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[This Blog is about technical intricacies and exploring non-aiki analogues of aiki principles, and other observations that may shed light on Aiki priciples and WHY they work the way they do. While the facts related are sound as far as they go, the application of everything here is speculative and subject verification in actual aikido practice. Choose wisely, therefore, and ask your own teacher. ]
Two dimensional state vectors at perpendicular angles (or any angle, really, but we are talking about juji + ) have a resultant that is the vector sum of the two i.e. -- a vector headed north and a vector headed west sum as a vector headed northwest. The magnitude of the vector is likewise a Pythagorean function (in the case of right angles (or a trigonometric function of the vector addition of any other angles).
But the human body is not a two dimensional object. The body as a whole can rotate in three axes about its center. Most human joints have more than one degree of freedom, some have two or three, even if some axes are more restricted, and one is a universal joint within its limits of rotation.
Dynamics of rotating objects require gyrodynamic analysis. Apply force to a rotating object and the resultant vector is ninety degrees out, on an axis that is not in the plane formed by the vector force and the axis of rotation of the object to which it is applied. This is counter-intuitive to the two-dimensional force assumptions that frame most people's walking-around knowledge, and counterintuitive to innate learning of most people's bodies. Gyrodynamic action also exists in vibrating as well rotating bodies.
To which the engineer says, most reasonably, that none of the joints in question rotate at a rate with sufficient momentum for classical gyrodynamic action. But what engineers puzzle over -- helo pilots live and die by and thus learn intuitively, wherefore the points I am making.
For all of our mechanical articulation, human beings are also not classical mechnical objects or mechanisms. We are exceedingly complex feedback engines. In short -- we can push back in quite disproportionate and confusing ways.
I have puzzled for years over the nature of the action involved in aikido technique. It is symbolized by the tachi sword and its deescendants, whose shape implies the spiral that gives it cutting efficiency. It is implied in the Red and White Jewels of O-Sensei's Doka. Jewels in classical Japanese reference mean the magatama shape, the comma-like elements of the tomoe, the same as reputed to be the shape of the Jewel of the Imperial Regalia. It is also the shape of the arm held in tegatana, the bent "unbendable" arm.
The principle of virtual work is another counterintuitive concept. To determine the dynamics of an complex articulated object that is too difficult to analyze in motion, assumes it is static and hardly moves at all. In more techincal words, it calculates the dynamic by an infinitesimal movement over an infinitesimal time. Without belaboring the specifics of the method of virtual work, suffice it to say that it is a very powerful tool for situations where other tools simply fail, miserably. Bernoulli's underlying assumptions about all things finding equilibrium also has resonance in aiki priniciples.
Aiki, ki musubi, uses the kinesthetic apparatus associated with every joint of the body. Most joints of the body are themselves complex affairs, and the body's articulated system of joints is yet more so. Two levels of virtual work analysis are necessary to fully assess the equilibrium conditions of the human body in dynamic action.
The human brain and body is analogous to a programmable analog computer. It is capable of calculations that are mathematically indistinguishable from the solution of difficult sytems of simultaneous equations that are the bread and butter of virtual work as a tool of engineering and physics.
Juji, as I have begun to understand it, is how aikido teaches to sense (or iinfer) and then to respond to the gyrodynamic rotation/oscillation in human movement. To describe my understanding, the brain/spirit/makoto learns in aikido training to provide resultant inputs to the attacker's joints along the axis of the gyrodynamic resultant, regardless whether "classical" gyrodynamics would seem to apply. The brain can posit a gyro dynamic according to the principle of virtual work. The result is spooky, tricky and very unnerving to the unprepared attacker's kinesthettic sytem, when everything goes wrong and yet he cannot feel exactly why.
The attacker intends his action to act in a single plane to maximize directed energy. If a motion rotates or oscillates it is admissible as a gyrodynamic input evenif it is only one oscillation or a very small rotation -- and the brain can treat it as it as such. By treating the attacking joint/body motion as a virtual gyro, the brain uses the principle of virtual work to create an output that is not a counterattack along or evasion from the incoming vector plane of rotation or oscillation (the more common martial response) but a gyrodynamic displacement of it by entering directly, and turning. The attack and the response in aiki are never in the same plane in a physical sense, as O-Sensei said "In Aikido there is never any attack."
I have not yet touched on the issue of magnitude, but radial ratios should give some idea of the manipulaiton of force amplification or dampening that are possible by such gyrodynamic means.