1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Why is the time integral of the Lagrangian minimal?

  1. May 31, 2012 #1
    I saw a proof recently that demonstrated that for

    F= ∫L(x, y(x), y'(x))dx, if y(x) is such that F is minimal (no other y(x) could produce a smaller F), then dL/dy - d/dx(dL/(dy/dx)) = 0.

    I understood the proof, and I was able to see that with a basic definition of Energy = (m/2)(dx/dt)^2 + V(x), that carrying out this calculation for the lagrangian = (m/2)(dx/dt)^2 - V(x), that doing this always results in m(d^2x/dt^2) + dV(x)/dr = ma - ma = 0.

    Now however, I'd like to know if this means that the time integral of T - V is minimized. Maybe I'm having a brain fart, but it seems like ∫T + Vdt would be minimized and not ∫T - Vdt because integrating the energy would give you ∫Tdt + ∫Vdt = E(t2 - t1), which would seem to indicate that for any given time interval, the value of the integral of energy is irrelevant of the particular equations for T and V, which would mean that no ∫T + Vdt is any larger than any other, which technically satisfies the requirement that it be minimum. I do not know what ∫T - Vdt looks like.
  2. jcsd
  3. May 31, 2012 #2
    The Hamilton's Principle states that the the action is NOT MINIMISED but has a stationary value. Think of it like this:

    When the derivative of a single-variable function is zero, that doesn't mean that the function is minimized or maximized in that point. A function which has a zero derivative but has not a minimal or maximal value is f(x) = x^3.

    Energy is not minimized. Try taking the Euler-Lagrange equation but put the energy instead of the Lagrangian. The Euler-Lagrange equation won't give you the correct Newton's 2nd Law.

    To clarify some things: the Euler-Lagrange equation is equivalent to Newton's Laws. That means that:

    With Newton's Laws you can derive Hamilton's Principle (see wikipedia for derivation)
    and from Hamilton's Principle you can derive Newton's Laws (see Mechanics, Landau Lisfhitz 1st chapter for derivation).

    Also, I cannot find any physical meaning for the Lagrangian. If I made any language mistakes, sorry, I'm Greek :P
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Why is the time integral of the Lagrangian minimal?