Forces appear and disaapear according to ones relative frames of reference and the constraints imposed on those relationships. Lagrange pt forward this philosophy for space, declaring there is no time only velocity. Hamilton countered by saying one could develop a science of pure time with allthe constraints and relations of Lagrange, and that is how the new mechanics was born. In the process lists of constraints and properties needed to be explicitly laid out, and the concept of a matrix, and then a similar concept of a vector was developed. But i was Grassmann's idea whih was the glue that held them all together for Physicists. Hamilton was too focussed on quaternions to see the wider need for a more general description of the complex relations in space, whereas Grassmann from the otset developed the most general philosophical terminology. All were dancing to the tune of Lagrange.
For one particle acted on by external forces, Newton's second law forms a set of 3 second-order ordinary differential equations, one for each dimension. Therefore, the motion of the particle can be completely described by 6 independent variables: 3 initial position coordinates and 3 initial velocity coordinates. Given these, the general solutions to Newton's 2nd law become particular solutions which determine the time evolution of the particle's behaviour after its initial state (t = 0).
The most familiar set of variables for position r = (r1, r2, r3) and velocity are Cartesian coordinates and their time derivatives (i.e. position (x, y, z) and velocity (vx, vy, vz) components. Determining forces in terms of standard coordinates can be complicated, and usually requires much labour.
An alternative and more efficient approach is to use only as many coordinates as are needed to define the position of the particle, at the same time incorporating the constraints on the system, and writing down kinetic and potential energies. In other words, to determine the number of degrees of freedom the particle has, i..e the number of possible ways the system can move subject to the constraints (forces which prevent it moving in certain paths). Energies are much easier to write down and calculate than forces, since energy is scalar while forces are vectors.
These coordinates are generalized coordinates, denoted qj, and there is one for each degree of freedom. Their corresponding time derivatives are the generalized velocities, . The number of degrees of freedom is usually not equal to the number of spatial dimensions: multi-body systems in 3 dimensional space (such as Barton's Pendulums, planets in the solar system, or atoms in molecules) can have many more degrees of freedom incorporating rotations as well as translations. This contrasts the number of spatial coordinates used with Newton's laws above.
The Lagrangian Mechanics is a philosophy as well as a method. Instead of peering at a problem from the inside, one looks at the problem from the outside, one takes the exterior or god viewpoint. From this wiewpoint the mechanics is that of a shaped space, and the shape is usually some box that is transformed. Newton was satisfied with modelling his mechanics on pulleys and gears, and his forces were typically tensile ropes. Thus Newton looked at the innards of a system. Lagrange looked at the shape of the whole system, what bounded and constrained the innards. Thus he was not very sure how the innards were connected, and found this by trial and error. He was thus able to proceed from the top down where Newton proceeded from the bottom up. Both proceses had their limitations, and the use of both could intuit both analysies. In this regard, Grassmann spent his time working on his Aysdehnungslehre trying to accomodate both approaches.
The one difference Einstein made unknowingly in using he lagrange and Hamilton techniques was to change the general shape from a box to a system of concentric and or interior spheres. It is how these spherical constraints effect yhe system of mechanics that Einstein tried to explore, and what physicists continue to explore today.
What they only lack is the vision to see the shape of the constraints, and the fractal nature of their interaction a all scales. The Mandelbulb is he closest model that artists have attained to so far.
For me, the inertial properties of space can only be addressed by Lagranges technique of analysis, and in this case the rotational force is a constraint. But in an inertial frame it is almost an invisible forcebeing swamped by the tangential and radial forces. In the Lagrangian frame, it is the most immoveable constraint, and without it mechanical systems fail.
However, the real problem lies in both the Lagrange and the inertial mechanics accepting mass as a dumb quantity that contributes nothing but the desired attributes of forces accelerations and velocities to mechanics, and thus separating mechanics from the true source of inertia which is the chemical and nuclear forces generated by charge.
Charge is the source of inertia, and that is not simply positive and negative!
Most of us do not notice the difference betwen the description of charge and force. We know that attractive and repulsive forces exist, and we locate them in matter. but when we haveparticles of matter that exhibit charge, we have to admit that attraction and repulsion can no longer be located in particulate matter. For to do so would be to give particles a switch, whereby they switch on attraction or repulsion according to the charge of their bedfellow.
No what we have is a particular spatial process that twists one way both expanding on the way ou asit rebounds from contracting on the way in. This bi polar twist allows these dynamic bubbles of toroidal activity to strongly attract or strongly repel, but otherwise to weakly associate as bigger bubbles. These solenoid type forms self organise and scale up . and who knows how small they scale down. The left and right hand rules distinguish two flavours of the spatial twiststed bubble, but not charge for charge is about contraction and condensation of space and rarefaction and expansion of space, and these always exist as a dipole.
Grassmann's law of 3 strecken is one of his strangest looking ideas until you realise it is based on the properties of Ruclidean parallel lines. The parallel lines are used to define parallelograms, but the strecken are the adjacent sides of that parallelogram. They are the exterior side of the parallelogram product or construction. the combinatorics of the construction look trivial until you interpolate a sign indicating the contra direction. Since the definition of the construction requires the sides to proceed from the same corner, the sign moves the construction around that corner as a centre. some strecken therefore have to take a negative sign to multiply. swapping the base strecken round however was more surprising, commutativity refers to a fixed basic realationship, if that basic relationship is mutable, there is no commutativity.
Finally identity of paralleograms give ab = -[ba].