This little app will show you how the modules are global vectors of momentum and force. At the moment we use classic a lot, depending on speed [1]: \[m = m_0 \gamma, \quad \gamma = 1 / \sqrt{1 - \beta^2}, \quad \beta = v/c \qquad (1.1)\] let's Start with the global vector of the momentum that we have already received here, \[\mathbf{P} = m_0 c \left\{\pm 1,\, \pm \beta,\, \pm \beta^2,\, \ldots,\, \pm \beta^n \right\} \qquad (1.2)\] and multiply it by itself: \[\mathbf{P}\cdot \mathbf{P} = (m_0 c)^2 \gamma^2 \qquad (1.3)\] it follows that the module of global vector momentum will be like this: \[|\mathbf{P}| = m_0 c \gamma \qquad (1.4)\]

As we know from physics, force is the derivative of the pulse on time: \[\mathbf{F} = {\Bbb{d} \mathbf{P} \over \Bbb{d}t} \qquad (1.5)\] this implies that the speed \(v\) depends on time, hence: \(\beta = \beta(t)\). Differentiate with respect to the global vector of the pulse, and get the global vector force (GVF): \[\mathbf{F} = m_0 a \left\{0, \pm 1,\, \pm 2\beta,\, \pm 3\beta^2,\, \ldots,\, \pm n\beta^{n-1} \right\} \qquad (1.6)\] where: \(a = c\beta^{'}_t \) acceleration. As we see, the coordinate with a single vector \(\mathbf{j_0}\) is equal to zero, i.e. the time coordinate is not involved in GVF, which is quite logical.

Formula (1.6) represents the second Newton's law [1] to a global vector.

Now find the module GVF from the formula (1.6). To do this, recall that the sum of the power series, which is obtained by scalar multiplication in the formula is as follows: \[\sum \limits_{n=1}^{\infty} n^2 x^{2(n-1)} = {1 + x^2 \over (1 - x^2)^3} \qquad (1.7)\] the Modulus of any vector, including GVF, is this: \[|\mathbf{F}| = \sqrt{\mathbf{F}\cdot \mathbf{F}} \qquad (1.8)\] Substituting into (1.8) formula (1.6), and turning immediately obtained a power series in (1.7), finally get the module global force vector: \[|\mathbf{F}| = m_{0} a\gamma^3 \sqrt{1 + \beta^2} \qquad (1.9)\] In the next section we will show the connection between the global vectors of momentum and strength with the energy.