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Maupertuis's principle applied to gravity field

  1. Jan 4, 2017 #1
    1. The problem statement, all variables and given/known data
    I learned how to deriving Maupertuis's principle from Hamilton's principle under conservative energy condition and feel this interesting.
    But when trying to derive a particle's behavior in gravitivity field,stuck....
    The direction of grativity is along ##x## axis,the particle is placed at ##(0,0,0)## as well as initial velocity equal to zero.At time ##t##,particle's position is ##(x(t),y(t),z(t))##.
    Because Maupertuis's principle derived from Hamilton's principle here,it is reasonable to fix ##t=0## and ##t={\Delta}t## just like we did with Hamilton's principle.
    2. Relevant equations
    ##\delta{x,y,z}(0)=\delta{x,y,z}(\Delta{t})=0##
    ##\delta{\int_0^{\Delta{t}}}p_x\dot{x}+p_y\dot{y}+p_z\dot{z}dt=m\delta{\int_0^{\Delta{t}}}{\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2dt=0 \ \ \ (1)##
    Conservative energy condition:##mgx=\frac{1}{2}m({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2)\ \ \ \ (2)##
    3. The attempt at a solution
    Substituite equation ##(2)## into ##(1)##,we got ##\delta\int_0^{\Delta{t}}2mgxdt=\int_0^{\Delta{t}}2mg\delta{x}dt=0## this means ##mg## should equal to zero,something wrong.....
     
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  3. Jan 4, 2017 #2

    TSny

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  4. Jan 4, 2017 #3
    I think in this way:
    ##W=-L+\sum\limits_ip_i{\dot{q}}_i##.Because for actual path, we have ##W=constant##, apply variation method to Lagrange function as well as constrain variation path with energy conservation condition:
    ##\delta\int_{t_1}^{t_2}Ldt=\delta\int_{t_1}^{t_2}L+Wdt=\delta\int_{t_1}^{t_2}\sum\limits_ip_i{\dot{q}}_idt=0##
    so,under energy conservation condition:
    ##\delta\int_{t_1}^{t_2}\sum\limits_ip_i{\dot{q}}_idt=0## for actual path.(I apply this to grativity field above,although this is not rigorous Maupertuis' principle)

    What's more,in De brogile's article:On the Theory of Quanta,which confusing me as well:
    Maupertuis' principle was deduced in this way:##\delta\int_{t_1}^{t_2}\sum\limits_ip_i{\dot{q}}_idt=\delta\int_{A}^{B}\sum\limits_ip_id{q_i}=0## with a Footnote added to German translation:To make this proof rigorous,it is neccessary,as it well known,to also vary ##t_1##,##t_2##;but because of the time independence of the result,our argument is not fase.
     
  5. Jan 4, 2017 #4

    TSny

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    This is done too quickly for me. On the far left you have ##\delta\int_{t_1}^{t_2}Ldt## which is an integral between two fixed times and the variation ##\delta## is for paths that generally have different energy ##W##. Then you equate this with ##\delta\int_{t_1}^{t_2}L+Wdt## in which you assume ##W## is constant. But when you do the variation at constant energy, the time ##t_2## will vary as you vary the path. So, I don't follow this line of argument.
    For Mauperuis' principle the variation ##\delta## is done at constant ##W##. But this is impossible to do if you keep ##t_1## and ##t_2## fixed during the variation. For a derivation of Maupertuis' principle from Hamilton's principle, see the Landau and Lifshitz Mechanics text, sections 43 and 44.
    https://archive.org/stream/Mechanics3eLandauLifshitz/Mechanics 3e-Landau,Lifshitz#page/n163/mode/2up

    For me, this is too sketchy.
     
  6. Jan 7, 2017 #5
    You are right.I must misunderstand something.
    Lagrangian function:##L=\frac{1}{2}m{\dot{q}_x}^2+\frac{1}{2}m{\dot{q}_y}^2+\frac{1}{2}m{\dot{q}_z}^2+mgq_x\ \ \ \ \ (1)##

    Energy conservation condition:##mgq_x=\frac{1}{2}m{\dot{q}_x}^2+\frac{1}{2}m{\dot{q}_y}^2+\frac{1}{2}m{\dot{q}_z}^2\ \ \ (2.1)## and ##mg\delta{q_x}=m\dot{q}_x\delta{\dot{q}_x}+m\dot{q}_y\delta{\dot{q}_y}+m\dot{q}_z\delta{\dot{q}_z} \ \ \ \ (2.2)##

    In my imagination,##\frac{\partial{L}}{\partial{q}_i}-\frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}_i}}=0## make ##\delta\int_{t_1}^{t_2}Ldt=0##,This could be right for equation ##(1)## and equavalent to ##\ddot{q}_x=g,\ddot{q}_y=\ddot{q}_z=0##

    Equation ##(2.2)## make ##\delta\int_{t_1}^{t_2}Wdt=0##
    So ##\delta\int_{t_1}^{t_2}(L+W)dt=0##,then we have ##\delta\int_{t_1}^{t_2}\sum\limits_ip_i\dot{q}_idt=0##

    But equation ##(2.1)## would make ##L=2mgq_x \ \ \ (3)##,
    ##\frac{\partial{L}}{\partial{q}_i}-\frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}_i}}=0## is not right argument for equation (3) now.
    It seems that energy conservation condition make ##\delta\int_{t_1}^{t_2}Ldt=\int_{t_1}^{t_2}2mg\delta{q}_xdt\ne0##.

    If varying ##t_2##,the ##\delta\int_{t_1}^{t_2}Ldt## would equal zero under energy conservation condition?
     
    Last edited: Jan 7, 2017
  7. Jan 7, 2017 #6

    TSny

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    Yes, this is correct if the ##\delta## symbol means to vary the paths such that the initial and final times (##t_1## and ##t_2##) are kept fixed for all paths. For this type of path variation, the energy ##W## generally will not be conserved (except for the true path).

    Here, you are assuming the energy ##W## remains constant for the varied paths. So, here you are using a different type of path variation than before. So, it might be good to use a different symbol ##\Delta## for this type of path variation. For this type of variation, you must allow the endpoint times ##t_1## and/or ##t_2## to vary as you vary the path.

    [EDIT: So, ##\Delta\int_{t_1}^{t_2}Wdt \neq 0## if you are varying paths keeping ##W## constant. Rather, you can see that you would get ##\Delta\int_{t_1}^{t_2}Wdt = W \Delta t_2 - W \Delta t_1##, where ##\Delta t_2## is the variation in the upper time limit when varying the path. Similarly for ##\Delta t_1##.]

    Here it looks like you are not considering that the two types of variation are different. [EDIT: From what I wrote above, this equation does not follow.]

    Lagrange's equations ##\frac{\partial{L}}{\partial{q}_i}-\frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}_i}}=0## assume that you are expressing ##L## as a function of both the ##q_i## and the ##\dot{q}_i## according to ##L = T - V##.

    The variation of the paths on the left side of the equation are the ##\delta## type of variation where energy is not conserved for the varied paths. But the integral involving ##2mg## assumes energy is conserved in the variation.

    A careful discussion of going from Hamilton's principle to Maupertuis' principle can be found in Goldstein's Mechanics, 3rd edition, starting on page 356. The text uses the ##\delta## versus ##\Delta## notation to distinguish the types of variation of path.
     
    Last edited: Jan 8, 2017
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