The Lagrangian of a particle of mass \( m \) is given by \( L = \frac{1}{2}m\dot{x}^2 - \lambda x^4 \), where \( \lambda \) is a positive constant. If the particle oscillates with total energy \( E \), then the time period of oscillations \( T \) is given by:
\[ T = a \int_0^{\left(\frac{E}{\lambda}\right)^{\frac{1}{4}}} \frac{dx}{\sqrt{\frac{2}{m}(E - \lambda x^4)}} \]
The value of \( a \) is ___ (in integer).
\[ T = a \int_0^{\left(\frac{E}{\lambda}\right)^{\frac{1}{4}}} \frac{dx}{\sqrt{\frac{2}{m}(E - \lambda x^4)}} \]
The value of \( a \) is ___ (in integer).
Correct Answer: 4
Solution and Explanation
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