# Equations of motion for a certain constrained system

As an exercise in Lagrangian and Hamiltonian mechanics, I am looking at a system with the following Lagrangian:

$$L=\dot R \cdot\dot R-\theta\dot R\cdot (SR)+\lambda(R\cdot R-1)$$

$$R$$ is a vector in 4D Euclidean space, constrained to lie on the unit hypersphere ($$\lambda$$ is the Lagrange multiplier for this constraint). $$S$$ is a skew-symmetric matrix that squares to $$-1$$. The Euler-Lagrange equations result in the following equations of motion:

$$\ddot R-\theta S\dot R-\lambda R=0$$ $$R\cdot R=1$$

My first difficulty is that I cannot figure out how to solve for $$\lambda$$ to determine the constraint forces. My second difficulty is in finding the general solution; I can find specific solutions of the form: $$R=e^{At} R_0$$ Where $$A$$ is assumed to be skew-symmetric so that the constraint is guaranteed to be satisfied. This has a unique solution $$A=\theta S/2$$ when $$\lambda=\theta^2/4$$ (which is problematic; there does not appear to be enough degrees of freedom to allow for arbitrary initial positions and velocities) and and infinite family of solutions for $$\lambda < \theta^2/4$$ (which I'm not sure how to combine to get a general solution).

Never mind those problems. Now I try to form the Hamiltonian. The conjugate momenta are:

$$P=\frac{\partial L}{\partial \dot R}=2\dot R-\theta SR$$

Which is invertible to find the velocity, and:

$$\pi=\frac{\partial L}{\partial \dot \lambda}=0$$

Which obviously is not. Not sure what to do about the constraint, I just evaluate it in computing the Hamiltonian and the equations of motion, so that term disappears. I get a Hamiltonian:

$$H=\dot R \cdot\dot R=\frac{1}{4}P\cdot P+\frac{1}{2}\theta P\cdot (SR)$$

Plugging this into the Hamiltonian equations, I get the result:

$$\dot P=\frac{1}{2}\theta SP$$ $$\dot R=\frac{1}{2}P+\frac{1}{2}\theta SR$$

And so:

$$\ddot R=\theta S\dot R+\frac{1}{4}\theta^2 R$$

Which is the same equations of motion that I got before, except that $$\lambda=\theta^2/4$$ specifically.

So that seemed to work. Except: I wanted to verify that these equations of motion preserve the constraint. Evaluating the Poisson bracket of the constraint with the Hamiltonian shows that it is proportional to $$P\cdot R$$, so that has to be zero in the initial conditions. But the Poisson bracket of $$P\cdot R$$ gives the expression:

$$\frac{1}{4}(P\cdot P+\theta P\cdot(SR)+\theta^2)$$

Which should be zero to ensure that the constraint is preserved - but it isn't zero for the unique solution that I found above for $$\lambda=\theta^2/4$$, even though that solution does preserve the constraint by construction!

So I'm trying to figure out (1) how to solve for the constant $$\lambda$$ and how to get a general solution to the equations of motion if possible, and (2) how to appropriately deal with the constraint in the Hamiltonian formalism, or (3) if I've just made a mistake somewhere. Hoping there is someone out there who can point me in the right direction. Thanks very much.

$$\lambda$$ isn't a constant. (I was misunderstanding the role of the Lagrange multiplier.) Then in fact it is easy to solve for, simply by taking the dot product of the equation: $$\ddot R-\theta S\dot R-\lambda R=0$$ with $$R$$.