Three particles of mass m each situated at x₁(t), x₂(t) and x₃(t) respectively are connected by two spring constants k and unstretched lengths $$l$$. The system is free to oscillate only in one-dimension along the straight line joining all the three particles. The Lagrangian of the system is

A) $$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_2} - l} \right)^2} + \frac{k}{2}\left( {{x_3} - {x_2} - l} \right)$$ B) $$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_3} - l} \right)^2} + \frac{k}{2}\left( {{x_3} - {x_2} - l} \right)$$ C) $$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_2} - l} \right)^2} - \frac{k}{2}{\left( {{x_3} - {x_2} + l} \right)^2}$$ D) $$L = \frac{m}{2}\left[ {{{\left( {\frac{{d{x_1}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_2}}}{{dt}}} \right)}^2} + {{\left( {\frac{{d{x_3}}}{{dt}}} \right)}^2}} \right] - \frac{k}{2}{\left( {{x_1} - {x_2} - L} \right)^2} - \frac{k}{2}{\left( {{x_3} - {x_2} - l} \right)^2}$$

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Correct Answer: Option A

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Three particles of mass m each situated at x1(t), x2(t) and x3(t) respectively are connected by two spring constants k and unstretched lengths $$l$$. The system is free to oscillate only in one-dimension along the straight line joining all the three particles. The Lagrangian of the system is

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Three particles of mass m each situated at x₁(t), x₂(t) and x₃(t) respectively are connected by two spring constants k and unstretched lengths $$l$$. The system is free to oscillate only in one-dimension along the straight line joining all the three particles. The Lagrangian of the system is