PACS: 02.40.Hw; 02.40.-k; 45.20.-d

1. Introduction

Recently the physics of particles and fields in curved surfaces in Euclidean space has become subject of interest, because many phenomena are reduced to one of them. For example, it is known that a liquid crystal on curved surfaces, look for smectic phases as parallel curved surface^{1}. We also know, that some electronic properties of certain two-dimensional materials, are explained modeling particles, living in the surface, satisfying the Schrödinger or the Dirac equation^{2}. Topological defects on surfaces^{3}, relativistic particle dynamics^{4}, and diffusion on surfaces^{5} are also some examples of how the interaction of particles and fields with surfaces through geometry, can be used to model several natural phenomena. Although from a mathematical point of view, variational problems related to curves and surfaces, has been addressed^{6}, they have recently attracted attention, because has been possible to interpret the Euler-Lagrange equations in physical terms, as a balance of internal forces and moments, see *e.g.*^{7} and^{8}.

Inspired by a recent formalism developed to investigate properties of elastic curves constrained on surfaces^{9}, in this work we obtain the basic, but generic, geometric elements of particles constrained on surfaces. Although it is well known the classical physics of particles, usually specific cases of particles moving on surfaces are solved. Here we present its description in terms of the geometric information of the surface, either intrinsic through the gaussian curvature or extrinsic information, encapsulated in the second fundamental form.

We obtain, aside to the equations of motion, the constraining force on the particle. The corresponding equations of small perturbations are also obtained, showing that they includes, even with no external forces, extrinsic information. Some general results if the surface has axial symmetry and the gravitational field taken into account are showed.

2. Lagrangian Classical Mechanics

Let us consider the action of a free particle, constrained on a surface through the vector Lagrange multiplier

Here

After integration by parts in the first term, the corresponding Euler-Lagrange equations are given by

Moreover, variation respect to the local variables,

where

and the equation for the constraint is obtained.

However, the Eq. (3), in not useful in this form. Instead, we write it in the Darboux frame **T** is the unit tangent to the curve and *s*^{10}. These equations defines de normal curvature

We also consider that the constraint does not depend on time, then the velocity of the particle is given by

where *s* the arc length along the trajectory of the particle. The relation of time *t*, with the arc length of the curve is given by

The second derivative can be obtained from

In the first term we can get

in the second line, the second fundamental form of the surface **T**, since it is a unit vector. Notice that we can identify

where

We can write this force in terms of the principal curvatures

where, as usual, **T** and the principal vector **v**
_{1}.

If there is an external force **F**, acting on the particle, we can decompose the equation of motion in the local basis and we can write

such that, the corresponding equations to solve are then given by

and the energy is conserved, *U* the corresponding potential relative to the external force field. Notice that if we use the first and the third equations, then

*i.e.* the force *t*. Therefore, a condition such that no constraining force exists is given by

This is a remarkable expression, which tell us that, in order that the constraining force vanishes, the ratio of non-tangential components of the external field must be equal to the ratio of the curvatures of the surface.

2.1 Perturbative expansion

To first order the Newton law (3) is given by (if the external force is a constant, *e.g* the gravitational field)

where the deformations are along the surface,

Then, since the energy is conserved, we see that

Thus we have that

If there is no external potential then

Therefore, with no external forces the tangential deformation of geodesics is a rigid translation,

In order to obtain

When projected into the tangent direction, the second term of the last equation is identically zero, as a consequence of the antisymmetry of the Riemann tensor. From Eq. (17) and (20), we can find

Since on two dimensional surfaces

where

Therefore to first order, the Newton law (3) implies that the constraining force along the perturbed path is given by

Whereas the transversal projection implies that

We see that it involves not only the transversal deformation

and

where we have defined

2. 2. Geodesic trajectories

On perturbed geodesic curves, the deformation on the constraining force, is given by

Without external forces it simplifies such that ^{12}

Besides intrinsic information, projections of the second fundamental form appear in this equation. Even without external forces, extrinsic information are required,

A simple example is given by a particle falling on the unit sphere, starting from rest at the north pole, under the gravitational field. Since **n** and the axis *z*. Due to the geodesic torsion vanishes we get

Nevertheless, we note that a first integral is,

2.2.1 Axisymmetric surfaces

If the surface has axial symmetry, we can parametrize it in the form

The infinitesimal element of distance is given by *l*. The tangent vectors **e**
_{a} are

The unit normal is then

The second fundamental form *K*
_{
ab
} has components,

and

The principal curvatures are found to be

and

With these basic elements we can describe a particle restricted to lie along this surface. Because the symmetry under rotations about the *z* axis,

The normal curvature along the trajectory is given by

This equation can be written in terms of *M*
_{
z
} . In the first term we can use Eq. (36), in the second one we can use the induced metric on the surface, to obtain a first integral of

where

Then we can write the normal curvature of geodesics

and therefore, the force

The velocity *v*(*t*), is determined by conservation of the energy

An example is the catenoid, a minimal surface that we can parametrize through

where *M*
_{
z
}

The curvature _{z}, see Fig. (2): If ^{11}.

An easy example is a particle falling along a meridian starting at *l*
_{0}, and

If the particle leaves the catenoid at *l*
_{
s
} , then

Being

2.2.2 Particle on a surface in a curved space

If the particle moves onto a surface embedded into a curved space, with coordinates

The first term involves the background metric,

where the Christoffel symbols compatible with

where *a*(*t*) = 0. The force

In this case the second fundamental form

It is certainly interesting to know the effect of nontrivial background, that we will present in future work.

3. Summary and conclusions

In this paper we have presented the geometric elements in the description of particles on surfaces. Projection of the Newton second law along the normal to surface, gives the constraining force, that involves the normal curvature of the surface. Geodesic curvature (times *v*
^{2}) gives the acceleration in the transversal direction to the movement. Classical perturbations of the path is equivalent to a field theory on curved surfaces. We show that from conservation of energy, a first order equation of the tangential deformation, is obtained. We also show that the equation of the transversal perturbation follows a Raychaudhuri-like equation that includes extrinsic information even if there is not external forces. If the particle moves onto a surface with axial symmetry, the results are parametrized in terms of the conserved projection of the angular momenta. The particular results in the case of geodesic curves are obtained. Using the formalism here presented, a problem to be addressed, is related to the motion of extended objects on surfaces, for elastic curves, the model must include the bending energy, quadratic in the curvature which has been extensively examined^{13}.