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Substitutions in Differential Equations As a Geometry of Their Symmetries

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Substitutions in Differential Equations As a Geometry of Their Symmetries THE TEACHING OF MATHEMATICS 2018, Vol. XXI, 1, pp. 1–14 SUBSTITUTIONS IN DIFFERENTIAL EQUATIONS AS A GEOMETRY OF THEIR SYMMETRIES Jelena Kati´c Abstract. We enlighten a geometrical background for familiar substitutions in certain ordinary differential equations. We explain how the existence of a symmetry of a differential equation provides a change of coordinates (i.e., a substitution) in which the initial equation becomes in quadrature form. MathEduc Subject Classification: I75 MSC Subject Classification: 97I70, 97I40 Key words and phrases: Ordinary differential equation; symmetries; substitution. 1. Introduction The simplest differential equation is one of the form dx = h(t); x(t ) = x ; dt 0 0 where h : I ! R is a continuous function defined on an interval I ⊆ R. Its only solution is obtained by a direct integration: Z t x(t) = x0 + h(¿) d¿: t0 The next very simple form is dx (1) = h(x); x(t ) = x ; dt 0 0 where h(x0) 6= 0. It follows from the assumptions that x(t) has the differentiable inverse in some neighbourhood of x0, so we can transform (1) into dt 1 1 = = ; t(x ) = t ; dx dx h(x) 0 0 dt which is again solvable by integrating. We accept the following definition. This work is partially supported by Ministry of Education and Science of Republic of Serbia, Project #ON174034. 2 J. Kati´c Definition 1. An ordinary differential equation is in quadrature form if it is of the first order, dx (2) = f(t; x) dt and if the right-hand side, i.e. the function f(t; x), depends only on t or on x. dx It is convenient to transform any differential equation dt = f(t; x) to the one in quadrature form; in some examples this can be done by a suitable choice of new variables, i.e. by a substitution. In this note we will try to lighten the background of certain substitutions from a geometrical point of view, i.e. to explain how some substitutions naturally arise from a symmetry of a differential equation. Let us recall some examples first. Example 2. (Homogeneous equation) Consider the equation dx ³x´ = g : dt t Using the substitution x s(t; x) = log t; y(t; x) = t x one transforms it to quadrature form. The substitution s(t; x) = t, y(t; x) = t transforms it to the equations with separable variables: dx = a(x)b(t): dt Example 3. (Generalized homogeneous equation) It is an equation of the form µ ¶ dx x x® = g : dt t t¯ The substitution x® s(t; x) = log t; y(t; x) = t¯ transforms it to quadrature form and the substitution x s(t; x) = t; y(t; x) = t transforms it to the equations with separable variables. Example 4. (Linear equation) The equation dx (3) = p(t)x(t) + q(t) dt can be transformed into the one in quadrature form using the substitution R s = t; y = xe¡ p(t) dt: Substitutions and symmetries 3 2. Substitutions Let us first precise some terminology. Let U be a domain (a connected open subset of Rn or of a manifold M) and let F = F (t; x) be a vector field defined on U which may also depend on t 2 I ⊆ R, meaning F : I £ U ! T M; F (t; x) 2 TxM; for all x 2 U; t 2 I: n » n If M = R we have the identification TxM = R , so F : I £ U ! Rn: Consider the differential equation dx (4) = F (t; x(t)): dt We have the following notions: ² the domain U is called the phase space of the equation (4) ² the product I £ U is called the extended phase space of the equation (4) ² the image of any solution x : I !U of the equation (4) is called the phase curve of the equation (4). A substitution in the autonomous differential equation dx (5) = F (x(t)) dt on a domain U is a change of local coordinates, i.e. a diffeomorphism defined on U which hopefully simplifies the form of (5). More precisely, let Á : U!V be a diffeomorphism between domains U and V. Let Á¤F denotes the push-forward of F by Á: ¡1 Á¤F (y) := dÁÁ¡1(y)(F (Á (y)); ¡1 where dÁÁ¡1(y) denotes the derivative of Á at the point Á (y). Proposition 5. The diffeomorphism Á maps the phase curves of the equa- tion (5) to the phase curves of the equation dy = Á F (y(t)): dt ¤ Proof. By differentiation we check that y(t) := Á(x(t)) is a phase curve of a vector field Á¤F . Let us now consider the non-autonomous case (4). The most important non- autonomous equation that we study in this note is the following simple equation: dx (6) = f(t; x); dt 4 J. Kati´c where f is a function of two variables defined at some domain in R2. It is often more elegant to study a non-autonomous case as an autonomous equation in the extended phase space. This means that we treat (t; x) 2 I £ U as a new function of ¿ and instead of the equation (4) we consider the following autonomous equation in the extended phase space: 8 dt <> = g(t; x) (7) d¿ > dx dx dt : = = F (t; x) ¢ g(t; x); d¿ dt d¿ for a suitable choice of g(t; x) (desirably of a constant sign, so that ¿ 7! t is a bijection). The simplest example of making an autonomous equation out of the non- autonomous one is the choice g(t; x) = 1; now (4) can also be written as dx˜ = Fe(˜x); wherex ˜ = (t; x) 2 I £ U; Fe(˜x) := (1;F (˜x)): d¿ The solutions of the equation (4) are the projections of the solutions of the equation (7) to the phase space U. The substitution is now a diffeomorphism Á : I £ U ! J £ V; where U; V ⊆ M (or Rn) and I;J ⊆ R. 3. Symmetries Definition 6. Let F be a vector field on a domain U. We say that a diffeo- morphism ' : U!U is a symmetry of the differential equation (5) if '¤F = F . In the non-autonomous case (4) a symmetry is a diffeomorphism of the extended phase space I £ U which is a symmetry for the autonomous equation (7). Remark 7. In the non-autonomous case (4), the symmetry ' : I £ U ! I £ U is a notion that depends on the choice of the function g in (7). Proposition 8. The symmetry ' of the autonomous system (5) maps the phase curves of given system to the phase curves of the same system. The same is true for the non-autonomous system (4). Proof. The assertion for (5) follows directly from Proposition 5. Regarding the system (4), from the autonomous case we conclude that ' : I £ U ! I £ U maps the phase curves of the extended system (7) to themselves. The phase curves of the initial system are solutions of (4). Since '¤(g(t; x);F (t; x) ¢ g(t; x)) = (g(t; x);F (t; x) ¢ g(t; x)); Substitutions and symmetries 5 we see that the projections of the phase curves are solutions to dx = F (t; x) ¢ g(t; x): d¿ dx dt But this is the reparametrized curve dt = F (t; x), since d¿ = g(t; x). Definition 9. Let '" be a symmetry of the equation (5) for every " 2 (¡"0;"0). We say that '² is a one-parameter local Lie group if the following holds: ² '0 is the identity map ² 's ± 't = 's+t ² for every x 2 U, the map " 7! '"(x) is smooth. If '" is defined for all " 2 R so that the above conditions holds, we have the action of the Lie group (R; +) on U. Recall that the action of the group (G; ±) on a set U is a mapping G £ U ! U; (g; x) 7! g ¢ x such that e ¢ x = x; f ¢ (g ¢ x) = (f ± g) ¢ x (where e is the neutral in G and ± is the group operation). Recall also that the orbit of a point x 2 U is defined as the set G ¢ x := fg ¢ x j g 2 Gg: Example 10. The one-parameter Lie group '" : R ! R defined as '" :(t; x) 7! (t; x + ") is a symmetry for the non-autonomous equation in quadrature form dx = f(t): dt Remark 11. Example 10 is very simple, actually, one does not need to notice any symmetry in order to solve an equation which is in quadrature form. However, this simple example is important since it notices a symmetry that an equation in quadrature form possesses – namely the symmetry with respect to the translation along x-axis.1 The orbits of this action are vertical lines in the plane. The purpose of this paper is to show how to find a substitution which transforms the orbits of a given (more complex) symmetry (of a more complex differential equation) to the straight lines (vertical or horizontal ones), and thus transforms the more complicated equation to the simple equation in Example 10. We will come back to this later, in Proposition 16 and Theorem 17. The following example illustrates the previous remark. 1 On the other hand, the one-parameter Lie group '" :(t; x) ! (t + "; x) is a symmetry for dx the equation dt = f(x). 6 J. Kati´c Example 12. (Homogeneous linear equation) The equation2 dx (8) = p(t) ¢ x; x(0) = x > 0 dt 0 is invariant with respect to the homothety x 7! ¸x: Of course, it is solvable as an equation with separated variables.
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