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## Homework Statement

A mass [tex]M_1[/tex] is suspended by a spring with constant [tex]k_1[/tex]. A second mass [tex]M_2[/tex] is suspended from the first by a spring with constant [tex]k_2[/tex]. The equilibrium height of the first is [tex]y_1[/tex] and the equilibrium height of the second is [tex]y_2[/tex].

I just want to write down the Lagrangian for now.

## Homework Equations

[tex]L=T-V[/tex] (i.e. Lagrangian is Kinetic energy minus Potential energy).

## The Attempt at a Solution

I'm using as generalized coordinates the displacements of the two masses from equilibrium [tex]\delta_1[/tex] and [tex]\delta_2[/tex]

I'm pretty sure that the kinetic energy parts are easy

[tex]T_1 = {1\over 2}M_1 \dot\delta_1^2[/tex] and [tex]T_2 = {1\over 2}M_2 \dot\delta_2^2[/tex]

Now for the potential energy for the second mass I think is

[tex]V_2 = -M_2 g (y_2+\delta_1+\delta_2) + {1\over 2}k_2(\delta_2-\delta_1)^2[/tex].

The first term is the gravitational part (taking the zero to be at the ceiling where the first spring is anchored) and the second term is the spring part.

But for the first mass, I'm tempted to write

[tex]V_1 = -M_1 g (y_1+\delta_1) + {1\over 2}k_1 \delta_1^2[/tex].

I think the spring part is wrong because [tex]M_1[/tex] is attached to spring 2, so there should be some contribution involving [tex]k_2[/tex]. This leads me to want to add another term like [tex]{1\over 2}k_2(\delta_2-\delta_1)^2[/tex], but I don't know if it should be positive or negative, and also I feel like I am 'double counting' the spring term in [tex]V_2[/tex]. For example, if spring 1 is streched and spring 2 is compressed, I expect [tex]V_1[/tex] to be increased by the compression of spring 2. I hope this is clear.