- #1
wormhole
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Hi
see the attached picture...
2 coupled masses, each suspended from spring in gravitational field...
also entire construction can vibrate only vertically...
I need to write lagrangian for this system in the following form:
[tex]
L=\frac{1}{2}\sum_{i=1,j=1}^2m_{ij}\dot{x}_i\dot{x}_j-\frac{1}{2}\sum_{i=1,j=1}^2k_{ij}x_ix_j
[/tex]
[itex]x_i[/itex] and [itex]x_j[/itex] are displacements of masses from equalibrium positions and springs are identical
the kinetic part of L is easy one but i stack with potential part...
i get to this expression(i always confused about right signs):
[tex]m_1g(l+x_1)+m_2g(l+x_1+l+x_2) + \frac{1}{2}m_1{x_1}^2+\frac{1}{2}m_2{x_2}^2[/tex]
- positive direction taken along g direction
- l is length of unstreched spring
every potential term in L must in [itex]k_{ij}x_ix_j[/itex] form but i don't see a way how do i get it from my expression for potential energy because there are g force terms containing only [itex]x_i[/itex] or [itex]x_j[/itex]
thanks
see the attached picture...
2 coupled masses, each suspended from spring in gravitational field...
also entire construction can vibrate only vertically...
I need to write lagrangian for this system in the following form:
[tex]
L=\frac{1}{2}\sum_{i=1,j=1}^2m_{ij}\dot{x}_i\dot{x}_j-\frac{1}{2}\sum_{i=1,j=1}^2k_{ij}x_ix_j
[/tex]
[itex]x_i[/itex] and [itex]x_j[/itex] are displacements of masses from equalibrium positions and springs are identical
the kinetic part of L is easy one but i stack with potential part...
i get to this expression(i always confused about right signs):
[tex]m_1g(l+x_1)+m_2g(l+x_1+l+x_2) + \frac{1}{2}m_1{x_1}^2+\frac{1}{2}m_2{x_2}^2[/tex]
- positive direction taken along g direction
- l is length of unstreched spring
every potential term in L must in [itex]k_{ij}x_ix_j[/itex] form but i don't see a way how do i get it from my expression for potential energy because there are g force terms containing only [itex]x_i[/itex] or [itex]x_j[/itex]
thanks
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