> In Lagrangian Mechanics you minimize the total action of a system to find its motion
Strictly, realizable paths are an extremum (maximum, minimum, or inflection point) of the action.
> it’s representation invariant
This form of representation invariance is true in Newtonian and Hamiltonian mechanics as well. It's a statement that physics is invariant under changes of coordinates. I think a better way of stating what you're after is that you work with a function of the whole state of the system rather than having to do the double entry bookkeeping for all the interactions among subsystems of what you're modeling.
> Studying the “shapes” of systems in this manner is part of larger field called Topology.
The geometry of configuration space (for Lagrangian mechanics) or phase space (for Hamiltonian mechanics) isn't really part of topology. For phase space, it's properly called symplectic geometry. For configuration spaces, it's just high dimensional Euclidean geometry. It is sometimes interesting to look at the topology of these spaces, but it's much more than just topology.
I'm glad to see a mention of SICM. That really is a great book. It might be worth pointing out that the field of mathematics that leads to the Euler-Lagrange equations goes under the name of "calculus of variations." I learned it from Weinstock's excellent old book by the same name.
Maybe also worth noting that you can't add non-conservative forces to Lagrangian mechanics in any generally accepted way, so if you have friction in your system, you're stuck in Newtonian.
> Strictly, realizable paths are an extremum (maximum, minimum, or inflection point) of the action.
At a realizable path, the action can be a local minimum or a saddle point but it's never a local maximum.
Tangentially, while "principle of least action" is a misnomer, it's not too severe of a misnomer, because realizable paths always minimize the action on a short enough time scale.
Also, a simple example of the action being a saddle point: a moving object constrained to the surface of a sphere, with no potential, which will move around a great circle.
> In Lagrangian Mechanics you minimize the total action of a system to find its motion
Strictly, realizable paths are an extremum (maximum, minimum, or inflection point) of the action.
> it’s representation invariant
This form of representation invariance is true in Newtonian and Hamiltonian mechanics as well. It's a statement that physics is invariant under changes of coordinates. I think a better way of stating what you're after is that you work with a function of the whole state of the system rather than having to do the double entry bookkeeping for all the interactions among subsystems of what you're modeling.
> Studying the “shapes” of systems in this manner is part of larger field called Topology.
The geometry of configuration space (for Lagrangian mechanics) or phase space (for Hamiltonian mechanics) isn't really part of topology. For phase space, it's properly called symplectic geometry. For configuration spaces, it's just high dimensional Euclidean geometry. It is sometimes interesting to look at the topology of these spaces, but it's much more than just topology.
I'm glad to see a mention of SICM. That really is a great book. It might be worth pointing out that the field of mathematics that leads to the Euler-Lagrange equations goes under the name of "calculus of variations." I learned it from Weinstock's excellent old book by the same name.
Maybe also worth noting that you can't add non-conservative forces to Lagrangian mechanics in any generally accepted way, so if you have friction in your system, you're stuck in Newtonian.