Chain Rule in Lagrangian Transformation

  • #1
Hello,
I'm trying to follow Goldstein textbook to show that the Lagrangian is invariant under coordinate transformation. I got confused by the step below
So
## L = L(q_{i}(s_{j},\dot s_{j},t),\dot q_{i}(s_{j},\dot s_{j},t),t)##

The book shows that ##\dot q_{i} = \frac {\partial q_{i}}{\partial s_{j}} \dot s_{j} + \frac{\partial q_{i}}{\partial t} ##,and I'm not sure where the ##\frac{\partial q_{i}}{\partial t} ## term come from? I've tried to look up chain rules for coordinate transformation but I can't find anything.

Please help!

Thank you :D
 

Answers and Replies

  • #2
Orodruin
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It is just the standard chain rule. Consider a function ##f(h,g)## where ##h## and ##g## are functions of some parameter ##t## and differentiate ##f## with respect to ##t##. The chain rule now states
$$
\frac{df}{dt} = \frac{\partial f}{\partial h} \frac{dh}{dt} + \frac{\partial f}{\partial g} \frac{dg}{dt}.
$$
Now, if ##g = t## you would obtain ##dg/dt = dt/dt = 1## and ##\partial f/\partial g = \partial f/\partial t## and therefore
$$
\frac{df}{dt} = \frac{\partial f}{\partial h} \frac{dh}{dt} + \frac{\partial f}{\partial t}.
$$

What is unclear is why a term ##(\partial q/\partial \dot s) \ddot s## is not included. You can only ignore this term if ##q## is a function only of ##s## and not of ##\dot s##.
 
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  • #3
@Oroduin,

You are correct. q is independent of ##\dot s##. I mistyped the question. And thank you for your answer.
 
  • #4
vanhees71
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If you want more general transformations, it's better to use the generalized Hamilton principle in terms of the Hamiltonian rather the Lagrangian formulation. This leads to the more general canonical transformations in phase space rather than the less general diffeomorphism invariance of the Lagrangian formalism in configuration space.

The Hamiltonian phase-space formalism turns out to be the most valuable form of classical mechanics from a fundamental-physics point of view since it reveals the symplectic structure of phase space and allows for a quasi algebraic formulation of the dynamics in terms of Poisson brackets, which brings the classical theory very close to its extension to quantum theory.
 

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