Boundary problem for system of the second order. Green function.

We consider a system of the second order boundary value problem of the

type

 

with the boundary conditions u (a) = and u (b) = (2)

and the continuity conditions of u and u’ at c and d. Here, f and g are continuous functions on [ a; b ] and [ c; d ], respectively. The parameters r; and

are real finite constants. Such type of systems arises in connection with the Green function. L[y] = 0, U1[y] = 0, U2[y] = 0. has only trivial solution.

Let (λ1, λ2) 6= (0, 0) be such that a1λ1 + a2λ2 = 0 and let φ1 be solution of L[y] = 0 satisfying φ1(a) = λ1 and φ′1(a) = λ2. Choose another solution φ2 of L[y] = 0 similarly. This way of choosing φ1 and φ2 make sure that both are non-trivial solutions.

Note that φ1 and φ2 form a fundamental pair of solutions of L[y] = 0, since we assumed that homogeneous BVP has only trivial solutions.

By Lagrange’s identity (5.20), we get d/dx[p(φ′1φ2 − φ1φ′2)= 0. This implies p(φ′1φ2 − φ1φ′2) ≡ c, a constant and non-zero, (5.44)

as a consequence of (φ′1φ2 − φ1φ′2) being the wronskian corresponding to a fundamental pair of solutions.

Green’s function is then given by

G(x, ξ):=1/c

 

21) Reduction of equation to canonical form.

Canonical form in a differential form that is defined in a natural (canonical) way; Finding a canonical form is called canonization. Canonical differential forms include the c anonical one-form and canonical symplectic form For instance, the expression f(x) dx from one-variable calculus is called a 1-form, and can be integrated over an interval [ a, b ] in the domain of f and similarly the expression: f (x, y, z) dx ∧ dy + g (x, y, z) dx ∧ dz + h (x, y, z) dy ∧ dz is a 2-form that has a surface integral over an oriented surface S:

Likewise, a 3-form f(x, y, z) dxdydz represents something that can be integrated over a region of space

Canonical one-form is a special 1-form defined on the cotangent bundle T * Q of a manifold Q. The exterior derivative of this form defines a symplectic form giving T * Q the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. In canonical coordinates, the tautological one-form is given by _idqⁱ

Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential (exact form), may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations.

The canonical symplectic form is given by ⁱdp_i

The extension of this concept to extended to general fibre bundles is known as the solder form.

 

22) Autonomic system properties. Trajectory, phase space.

A system of ordinary differential equations which does not explicitly contain the independent variable (time). The general form of a first-order autonomous system in normal form is:

= () j=1,…,n

or, in vector notation, =f(x). (1)

A non-autonomous system = f(t,x) can be reduced to an autonomous one by introducing a new unknown function =t. Historically, autonomous systems first appeared in descriptions of physical processes with a finite number of degrees of freedom. They are also called dynamical or conservative systems

A complex autonomous system of the form (1) is equivalent to a real autonomous system with 2n unknown functions

(Re x)= Re f(x), (Imx)=Im f(x).

The essential contents of the theory of complex autonomous systems — unlike in the real case — is found in the case of an analytic f(x)

Consider an analytic system with real coefficients and its real solutions. Let x Φ (t) be an (arbitrary) solution of the analytic system (1), let ∆ =( ) be the interval in which it is defined, and let x(t; , ) be the solution with initial data x = . Let G be a domain in and f (G). The point G is said to be an equilibrium point, or a point of rest, of the autonomous system (1) if f() 0. The solution, Φ(t) , t corresponds to such an equilibrium point

Local properties of solutions.

1) If Φ(t) is a solution, then Φ(t+c) is a solution for any c .

2) Existence: For any , G,a solution x(t; , ) exists in a certain interval .

 

3) Smoothness: If f⋲ , then Φ(t)⋲ .

4) Dependence on parameters: Let, f=f(x,α),α⋲ C R,where is a domain; if f⋲ (G* ),p≥1, x(t; , )⋲ (∆* )

5) Let be a non-equilibrium point; then there exist neighbourhoods V,W of the points , f()respectively, and a diffeomorphism y=h(x): VàW such that the autonomous system has the form = const in W.

 

A substitution of variables x = Φ(y) in the autonomous system (1) yields the system (2)

f(Φ(y)), where (y) is the Jacobi matrix.

where is the Jacobi matrix.

 

23) Solution properties of autonomic systems.