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I am thinking of what the Euler-Lagrange equation, $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0 $$ specifically represents in satisfying the conditions to put a particle "on-shell" in terms of the Classical Action (1).

I know that $\frac{\partial L}{\partial \dot{x}} = p$ I also know that putting a particle "on-shell" is equivalent to stating, in natural units, $E^2 = p^2 + m^2$ (2). In this context, what physical quantities do, the following represent? $$ \tag{1} \frac{\partial L}{\partial \dot{x}} $$ $$ \tag{2} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) $$ and $$ \tag{3} \frac{\partial L}{\partial x} $$

References

  1. The Euler–Lagrange Equations and Noether’s Theorem, Springer, (top of page 26)
  2. On-Shell and Off-Shell, Wikipedia
Alexander McFarlane
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    It seems that the question (v3) is spurred by different uses of the word on-shell, see e.g. my Phys.SE answer here. – Qmechanic Jan 03 '16 at 13:22
  • @Qmechanic Sorry I grabbed a coffee and edited it to make more sense of my thoughts! :) V4 is the final version as I really want to know the dimensions of the E-L equation – Alexander McFarlane Jan 03 '16 at 13:24

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