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If we consider physical systems where we have initial conditions, such as a rolling circle having initial initial velocity $v$, what does our Lagrangian look like?

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The initial conditions come in when you go to solve the system’s differential equations of motion for your particular case. The Lagrangian itself is defined without reference to the initial conditions, and it is used to find those equations of motion via the Euler-Lagrange Equation.

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You do not need the initial condition to write down the Lagrangian. The beauty of Lagrangian is that it gives you the differential equation of motion. When you want to solve the differential equation of motion, you need the initial condition.

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