Want to derive the euler- lagrange equations using newtons second law?

Question

Anyone know where I can find a good derivation?

Answer 1

(NB dots .... are used as formating place holders except where a product a.b is implied)



The calculus of variations may be used to find the stationary values of an integral of the form: -

..... x₁

I = ∫ f(y, y').dx

.... x₀



where f(y, y') is a function of y and its first derivative. If we consider a small variation δy(x) in the function f(y), subject to the condition that the values of y at the end-points are unchanged: -



δy(x₀) = 0, δy(x₁) = 0



To first order, the variation in f(y, y') is: -



δf = δf.δy + δf.δy'

....... __ ..... __

....... δy ..... δy'



Where: -



y' = d(δy)

..... ____

..... dx



Thus, the variation of the integral I is: -



..... x₁

δI = ∫ |δf.δy + δf. . d δy. |.dx

.... x₀|__ ..... __ . _ ..... |

........ |δy ..... δy' dx .... |



In the second term, we may integrate by parts. The integrated term, namely: -



| δf.δy |x₁

| __ .. |

|δy' .. |x₀



vanishes at the limits because of the conditions δy(x₀) = 0, δy(x₁) = 0. Hence, the integral becomes: -



..... x₁

δI = ∫ |δf. - . d . | δf. | |.δy(x).dx

.... x₀|__ ... __ | __ | |

........ |δy .. dx | δy' | |



Thus, for I to be stationary, the variation δI must vanish for a small variation δy(x). Thus, we requires: -



δf. - . d . | δf. | = 0

__ ... __ | __ |

δy ... dx | δy' |



This known as the Euler-Lagrange equation!





Now, the Lagrangian function L, in terms of coordinates x, y, z, is: -



L = T - V = m | | dx |² + | dy |² + | dz |² | - V(x, y, z)

................. _ | | __ | ... | __ | ... | __ |. |

................. 2 | | dt | .... | dt | .... | dt |. |





Where derivatives for (say) z, ż are: -



∂L = mż = p(z), ∂L =F(z) . = . ma(z)

__ .................. __

∂ż .................. ∂z.



Or, Newton's second law of motion!





Thus, the equation of motion for z components is: -



dp(z) = F(z)

____

dt



This may be written as: -



d . | ∂L | = ∂L

__ | __ | .. __

dt. | ∂ż | .. ∂z



Which is the Euler-Lagrange equation for the Action integral of Hamilton's principle of least action.



..... t₁

I = ∫ L.dt

.... t₀

Answer 2

you have to decide whether your FBD will be one system or two.is it in equilibrium? I don't know if one of the pulleys is hanging off a surface and the other one is on it. But if it is, then the mass on the table would have gravity, normal force and tension make your x axis parallel with the surface. for the hanging pulley, point the x-axis downward so that the two forces, tension and mass, will equal each other in the x-axis, t=mg

Answer 3

wikipedia