Suppose a particle is moving on a surface of a sphere,then it contains a holonomic constraint and so the three Cartesian co-ordinates are available with a constraint equation(equation of surface in Cartesian co-ordinate system). Then there must exist two generalized co-ordinates in form of which Lagrange equation can be constructed.Then is it true to say that the generalized co-ordinate system in this particular case represent Riemannian manifold and so Lagrange equation is equation of motion on this Riemannian manifold?I should made one more important specification that work done by forces of constraints are zero.
[Physics] A particular case when Lagrange equation is equivalent to equation of motion on a Riemannian manifold
classical-mechanicsconstrained-dynamicsdifferential-geometrylagrangian-formalism
Related Solutions
The full question (v13) is rather broad, but here are some comments/feedback:
Traditionally virtual displacements are frozen in time $t$. The $t$-differentiations in eqs. (3) & (5) are misleading at best (depending on what the notation $t$ is supposed to represent).
Eq. (4) is Newton's 2nd law if ${\bf F}_i$ denotes the total force on the $i$th point particle, i.e. a sum of "applied" and constraint forces. One of the main points is to try to eliminate the constraint forces from the formalism, at least for the holonomic constraints. OP seems to have made no progress in this.
It should be noted that a $\partial g / \partial \dot q_j$-term in OP's final eq. (8) is not appropriate for a general non-holonomic constraint $g$, but only for a so-called semi-holonomic constraint, $$ g(q,\dot{q},t)~\equiv~\sum_j a_j(q,t)\dot{q}^j+a_t(q,t)~\approx~0, $$ which by definition is an affine function of $\dot{q}^j$.
In case OP is following Ref. 1, note that the treatment of Lagrange equations for non-holonomic constraints in Ref. 1 is inconsistent with Newton's laws, and has been retracted on the errata homepage for Ref. 1. See Ref. 2 for details.
In case OP is following Ref. 3, note that Ref. 3 only deals with holonomic constraints in Chapter 2. (This is explicitly mentioned on the middle of p. 50.) Ref. 3 tentatively introduces non-holonomic constraints in the beginning of Section 3.1.2, only to later rejects the approach as unphysical.
References:
H. Goldstein, Classical Mechanics; 3rd ed; Section 2.4. Errata homepage. (Note that this criticism only concerns the treatment in the 3rd edition; the results in the 2nd edition are correct.)
M.R. Flannery, The enigma of nonholonomic constraints, Am. J. Phys. 73 (2005) 265.
J.V. Jose & E.J. Saletan, Classical Dynamics: A Contemporary Approach, 1998; Sections 2.1.1, 2.2.1 & 3.1.2.
The main difference between "real" displacements and "virtual" displacements arise when the constraint itself is time-dependent.
For example, you have an incline that is accelerating (I am saying accelerating because a moving side can be cancelled with galilei transform, accelerating one cannot; but you should imagine simply a moving incline). Then because the slide will move, any instantenous (!!!!) real displacement $d\mathbf{r}$ will not be tangential to the incline. It will consist of a $\delta\mathbf{r}$ displacement that is tangential along the incline, and a displacement $\Delta\mathbf{r}$ caused by the motion of the incline, so $d\mathbf{r}=\delta\mathbf{r}+\Delta\mathbf{r}$.
The $\delta\mathbf{r}$ here is what is called a virtual displacement. Because it doesn't take into account the time-evolution of the constraint (the motion of the incline), it can be calculated by "freezing time" and in the frozen landspace, you evaluate the possible displacements of the system that are allowed by the contraints (eg. in this case, tangent to the incline), hence the Calkin definition.
Best Answer
If you look at a particle constrained to move on the surface of a sphere, and the motion is frictionless, then you can use the usual geometric formalism of classical mechanics to describe the situation. The sphere constitutes the configuration space for the system (traditionally denoted $Q$), and in this case is two dimensional. The Lagrangian $L$ is a function which is dependent, not only on the two coordinates of the configuration space ${q_i}$, but also on the two velocities ${v_i}$. So in this case $L$ is a function on the tangent bundle of the sphere.
The motions of the particle are given by the Euler-Lagrange equations, in turn given by extrema of the action $$I=\int L(q_i,v_i)dt$$
So in this picture, you can work intrinsically on the sphere and forget that it is embedded in $\mathbb{R^3}$, and that it arose as a constraint surface.
The first couple of chapters of Nick Woodhouse's book give a good description of the geometric treatment of classical mechanics.