Mathematical Model of Gravitational and Electrostatic Forces
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Mathematical Model of Gravitational and Electrostatic Forces Alexei Krouglov Email: [email protected] ABSTRACT Author presents mathematical model for acting-on-a-distance attractive and repulsive forces based on propagation of energy waves that produces Newton expression for gravitational and Coulomb expression for electrostatic forces. Model uses mathematical observation that difference between two inverse exponential functions of the distance asymptotically converges to function proportional to reciprocal of distance squared. Keywords: Gravitational Force; Electrostatic Force; Field Theory 2 1. Introduction Author described his Dual Time-Space Model of Wave Propagation in work [2]. He tries to express physical phenomena based only on concept of energy. Model asserts that when at some point in space and in time local energy value is different from global energy level in surrounding area, energy disturbance is created at this point. Energy disturbance propagates both in time and in space as energy wave, and oscillates. Author used the model to describe both propagation of energy waves in physics [3] – [5] and fluctuations of demand and supply in economics [6], [7]. Author presented in [3] mathematical model for attractive and repulsive forces that demonstrated how gravitational and electrostatic forces are created on a distance based on instrument of Dual Time-Space Model of Wave Propagation. However paper [3] didn’t explain one important point. It described forces acting on a distance as being proportional to inverse exponential function of distance. Lately author realized that difference between two inverse exponential functions of distance is asymptotically converging to function that is proportional to reciprocal of distance squared. Thus result provided by the model is in agreement with experimentally obtained formulas for forces acting on a distance such as gravitational and electrostatic forces. In current paper author presents quantitative mathematical model for acting-on-a- distance static forces. 3 2. Model Methodology As it was described in [2], each point in space and in time is characterized by two energy values – local energy value in point U (x,t) and global energy level in neighboring area Φ(x,t). The difference between quantities U (x,t) and Φ(x,t) creates energy disturbance, which propagates both in time and in space according to ordinary differential equations presented below. Differential equations in time domain are, dP()x ,t 0 = −λ (U ()x ,t − Φ(x ,t)) (1) dt t 0 0 d 2U ()x ,t dP()x ,t 0 = µ 0 (2) dt 2 t dt dΦ()x ,t dP()x ,t 0 = −ν 0 (3) dt t dt where x0 ∈ ℜ , λt , µt ,ν t ≥ 0 are constants, and P(x0 ,t) is dual variable evaluated in point x0 at time t . Briefly speaking, meaning of equations (1) – (3) is as follows. Second derivative of energy value U (x,t) with respect to time is inversely proportional to energy disturbance U ()x,t − Φ(x,t) . First derivative of energy level Φ(x,t) with respect to time is directly proportional to energy disturbance U ()x,t − Φ(x,t) . We can rewrite equations (1) – (3) to describe dynamics of changes for energy disturbance E()x0 ,t = U (x0 ,t)− Φ(x0 ,t) in time, d 2 E()x ,t dE()x ,t 0 +ν λ 0 + µ λ E()x ,t = 0 (4) dt 2 t t dt t t 0 4 Equation (4) is an ordinary differential equation of the second order, and can be resolved by regular methods (for example, see [1]). Note that value E()x0 ,t → 0 when t → +∞ for all x0 . Differential equations in space domain are, dP()x,t 0 = −λ (U ()x,t − Φ(x,t )) (5) dx x 0 0 d 2U ()x,t dP()x,t 0 = µ 0 (6) dx 2 x dx dΦ()x,t dP()x,t 0 = −ν 0 (7) dx x dx + where t0 ∈ℜ , λx , µ x ,ν x ≥ 0 are constants. Similarly as above, meaning of equations (5) – (7) is as follows. Second derivative of energy value U (x,t) with respect to the distance in space is inversely proportional to energy disturbance E(x,t). First derivative of energy level Φ(x,t) with respect to the distance in space is directly proportional to energy disturbance E()x,.t We can rewrite equations (5) – (7) to describe dynamics of changes for energy disturbance E(x,t0 ) in space, d 2 E()x,t dE()x,t 0 +ν λ 0 + µ λ E()x,t = 0 (8) dx 2 x x dx x x 0 Equation (8) is also an ordinary differential equation of the second order. Here again value E()x,t0 → 0 when x → +∞ for all t0 > 0 . One of primary characteristics of current model is that it uses three variables to describe dynamics of wave propagation. Nonetheless present paper is devoted to static 5 phenomena, particularly to formation of static forces acting on a distance between solid bodies. Dynamics of wave propagation in space will involve a special study. 3. Transformation of Global Energy Level near Solid Body Author considers in current paper the static situation t ≡ t0 . It means that energy disturbance is not propagated in time. Here we deal with solid bodies – it means from the model’s point of view that energy disturbance is not propagated in space with regard to local energy values U (x,t). Thus we only consider transformation of global energy level Φ()x,t . Mathematically speaking, static situation means that equations (1) – (3) have null coefficients λt , µt ,ν t = 0 . Solid body means null coefficient µ x = 0 in equations (5) – (7) (other coefficients in equations (5) – (7) are nonnegative λx ,ν x ≥ 0 ). Therefore equations (5) and (7) can be combined into the following one, dΦ()x,t = −ν λ (Φ()x,t −U (x,t)) (9) dx x x Since we are dealing with solid body, local energy value U (x,t) can’t propagate behind body’s boundaries x0 ∈ ℜ . The situation is illustrated on Figure 1, which shows transformation of energy level in space near boundary of solid body. Figure 1a shows situation when no energy disturbance is present and local energy values in space match global energy level in surrounding area. Figure 1b displays situation on the boundary of solid body where local energy value abruptly changes its 6 magnitude. That change creates energy disturbance, which transforms global energy level behind the boundaries of solid body. To calculate transformation of global energy level one has to solve equation (9) for x1 > x0 . We have following initial conditions at the body’s boundary x0 ∈ ℜ for ∀t > 0 , U ()x0 ,t ≡ Φ()x0 ,t ≡ U 0 . We assume that U ()x1 ,t ≡ U1 is constant for x1 > x0 and ∀t > 0 , and equation (9) for global energy level Φ(x,t) can produce following equation for energy disturbance E()x,t = U1 − Φ()x,t , dE()x,t = −ν λ E(x,t) (10) dx x x where x > x0 , ∀t > 0 , and E()x0 ,.t = U (x1 ,t)()− Φ x0 ,t = U1 −U 0 = −∆U Thus solution of equation (10) is as follows for x > x0 , E()x,t = E0 exp{−ν x λx (x − x0 )} (11) where E0 = −∆U . Therefore solution of equation (9) is for x > x0 , Φ()x,t = (U 0 −U1 )exp{−ν xλx (x − x0 )}+U1 (12) since ∆U = −E0 = U 0 −U1 . Figure 1c illustrates how global energy level is transformed near the boundary of solid body according to equation (12). 7 4. Forces near Solid Body at Rest As we know, concept of physical forces is related with ideas of work and distance, and particularly – multiplication of average force applied to body by distance equals to the work, where work equals to change of body’s energy. Following to discussion above, when a solid body has energy disturbance, it creates energy disbalance in surrounding area. If energy disbalance occurs on the boundary of solid body, the body is trying to move to another position in order to reduce this energy disbalance. Thus energy disturbance creates physical force. The magnitude of that force is proportional to changes of absolute value of energy disturbance with respect to distance in space. Force points to direction of decrease of absolute value of energy disturbance. Next definition of force is used in the model. Definition. Instantaneous change of absolute value of energy disturbance at the d E()x ,t boundary of solid body 0 in point x with respect to distance in space is called dx 0 force F(x0 ,t) applied to solid body in point x0 and directed to the decrease of absolute value of energy disturbance, d E()x ,t d U ()()x ,t − Φ x ,t F()x ,t ≡ −k 0 = −k 0 0 (13) 0 dx dx where k > 0 is coefficient of proportionality. Let’s look how this definition of force works for solid body that is placed at rest. The situation with solid body at rest is illustrated on Figure 2. Figure 2a displays local energy values U (x,t) and global energy level Φ()x,t near solid body. Figure 2b presents distribution of energy disturbance E(x,t) near solid 8 body. Figure 2c examines adjusted distribution of energy disturbance E(x,t) near solid body, which takes into account transformation of global energy level on each side of the solid body’s border (this correction of global energy level is not further used in the paper). Thus if points in space x0 ∈ ℜ and x2 ∈ ℜ , where x0 > x2 , are respectively right and left boundaries of solid body, transformation of global energy level Φ(x,t) behind right and left boundaries of solid body is described as, ⎧∆U exp{}−ν x λx ()x − x0 +U1 if x > x0 Φ()x,t = ⎨ (14) ⎩ ∆U exp{}ν x λx ()x − x2 +U1 if x < x2 Then forces F(x0 ,t) and F(x2 ,t) acting on solid body at its boundaries are as follows (function E()x,t = U1 − Φ(x,t) is decreasing to the right of body’s boundary x > x0 , and is increasing to the left of body’s boundary x < x2 ), E()x,t − E()x0 ,t Φ()x,t − Φ(x0 ,t) dΦ()x0 ,t F()x0 ,t = − lim k = − lim k = −k =ν x λx k ∆U x→x x→x 0 x − x0 0 x − x0 dx x>x0 x>x0 (15) E()x,t − E()x2 ,t Φ()x,t − Φ(x2 ,t) dΦ()x2 ,t F()x2 ,t = − lim k = − lim k = −k = −ν x λx k ∆U x→x x→x 2 x − x2 2 x − x2 dx x<x2 x<x2 (16) Since we assumed above that solid body is at rest (i.e.