Consider a simple two particle system with two point masses of mass m at x1 and x2 with a potential energy relative to each other which depends on the difference in their coordinates V = V(x1-x2)
The lagrangian is:
L = ½m(x1')2 + ½m(x2')2 + V(x1-x2)
Obviously their total momentum is conserved d/dt(mx1' + mx2') = 0, which can be verified by plugging in to the lagrangian. But there is no cyclic coordinates in the lagrangian. Is it possible to put it in a form where this hidden cyclic coordinate is shown?
The lagrangian is:
L = ½m(x1')2 + ½m(x2')2 + V(x1-x2)
Obviously their total momentum is conserved d/dt(mx1' + mx2') = 0, which can be verified by plugging in to the lagrangian. But there is no cyclic coordinates in the lagrangian. Is it possible to put it in a form where this hidden cyclic coordinate is shown?