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We formulate Mach's principle and show that the inertial force is the gravitational induction force the Universe exerts on an accelerated body. We show that inertial mass depends on cosmic time, which has measurable consequences.

According to the principle dynamic equilibrium, we understand the force of inertia is a
force that acts on whatever body that accelerated with respect to an inertial frame. It is, therefore, a real force, observed in whatever reference frame. We identify this force with force gravitational induction produced by the whole of the Universe. Therefore, the Universe is an inertial reference frame. In developing the theory, we find that the gravitational induction force produced by the entire Universe is proportional to acceleration and in the opposite sense, as is the force of inertia. In making this identification, we find that the inertial and gravitational mass are proportional, with a coefficient of proportionality depending on the cosmic time.

We study the application of the laws of mechanics in inertial and non-inertial reference frames. We verify that in the usual formulation of mechanics, there are no centrifugal forces. We understand the force of inertia as a force that acts on everybody that is accelerated with respect to an inertial reference frame. The so-called fictitious forces are not considered as forces since other bodies do not produce them. We indicate the superiority of the principle of dynamic equilibrium or D'Alembert's law over the usual way of expressing the second law of mechanics. In the second part of this research, Mach's principle develops and it is shown that inertia is produced by the action of the gravitational induction forces of the whole Universe.

We find that the force of inertia acting on an accelerated body is the result of the action of the gravitational induction force produced by the relative movement of the Universe as a whole, which fully confirms the Mach's Principle. The calculations are developed with the linearized theory of General Relativity.

We show that the forces of inertia acting on the accelerated bodies are forces of gravitational induction exerted by the whole of the Universe. Therefore, the phenomenon of inertia and the inertial mass of a body have a cosmic origin, as demanded by the Mach's principle. The calculations will be applied to a vector gravitational field theory. In a second part of this research we will apply these results to the General Theory of Relativity.

We will deduce the inductive forces of a vectorial gravitational theory and we will study whether these forces can be identified with the forces of inertia that act on a body when it is accelerated.

We will deduce the induction forces obtained from the field equation of Nordstrom's scalar gravitational theory and investigate whether they can explain the origin of the forces of inertia acting on a body when it is accelerated.