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In this paper, the form invariance of second-order linear nonholonomic systems in the phase space is studied. The definition and criterion of the form invariance of secondorder linear nonholonomic systems in the phase space is given. The structure equations and the conserved quantities of the form invariance are obtained. and an example is given to illustrate the application of the results.

The s-wave bound states of Klein-Gordon equation and Dirac equation with scalar and vector reflectionless-type potentials are obtained.

A gradient representation and a second order gradient representation of the mechanics system are studied. The differential equations of motion of the holonomic and nonholonomic mechanics systems are expressed in the canonical coordinates. A condition under which the system can be considered as a gradient system is given. A condition under which the system can be considered as a second order gradient system is obtained. Two examples are given to illustrate the application of the result.