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<<Moderator's note: Moved from a technical forum, no template.>>
Description of the system:
The masses m_{1} and m_{2} lie on a smooth surface. The masses are attached with a spring of non stretched length l_{0} and spring constant k. A constant force F is being applied to m_{2}.
My coordinates:
Left of m_{1} there is a stationary wall. The distance between m_{1}'s left surface and the wall is represented as y and the distance between the right and left surfaces of m_{1} and m_{2}, respectively, is represented as x.
What my brain says:
The kinetic energy of this system is equal to the sum of kinetic energies of the two masses, i.e,
$$\frac{1}{2}m_1 \dot{y}^2 + \frac{1}{2}m_2 (\dot{x}+\dot{y})^2$$
and the potential energy is just the spring potential, i.e,
$$\frac{1}{2} k (xl_0)^2$$
Therefore, the Lagrangian seems to be,
$$L=\frac{1}{2}m_1 \dot{y}^2 + \frac{1}{2}m_2 (\dot{x}+\dot{y})^2  \frac{1}{2} k (xl_0)^2$$
Now, the generalized force is
$$F\cdot \frac{\partial {(x+y)}}{\partial x}$$
So, the equations of motion seem to be,
$$\frac{d}{dt}(\frac{\partial L}{\partial {\dot x}})  \frac{\partial L}{\partial { x}}=F$$
and
$$\frac{d}{dt}(\frac{\partial L}{\partial {\dot y}})  \frac{\partial L}{\partial { y}}=0$$
Is this correct? Or am I missing something? Please help me figure out the mistakes.
Description of the system:
The masses m_{1} and m_{2} lie on a smooth surface. The masses are attached with a spring of non stretched length l_{0} and spring constant k. A constant force F is being applied to m_{2}.
My coordinates:
Left of m_{1} there is a stationary wall. The distance between m_{1}'s left surface and the wall is represented as y and the distance between the right and left surfaces of m_{1} and m_{2}, respectively, is represented as x.
What my brain says:
The kinetic energy of this system is equal to the sum of kinetic energies of the two masses, i.e,
$$\frac{1}{2}m_1 \dot{y}^2 + \frac{1}{2}m_2 (\dot{x}+\dot{y})^2$$
and the potential energy is just the spring potential, i.e,
$$\frac{1}{2} k (xl_0)^2$$
Therefore, the Lagrangian seems to be,
$$L=\frac{1}{2}m_1 \dot{y}^2 + \frac{1}{2}m_2 (\dot{x}+\dot{y})^2  \frac{1}{2} k (xl_0)^2$$
Now, the generalized force is
$$F\cdot \frac{\partial {(x+y)}}{\partial x}$$
So, the equations of motion seem to be,
$$\frac{d}{dt}(\frac{\partial L}{\partial {\dot x}})  \frac{\partial L}{\partial { x}}=F$$
and
$$\frac{d}{dt}(\frac{\partial L}{\partial {\dot y}})  \frac{\partial L}{\partial { y}}=0$$
Is this correct? Or am I missing something? Please help me figure out the mistakes.
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