> * There is a function L(q,q'), the Lagrangian. How do I compute this function? Does it have an analytical expression? How does it depend on p or F? L can be

> * There is a function L(q,q'), the Lagrangian. How do I compute this function? Does it have an analytical expression? How does it depend on p or F? L can be a function which changes with the system(a spring or a free particle will have different formulae), just like the Force in classical physics changes with the system. The goal here is to find properties independent of any particular expression for L only assuming that L is invariant under certain symmetries. > * There is a parameter s, "sending q to some new position q(s)". Do you mean q is actually a function q(s)? Do I define q(s) any way I want, or do I solve q(s) given the constraints of the system? There is a bit of overloading of notation which can cause some confusion, q is not a function of s, but s is a parameter for a symmetry Tₛ which moves points in the space, here the line (For example, Tₛ(q)= q+sv is the translations in the v direction parametrized by s). So, one can read q(s) as Tₛ(q) and d(q(s))/ds as the derivative d(Tₛ(q))/ds. For the translation, the derivative will be v at all points, but it can be different at different points for instance when we are in a plane and Tₛ(q) is rotating q around the origin by s units, then the derivative with respect to s will be a tangent vector to the circle containing q. This derivative vector field is called the infinitesimal symmetry, as the symmetry Tₛ is just the flow along the vector field. BTW, you are in good company with your question. Many mathematicians and cs people also get confused with physics notation where the type of functions is not clear (where the domain and range of the function is not clear). This issue is prominent when deriving the Euler-Lagrange equations. Read the preface of the referenced book http://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/book-Z-H-4.html#%_chap_Temp_2 http://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/boo...