Scalar potential Cartesian coordinates

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

Px, Py and pz are the components of the linear momentum of the particle, in a cartesian coordinate system which we need not define further at this stage. The electromagnetic field is described by the vector potential A and scalar potential 4>. Using the operator form for the linear momentum gives us the semi-classical expression,... 

It is important to realize that the potential energy U is still a scalar, i.e., realvalued function, but depends on all 3N Cartesian coordinates of the N-particle... 

The details of electrical properties require some basic information about electric fields. Fields are usually thought of as Cartesian vectors. They arise from or may be defined from a scalar function, the electric potential, V. The x-component of the electric field is dV/dx, which we will d gnate as Vx. If the potential V is independent of x, then there is no x-component of die field. If V has only a linear dependence on x, the field is uniform in die x-direction. Of course, the elearic potential may depend on the spatial coordinates in some... 

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