Newtonian motion of a particle confined to a smooth surface
Recently, I've been considering a model wherein a lone particle with constant mass m is confined to a surface $F$: $\mathbb{R}^m \to \mathbb{R}^n$ , where $m < n$. I declare this surface to be strictly stationary, even resisting rigid motions. We then declare the particle's initial position to be $w(t_0) = F(u(t_0))$ with $w$ in $\mathbb{R}^n$ and $u$ in $\mathbb{R}^m$, and we declare its initial velocity to be $v(t_0) = w'(t_0) = F'(u(t_0))*u'(t_0)$, where $F'$ is the Jacobian of $w$ with respect to $u$. So, $w(t_0+dt) = F(u(t_0) + u'(t_0)dt)$, to ensure our particle is still confined to $F$ at $t_0+dt$.
However, the velocity vector cannot be unchanged from $t_0$ to $t_0+dt$, since the span of $w'(t_0)$ does not generally coincide with that of $F'(u(t_0) + u'(t_0)dt)$, i.e., the "old" velocity is not tangent to $F$ at the "new" position. To meet this requirement of tangency, I have proposed (axiomatically) that the true velocity of the particle at $t_0+dt$ is the projection of the "old" velocity $v(t_0)$ onto the span of the "new" Jacobian matrix $F'(u(t_0) + u'(t_0)dt)$, as to minimize any instantaneous acceleration. In other words, $$w'(t_0 + dt) := A(A^T A)^{-1}A^Tw'(t_0)$$ where $A = F'(u(t_0) + u'(t_0)dt)$; the $^T$ and $^{-1}$ postfixes denote matrix transposition and inversion, respectively.
Given this axiom, it is clear that $w'(t_0) \neq w'(t_0+dt)$, implying that some instantaneous acceleration $$a = \frac{w'(t_0+dt) - w'(t_0)}{dt}$$ is imposed on the particle.
These are my primary questions:
* Where does this acceleration come from? Following an energy-conserving (Newtonian) model, some force should be present to impart any supposed acceleration to our particle. This would imply that something in the system loses energy. 563
Would we achieve conservation of energy by simply subtracting this energy from the momentum of the particle in question?
* Is my proposed axiom one which is physically (in a Newtonian sense) valid? Am I going down the wrong rabbit hole when I assume that minimization of instantaneous acceleration is a correct way to describe the motion of a particle confined to a surface?
I would greatly appreciate any sort of insight into how I could continue with and/or improve upon this model. If this model is totally invalid, or if there is a better way to approach this problem, any point in a better direction would be equally appreciated.
P.S. This is my first ever post to any StackExchange site, so I apologize for any formatting/communication errors. I will be more than happy to clarify any uncertainties regarding my model as needed. Thanks!
<PERSON> (edit) My intentions are that the particle not be acted upon by any external force (not even friction) other than that of constrain. Additionally, I intend for the particle to be bounded to the surface strictly, ie it is located on the surface at any given time.
As a final thought, why doesn't a force orthogonal to the particle's current velocity affect its kinetic energy? I suspect it has something to do with centripetal acceleration. I'd prefer to use linear functions to describe the particle's motion, but if trig functions are the only practical option then so be it. 319