{Omega} mathcal{L}(x_1, dots , x_n, f, f_{x_1}, dots , f_{x_n}), mathrm{d}mathbf{x},! ~;~~
f_{x_i} := cfrac{partial f}{partial x_i}

is extremized only if f satisfies the partial differential equation

 frac{partial mathcal{L}}{partial f} - sum_{i=1}^{n} frac{partial}{partial x_i} frac{partial mathcal{L}}{partial f_{x_i}} = 0. ,!

When n = 2 and mathcal{L} is the energy functional, this leads to the soap-film minimal surface problem.

[edit]Several functions of several variables

If there are several unknown functions to be determined and several variables such that


I[f_1,f_2,dots,f_m] = int_{Omega} mathcal{L}(x_1, dots , x_n, f_1, dots, f_m, f_{1,1}, dots , f_{1,n},  dots, f_{m,1}, dots, f_{m,n}) , mathrm{d}mathbf{x},! ~;~~
f_{j,i} := cfrac{partial f_j}{partial x_i}

the system of Euler–Lagrange equations is[3]


begin{align}
frac{partial mathcal{L}}{partial f_1} - sum_{i=1}^{n} frac{partial}{partial x_i} frac{partial mathcal{L}}{partial f_{1,i}} & = 0 
frac{partial mathcal{L}}{partial f_2} - sum_{i=1}^{n} frac{partial}{partial x_i} frac{partial mathcal{L}}{partial f_{2,i}} & = 0 
dots quad dots qquad dots  & 
frac{partial mathcal{L}}{partial f_m} - sum_{i=1}^{n} frac{partial}{partial x_i} frac{partial mathcal{L}}{partial f_{m,i}} & = 0.
end{align}

[edit]Single function of two variables with higher derivatives

If there is a single unknown function f to be determined that is dependent on two variables x1 and x2 and if the functional depends on higher derivatives of f up to n-th order such that


begin{align}
I[f] & = int_{Omega} mathcal{L}(x_1, x_2, f, f_{,1}, f_{,2}, f_{,11}, f_{,12}, f_{,22},
dots, f_{,22dots 2}), mathrm{d}mathbf{x} 
& qquad quad
f_{,i} := cfrac{partial f}{partial x_i} ; , quad
f_{,ij} := cfrac{partial^2 f}{partial x_ipartial x_j} ; , ;; dots
end{align}

then the Euler–Lagrange equation is[3]


begin{align}
frac{partial mathcal{L}}{partial f}
& - frac{partial}{partial x_1}left(frac{partial mathcal{L}}{partial f_{,1}}right)
- frac{partial}{partial x_2}left(frac{partial mathcal{L}}{partial f_{,2}}right)
+ frac{partial^2}{partial x_1^2}left(frac{partial mathcal{L}}{partial f_{,11}}right)
+ frac{partial^2}{partial x_1partial x_2}left(frac{partial mathcal{L}}{partial f_{,12}}right)
+ frac{partial^2}{partial x_2^2}left(frac{partial mathcal{L}}{partial f_{,22}}right) 
& - dots
+ (-1)^n frac{partial^n}{partial x_2^n}left(frac{partial mathcal{L}}{partial f_{,22dots 2}}right) = 0
end{align}

[edit]Notes

  1. ^ Fox, Charles (1987). An introduction to the calculus of variations. Courier Dover Publications. ISBN 978-0-486-65499-7.
  2. ^ A short biography of Lagrange
  3. a b c Courant, R. and Hilbert, D., 1953, Methods of Mathematical Physics: Vol I, Interscience Publishers, New York.
  4. ^ Weinstock, R., 1952, Calculus of Variations With Applications to Physics and Engineering, McGraw-Hill Book Company, New York.

[edit]References

[edit]See also

Help improve this page

 
 

Did you find what you were looking for?

Yes
No
 
 
 
 






上一篇:没有了    下一篇:没有了

Copyright ©2019    计算数学达人 All Right Reserved.

技术支持:自助建站 | 领地网站建设 |短信接口 |燕窝 版权所有 © 2005-2019 lingw.net.粤ICP备16125321号 -5