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Consider a holonomic system where I have ##n## not independent variables and one constraint ##f(q1,q2,...,qN,t)=0##. One can rewrite the minimal action principle as:
##\frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial q'_i} - \lambda \frac{\partial f}{\partial q_i} = 0 ##
My question: how can one check if lambda is a constant or a function? Because in general it does not have to be a constant in this reasoning according to my understanding.
Example:
The harmonic oscillator given by the constaint that the distance of the mass to the origin is constant. The small angle approximation is allowed. So here I write out the equations and I indeed do find a harmonic oscillation under the assumption that ##\lambda## is constant. How to show that this is indeed constant? If I try doing this and do some small angle approximations using the tension I eventually find that the ##y##-coordinate is constant so I'm probably doing something wrong.
Is the lambda not constant when not using the small angle approximation?
##\frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial q'_i} - \lambda \frac{\partial f}{\partial q_i} = 0 ##
My question: how can one check if lambda is a constant or a function? Because in general it does not have to be a constant in this reasoning according to my understanding.
Example:
The harmonic oscillator given by the constaint that the distance of the mass to the origin is constant. The small angle approximation is allowed. So here I write out the equations and I indeed do find a harmonic oscillation under the assumption that ##\lambda## is constant. How to show that this is indeed constant? If I try doing this and do some small angle approximations using the tension I eventually find that the ##y##-coordinate is constant so I'm probably doing something wrong.
Is the lambda not constant when not using the small angle approximation?