- #1

zhouhao

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## Homework Statement

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.

## Homework 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)##

## 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...