Fractional differential equations in National Sports Training in Colleges and Universities

In sports training in colleges and universities, people generally equate a team's offensive ability with the striker's level of ‘one-to-one’ [1]. In the fierce competition in football, the forward is an essential factor in winning, but the passing and receiving cooperation from the backcourt and midfielders is also essential.

Network topology has been widely used in communication networks, computer networks, and operations research. Here we use line graph analysis in network theory to discuss passing and receiving in football [2]. We first define the generalized path (1) L(ki)=∏i=1nki L({k_i}) = \prod\limits_{i = 1}^n {k_i} Here, k_i represents a quantity that affects the generalized path. In a football game, we mainly consider ‘the geometric distance between two points, the speed of the ball, the accuracy of passing, the ability to receive the ball, and the defense of the passing and receiving points.’ When multiple people pass and receive the ball multiple times, L(k_i) forms a distance matrix, which follows the following rules (2) L2(ki)=L(ki)*L(ki)L3(ki)=L2(ki)*L(ki)⋯⋯ {L^2}({k_i}) = L({k_i})*L({k_i}){L^3}({k_i}) = {L^2}({k_i})*L({k_i}) \cdots \cdots We define (3) C*D=E=[ejk]m×n C*D = E = [{e_{jk}}{]_{m \times n}} Among them C = [c_jk]_m×l, D = [d_jk]_l×n, ejk=mincjl≠0,dlk≠0{cj1*d1k,cj2*d2k,⋯⋯cjl*dlk} {e_{jk}} = \mathop {\min }\limits_{{c_{jl}} \ne 0,{d_{lk}} \ne 0} \{ {c_{j1}}*{d_{1k}},{c_{j2}}*{d_{2k}}, \cdots \cdots {c_{jl}}*{d_{lk}}\} .

For example, C = [1, 2, 3, 4, 0], D = [0, 2, 1, 2, 0]^T, ejk=mincjl≠0,dlk≠0{0,2,4,6,0} {e_{jk}} = \mathop {\min }\limits_{{c_{jl}} \ne 0,{d_{lk}} \ne 0} \{ 0,2,4,6,0\} .

We use Figure 1 to show the generalized path diagram of the pass when attacking. P₁ represents the node formed by the defender of the possession side and the goalkeeper. P₂, P₃ corresponds to midfielders and full-backs. P₄ stands for striker. P₅ refers to the goal. Then (4) L=[0320000313000100000100000], L2=[1jk]=L*L=[0063600042000020000000000] L = \left[ {\matrix{ 0 & 3 & 2 & 0 & 0 \cr 0 & 0 & 3 & 1 & 3 \cr 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right],\quad {L^2}{ = [1_{jk}}] = L*L = \left[ {\matrix{ 0 & 0 & 6 & 3 & 6 \cr 0 & 0 & 0 & 4 & 2 \cr 0 & 0 & 0 & 0 & 2 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right]

Fig. 1

The same (5) L3=L2*L=[0007400005000000000000000], L4=L3*L=[0000800000000000000000000], L5=L4*L=0 {L^3} = {L^2}*L = \left[ {\matrix{ 0 & 0 & 0 & 7 & 4 \cr 0 & 0 & 0 & 0 & 5 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right],\quad {L^4} = {L^3}*L = \left[ {\matrix{ 0 & 0 & 0 & 0 & 8 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right],\quad {L^5} = {L^4}*L = 0 To find the shortest distance from P_j to P_k, we define (6) C⊕D=[ejk]m×n C \oplus D = [{e_{jk}}{]_{m \times n}} among them (7) C=[ajk]m×nD=[djk]m×n,ejk={0,cjk=djk=0cjk,cjk>0,djk=0djk,cc8=0,djk>0min(cjk,djk),cjk≠0djk≠0Lmin=L⊕L2⊕L3⊕L4⊕L5=[0323400313000120000100000]=[1jk]min \matrix{ {C = [{a_{jk}}{]_{m \times n}}D = [{d_{jk}}{]_{m \times n}},{e_{jk}} = \left\{ {\matrix{ {0,{c_{jk}} = {d_{jk}} = 0} \hfill \cr {{c_{jk}},{c_{jk}} > 0,{d_{jk}} = 0} \hfill \cr {{d_{jk}},c{c^8} = 0,{d_{jk}} > 0} \hfill \cr {\min ({c_{jk}},{d_{jk}}),{c_{jk}} \ne 0{d_{jk}} \ne 0} \hfill \cr} } \right.} \hfill \cr {{L_{\min }} = L \oplus {L^2} \oplus {L^3} \oplus {L^4} \oplus {L^5} = \left[ {\matrix{ 0 & 3 & 2 & 3 & 4 \cr 0 & 0 & 3 & 1 & 3 \cr 0 & 0 & 0 & 1 & 2 \cr 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right] = \mathop {{{[1}_{jk}}]}\limits_{\min } } \hfill \cr } [1jk]min \mathop {{{[1}_{jk}}]}\limits_{\min } represents the shortest distance from P_j to P_k. The ultimate goal of passing and receiving is to get the (best) shot. There are many factors to consider when passing the ball. A node that is geometrically connected is not necessarily a valid point. From formula (1), it can be known that any factor other than the geometric distance is zero, and the generalized path is the point of leading and receiving the ball, which is the weak point [3]. From formula (2), we can know the importance of midfielders. Because P₁, P₅ is constant for the stadium. The value of [1jk]min \mathop {{{[1}_{jk}}]}\limits_{\min } mainly depends on the direction and size of P₂, P₃.

The best-generalized path gives us the whereabouts of the ball in the pass. But how to form the best path is primarily affected by the defensive team's defensive lineup. Let us illustrate this point with a simple diagram [4]. The three U₁, U₂, U₃ secant lines in Figure 2 below represent the three defense lines of the defender. And define (8) Ui=∑PkPk+1U1=5,U2=5,U3=4 \matrix{ {{U_i} = \sum {P_k}{P_{k + 1}}} \hfill \cr {{U_1} = 5,{U_2} = 5,{U_3} = 4} \hfill \cr }

We introduce the concept of generalized attack intensity and define the following formula. The attack intensity in football refers to the attack intensity, the reliability of passing and receiving the ball, and the degree of threat to the defense [5]. (9) fjk↔Pjk {f_{jk}} \leftrightarrow {P_{jk}}

A linear model can represent the generalized attack intensity secant (10) minfjkU=∑fjk \mathop {\min }\limits_{{f_{jk}}} U = \sum {f_{jk}}

The minimum cut is equivalent to the maximum flow, so the generalized attack intensity cut is (11) minfjkU=∑fjk−∑fjk{U,j≠s0,j≠s,t−U,j=t \mathop {\min }\limits_{{f_{jk}}} U = \sum {f_{jk}} - \sum {f_{jk}}\left\{ {\matrix{ {U,j \ne s} \hfill\cr {0,j \ne s,t}\hfill \cr { - U,j = t} \hfill\cr } } \right. s, t represents a confident start and endpoint, respectively. Through the secant analysis, we know two ways to increase the attack intensity: one is to increase the attack line, and the other is to reduce the path to the attack point P₅. This is the value that forms the minimum secant U₃ to form the maximum attack intensity [6].

Fig. 2

We can write the dynamic equations of the model by the Kane method (12) Fj+Fj*=0 {F_j} + F_j^* = 0

We replace the extra force acting on the system with an equal force system. Then the generalized active force F_j can be written as (13) Fj=∑K=115(∂Vk∂x˙jFk+∂ωk∂x˙jMk) {F_j} = \sum\limits_{K = 1}^{15} \left( {{{\partial {V_k}} \over {\partial {{\dot x}_j}}}{F_k} + {{\partial {\omega _k}} \over {\partial {{\dot x}_j}}}{M_k}} \right) F_k and M_k represent the ‘equivalent’ force and force matrix acting on the link B_k. The line of action of F_k passes through the center of the link. G_k ·V_k and ω_k are respectively called the velocity and angular velocity through the point G_k in the inertial reference frame. ∂Vk∂x˙j {{\partial {V_k}} \over {\partial {{\dot x}_j}}} and ∂ωk∂x˙j {{\partial {\omega _k}} \over {\partial {{\dot x}_j}}} are respectively referred to as ‘change rate of position deviation’ and ‘change rate of azimuth deviation.’ M_k is (14) Mk=Mkext+∑Mkl {M_k} = M_k^{ext} + \sum M_k^l M_k is the moment produced by the external force acting on B_k. Mkl M_k^l is the force applied by B_l to B_k. It is generally caused by force B_l exerted on B_k. Therefore, rewriting (14) has (15) Mk=Fjext+Fjint {M_k} = F_j^{ext} + F_j^{\mathop {{\rm{int}}}} among them (16) Fjext=∑K=115(∂Vk∂x˙jFk+∂ωk∂x˙jMkext) F_j^{ext} = \sum\limits_{K = 1}^{15} \left( {{{\partial {V_k}} \over {\partial {{\dot x}_j}}}{F_k} + {{\partial {\omega _k}} \over {\partial {{\dot x}_j}}}M_k^{ext}} \right) (17) Fjint=∑K=115(∑∂ωk∂x˙jMk) F_j^{\mathop {{\rm{int}}}} = \sum\limits_{K = 1}^{15} \left( {\sum {{\partial {\omega _k}} \over {\partial {{\dot x}_j}}}{M_k}} \right)

Due to (18) ∂ωk∂x˙j=ωijknoi, ∂Vk∂x˙j=υijknoi {{\partial {\omega _k}} \over {\partial {{\dot x}_j}}} = {\omega _{ijk}}{n_{oi}},\quad {{\partial {V_k}} \over {\partial {{\dot x}_j}}} = {\upsilon _{ijk}}{n_{oi}}

Equation (17) can be rewritten as (19) Fjext=υkjkFki+ωkjkMkiext F_j^{ext} = {\upsilon _{kjk}}{F_{ki}} + {\omega _{kjk}}M_{ki}^{ext} Where F_ki and Mkiext M_{ki}^{ext} are the components of F_k and Mkext M_k^{ext} in the n_oi direction, respectively [7]. Let's analyze the generalized inertial force. From Eq. (14), we can get: (20) Fj*=∑k=115(∂Vk∂x˙jFk*+∂ωk∂x˙jMk*) F_j^* = \sum\limits_{k = 1}^{15} \left( {{{\partial {V_k}} \over {\partial {{\dot x}_j}}}F_k^* + {{\partial {\omega _k}} \over {\partial {{\dot x}_j}}}M_k^*} \right)

Among them Fj*=−mkak=−mk(υijkx¨j+υ˙ijkx˙j)noi, Tk*=−Ik⋅ak−ωk×Ik⋅ωk F_j^* = - {m_k}{a_k} = - {m_k}({\upsilon _{ijk}}{\ddot x_j} + {\dot \upsilon _{ijk}}{\dot x_j}){n_{oi}},\quad T_k^* = - {I_k} \cdot {a_k} - {\omega _k} \times {I_k} \cdot {\omega _k}

The inertial dyadic I_k is given by (21) Ik=Imnknomnon {I_k} = {I_{mnk}}{n_{om}}{n_{on}}

Among them, the value of element I_mnk related to n_om and n_mnk is related. It has the same value as n_kr and n_ks related element Irskk(k=1,2,3;m,n,r,s,k) I_{rsk}^k(k = 1,2,3;m,n,r,s,k) . O represents the origin of the reference coordinate of the inertial reference system [8]. The expression is as follows: (22) Imnk=nomnonnksnrskk {I_{mnk}} = {n_{om}}{n_{on}}{n_{ks}}n_{rsk}^k (23) S0Kmr=nomnkr, S0Kns=nonnks, Imnk=S0KmrS0KnsIrsk {S_0}{K_{mr}} = {n_{om}}{n_{kr}},\quad {S_0}{K_{ns}} = {n_{on}}{n_{ks}},\quad {I_{mnk}} = {S_0}{K_{mr}}{S_0}{K_{ns}}{I_{rsk}}

Organized (24) Tk*=−(Imnknomnon)[(w˙ijkx¨j+ωijkx¨j)noi]−(ωijkx˙jnoi)×Imnknomnonωijkx˙jnoiTk*=−[Imnk(ω˙njkx¨j+ωnjkx¨j)+eiqmωipkωnjkx˙px˙jIqnk]nom \matrix{ {T_k^* = - ({I_{mnk}}{n_{om}}{n_{on}})\left[ {\left( {{{\dot w}_{ijk}}{{\ddot x}_j} + {\omega _{ijk}}{{\ddot x}_j}} \right){n_{oi}}} \right] - \left( {{\omega _{ijk}}{{\dot x}_j}{n_{oi}}} \right) \times {I_{mnk}}{n_{om}}{n_{on}}{\omega _{ijk}}{{\dot x}_j}{n_{oi}}} \hfill \cr {T_k^* = - [{I_{mnk}}({{\dot \omega }_{njk}}{{\ddot x}_j} + {\omega _{njk}}{{\ddot x}_j}) + {e_{iqm}}{\omega _{ipk}}{\omega _{njk}}{{\dot x}_p}{{\dot x}_j}{I_{qnk}}]{n_{om}}} \hfill \cr }

Among them Tk*=∑k=115{mkυijk(υipkx¨j+υ˙ipkx˙j)+ωijk[Iink(ωnpkx¨p+ω˙npkx˙p)+emqiωmpkωnhkωnhkx˙px˙hIqnk]} T_k^* = \sum\limits_{k = 1}^{15} \left\{ {{m_k}{\upsilon _{ijk}}\left( {{\upsilon _{ipk}}{{\ddot x}_j} + {{\dot \upsilon }_{ipk}}{{\dot x}_j}} \right) + {\omega _{ijk}}\left[ {{I_{ink}}({\omega _{npk}}{{\ddot x}_p} + {{\dot \omega }_{npk}}{{\dot x}_p}) + {e_{mqi}}{\omega _{mpk}}{\omega _{nhk}}{\omega _{nhk}}{{\dot x}_p}{{\dot x}_h}{I_{qnk}}} \right]} \right\} i, m, n, q and p are summed from 1 to 3. Both j and h are summed from 1 to 40. The kinetic equation (14) can be expressed in the following concise form (25) Ajpx¨p=fj+Fjext+Fjint {A_{jp}}{\ddot x_p} = {f_j} + F_j^{ext} + F_j^{{\rm{int}}} Where p is summed from 1 to 40. A_jp and f_j are given by (26) Ajp=∑k=115[mkυijkυipk+Iinkωijkωnpk] {A_{jp}} = \sum\limits_{k = 1}^{15} \left[ {{m_k}{\upsilon _{ijk}}{\upsilon _{ipk}} + {I_{ink}}{\omega _{ijk}}{\omega _{npk}}} \right] (27) fj=∑k=115[mkυijkυ˙ipkx˙p+Iinkωijkω˙npkx˙p+emqiIqnkωmpkωnhkx˙px˙h] {f_j} = \sum\limits_{k = 1}^{15} \left[ {{m_k}{\upsilon _{ijk}}{{\dot \upsilon }_{ipk}}{{\dot x}_p} + {I_{ink}}{\omega _{ijk}}{{\dot \omega }_{npk}}{{\dot x}_p} + {e_{mqi}}{I_{qnk}}{\omega _{mpk}}{\omega _{nhk}}{{\dot x}_p}{{\dot x}_h}} \right]

Equation (24) forms 40 second-order nonlinear differential equations. Use the Runge-Kutta method to find numerical solutions to these equations. We can find 40 generalized coordinates. If some equations of these generalized coordinates are known, a part of the Eq. (24) becomes an algebraic equation for finding the sum of unknown forces [9]. All the coefficients in the equation can be obtained by calculating the four coefficient matrices of ω_ijk, υ_ijk, υ˙ijkx˙j {\dot \upsilon _{ijk}}{\dot x_j} and ω˙ijkx˙j {\dot \omega _{ijk}}{\dot x_j} . So we can solve the dynamic equation on the computer.

To obtain the dynamic equation of the constraint system, we only need to multiply the left side of the dynamic equation of the original system by transposing the coefficient matrix of the constraint equation [10]. Therefore, from Eq. (24), the dynamic equation of the multi-rigid body mechanical model of the human body under constraints can be obtained: (28) a(Fj+Fj*)=0 aAjpx¨p=a(fj+Fjext+Fjint) \matrix{ {a\left( {{F_j} + F_j^*} \right) = 0} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,a{A_{jp}}{{\ddot x}_p} = a\left( {{f_j} + F_j^{ext} + F_j^{{\rm{int}}}} \right)} \hfill \cr } Where a is the transposition of the coefficient matrix of the constraint equation. In the formula, A_jp, f_j, Fjext F_j^{ext} and Fjint F_j^{{\rm{int}}} are determined by formulas (18–21).

The election here is one of the ten classic World Cup events-a-goal matches between Argentina and the Netherlands. The picture is the eight sports characteristic pictures from the attack from the middle of the ball to the goal. Through image reconstruction, a panoramic view of the whole process is obtained [11]. From the reconstructed panorama, the combination line of passing and receiving for the offensive midfielder is obtained as shown in Figure 3:

Fig. 3

According to the offensive strength, the offensive midfielder's adequate passing directions, namely, to the right and pass back. The defense of the three forwards on the offensive side is successful for the defense. Still, the defensive guards are too densely positioned to create opportunities for the proper offensive back [12]. Although a ‘6-3 situation is formed in the local area, there is unnecessary defensive overlap, resulting in insufficient defensive space. At this time, the three strikers of the offensive side are effectively blocked, and the goalkeeper's proper position is not a fatal threat to the defensive side.

Figures 4 and 5 are the characteristic pictures of the ‘back bow’ on the pole in the shooting action using the improved dynamic model and the original model [13]. It can be seen that the hip movement, which is the crucial part of the technical movement, more realistically describes the characteristics of the movement by increasing the degree of freedom.

Fig. 4

Fig. 5

The essential physical fitness and technical level of a football player are essential. It is an integral part of the technical level of the entire team. However, the most considerable charm of football is the ‘acting’ of talented players and the unpredictable results. The team's tactical style and performance are also the most attractive. Research has found that establishing differential equation models in football can help us use mathematical ideas, methods, and knowledge to solve practical problems.