Dynamical Systems and Lagrangian Spaces

V. Obădeanu and I.D. Albu

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract For the general case of the dynamical systems which are not necessarily Lagrangian, a geometrical method for the determination of a second order Lagrangian (linear with respect to accelerations) is described.

Mathematics Subject Classification: 58F05, 70Hxx Key words: sprays, first order and second order Lagrangians

1 Introduction

Several physical theories use second order Lagrangians. Generally speaking any geometric formulation of the higher order Lagrangian systems is developed in the framework of higher order tangent bundles by using their underlying canonical structures.

In this Note we discuss the dynamical systems on manifolds locally expressed by second order differential equations of the form

(1)  \frac{d^{2}x^{i}}{dt^{2}} = S^{i} (x, \frac{dx}{dt}) , i = \overline{1, n}.

It is known that for such a dynamical system there exists a Lagrangian, which is linear with respect to accelerations [6], [7] that is

(2)  L(x, ẋ, ẍ) = A_{i}(x, ẋ)ẍ^{i} + B(x, ẋ).

Our aim is to present a geometrical manner to obtain a Lagrangian for (1). In order to do this we use the following arguments: 1) The existence of an infinite Riemannian structures on the tangent manifold of a Lagrange space (in sense of [3]); 2) For a first order dynamical system,

(3)  \frac{dx^{i}}{dt} = f^{i}(x), i = \overline{1, n},

on a Riemannian space (M, g), there exists a nondegenerate Lagrangian of the form [5], [8]

(4)  L(x, ẋ) = \frac{1}{2}g_{ij}(x)(ẋ^{i} − f^{i})(ẋ^{j} − f^{j}), .

Let us remark that the result is still valid if g is an indefinite Riemannian metric on M.

Balkan Journal of Geometry and Its Applications, Vol.5, No.2, 2000, pp. 75-80 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2 Geometrical structures associated to sprays

Let M be an n-dimensional differentiable manifold. A spray on M is a vector field S on the tangent manifold T M such that

JS = C,

where C ∈ X (T M) is the canonical vector field on T M and J ∈ X_{1}^{1}(T M) is the natural almost tangent structure on T M. Locally,

J = dx^{i} ⊗ \frac{∂}{∂y^{j}} , C = y^{i} \frac{∂}{∂y^{i}}

and

(5)  S = y^{i} \frac{∂}{∂x^{i}} + S^{i} \frac{∂}{∂y^{i}} ,

where (x^{i}, y^{i}) are the local coordinates on T M and \{\frac{∂}{∂x^{i}} , \frac{∂}{∂y^{i}} \} is the natural local

basis.

To give a second order system (1) on M is equivalent to give a spray S on M, and the trajectories of S are the solutions of (1).

The geometry induced by S follows from the existence of the nonlinear connection N associated to S. Locally N is determined by the coefficients

(6)  N_{j}^{i} = −\frac{1}{2} \frac{∂S^{i}}{∂y^{i}} .

The local basis adapted to the horizontal distribution HT M of N and to the vertical distribution V T M on T M is given by

(7)  \{\frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − N_{i}^{j} \frac{∂}{∂y^{j}} , \frac{∂}{∂y^{i}} \}

and the dual basis of (7) is

(8)  {dx^{i}, δy^{i} = dy^{i} + N_{j}^{i}dx^{j}}.

A pair (M, L) is called a Lagrange space if L ∈ C^{∞}(T M) is a nondegenerate or

regular Lagrangian, that is det ∥\frac{∂^{2}L}{∂y^{i}∂y^{j}} ∥ ≠ 0. ([4]). On a Lagrange space (M, L) there

is a canonical spray (the Lagrangian spray associated to L) locally given by (5) with

(9)  S^{i} = g^{ij} (\frac{∂L}{∂x^{j}} − \frac{∂^{2}L}{∂x^{k}∂y^{j}} y^{k});

g^{ij} are the components of the inverse matrix of ∥g_{ij}∥ = ∥\frac{∂^{2}L}{∂y^{i}∂y^{j}} ∥ .

A tensor field g of type (0, 2) on T M is called a d-tensor field of type (0, 2) on M if the components g_{ij}(x, y) of g change by the relations

(10)  g_{i′j′} = \frac{∂x^{i}}{∂x^{i′}} \frac{∂x^{j}}{∂x^{j′}} g_{ij},

with respect to a local coordinate transformation x^{i} = x^{i}(x^{1′}, …, x^{n′}) on M, i.e., g is a vertical tensor field of type (0, 2) on M or g = g_{ij}(x, y)dx^{i} ⊗ dx^{j}.

A pair (M, g) where g is a nondegenerate symmetric d -tensor field of type (0, 2) on M is called a generalized Lagrange space with the generalized Lagrange metric g ([4]). A Riemannian metric, a Finslerian metric and the fundamental tensor g =

\frac{∂^{2}L}{∂y^{i}∂y^{j}} dx^{i} ⊗ dx^{j} of a Lagrange space supply such generalized Lagrange metrics.

3 Determination of Lagrangians for a dynamical system

In the sequel we shall present a geometrical method for the determination of a Lagrangian of the form (2) in the general case of the dynamical systems which are not necessarily Lagrangian in classical sense. Proposition 1. Any dynamical system of the form (1) defined on a generalized Lagrange space admits a Lagrangian of the form (2) such that the corresponding Euler-Lagrange equations were linear combinations of the equations (1) and their derivatives. Proof. Let (M, g) be a generalized Lagrange space and let g_{ij}(x, y) be the components of g in a local chart. Also let S be the spray which corresponds to the system (1). S is of the form (5), where the components S^{i}(x, y) are the right-hand sides of (1). Lemma. (M, g, S) defines a canonical indefinite Riemannian metric on T M,

(11)  G = 2g_{ij}dx^{i} ⊗ δy^{j},

where {δy^{i}, dx^{i}} is the co-basis adapted to the nonlinear connection N associated to S via (6). Proof. G is globally defined on T M and nonsingular. The quadratic form defined by G is dσ = 2g_{ih}N_{j}^{h}dx^{i}dx^{j} + 2g_{ij}dx^{i}dy^{j} and the matrix of the components of G is

(12)  ∥G_{ab}∥ = ∥ \begin{matrix}g_{ih}N_{j}^{h} + g_{jh}N_{i}^{h} & g_{ij} \\ g_{ij} & 0 \end{matrix} ∥ .

G may be considered as the horizontal lift of g with respect to N.

Going back to our problem let us consider a first order system on T M associated to (1), namely

(13)  \frac{dy^{i}}{dt} = S^{i}(x, y), \frac{dx^{i}}{dt} = y^{i}, i = \overline{1, n},

or, in a homogeneous writing,

(14)  ż^{a} = F^{a}(z), a = \overline{1, 2n},

where (z^{a}) = (x^{i}, y^{i}), F^{i} = S^{i}, F^{n+i} = y^{i}, i = \overline{1, n}. Obviously the solutions of (1) are the projections of the solutions of (13). Using the structure of T M and the result from [5] we obtain for the system (14) a Lagrangian L : T T M → R of the form

(15)  L(z, ż) = \frac{1}{2}G_{ab}(z)(ż^{a} − F^{a})(ż^{b} − F^{b}), a, b = \overline{1, 2n},

where G_{ab} are given by (12). Thanks to the special form of the matrix ∥G_{ab}∥, the expression of L in the initial variables is (2). (T M, L) is a Lagrange space whose geodesics are combinations of the solutions of (13), respectively the Euler-Lagrange equations associated to L are linear combinations of the equations (1) and their derivatives.

Let (M, L) be an n -dimensional Lagrange space and let S be the canonical Lagrangian spray associated to L. S is determined by the relations (9) and the corresponding second order differential equations (1) coincide with the Euler-Lagrange equations

(16)  \frac{d}{dt} (\frac{∂L}{∂y^{i}} ) − \frac{∂L}{∂x^{i}} = 0, i = \overline{1, n},

where y^{i} = \frac{dx^{i}}{dt} = ẋ^{i}. The trajectories of S, respectively the solutions of (1) are the geodesics of (M, L). Finally, the tangent manifols T M of the Lagrange space (M, L) is endowed with a canonical indefinite Riemannian metric. This metric is defined by

(M, \frac{∂^{2}L}{∂y^{i}∂y^{j}} dx^{i} ⊗ dx^{j}, S) via Lemma.

Example. The Morse-Feshbach equations

\{ \begin{matrix}ẍ^{1} + bẋ^{1} + kx^{1} = 0, \\ ẍ^{2} − bẋ^{2} + kx^{2} = 0, (b, k ∈ R, k > 0) \end{matrix}

on a 2-dimensional manifold M_{2}, supply a Lagrangian spray

S = y^{1} \frac{∂}{∂x^{1}} + y^{2} \frac{∂}{∂x^{2}} − (by^{1} + kx^{1})\frac{∂}{∂y^{1}} + (by^{2} − kx^{2})\frac{∂}{∂y^{2}}

on M_{2}. The Lagrangian which generates S is L : T M → R, L(x, y) = y^{1}y^{2} + \frac{b}{2}(x^{1}y^{2} −

x^{2}y^{1}) − kx^{1}x^{2}. The fundamental tensor of (M, L) is g = 2dx^{1} ⊗ dx^{2} and the metric on T M has the components

∥G_{ab}∥ = ∥ \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ∥ .

Now we consider the most general case when a second order system of the form (1) is given on a n-dimensional manifold M. In general the solutions of such a system satisfy so called Euler-Lagrange equations of type II,

(17)  \frac{d}{dt} (\frac{∂L_{0}}{∂y^{i}} ) − \frac{∂L_{0}}{∂x^{i}} = Q_{i}(x, y), y^{i} = \frac{dx^{i}}{dt} , i = \overline{1, n},

where L_{0} ∈ C^{∞}(T M) is a nondegenerate first order Lagrangian and Q_{i} ∈ C^{∞}(T M), i = \overline{1, n}.

Proposition 2. Any dynamical system of the form (1) on M satisfying the above condition admits a Lagrangian of the form (2) such that the corresponding Euler-Lagrange equations were linear combinations of the equations (1) and their derivatives. Proof. Under the supposed conditions, the spray S defined by (1) on M admits a decomposition S = S_{0}+S′, where S_{0} is the Lagrangian spray associated to L_{0}. (M, L_{0})

is a Lagrange space and g = \frac{∂L_{0}}{∂y^{i}∂y^{j}} dx^{i} ⊗ dx^{j}, i, j = \overline{1, n}, defines the fundamental

tensor field of (M, L_{0}). According to Lemma there is an indefinite Riemannian metric G_{0} on T M, determined by g and the nonlinear connection associated to S_{0}, therefore intrinsically jointed to S. From now on the proof is similar to that of Proposition 1. Example. The spray defined on a 2-dimensional manifold M_{2} by the equations

(18)  \{ \begin{matrix}ẍ^{1} + εx^{1} = 0 \\ ẍ^{2} + εẋ^{1} = 0, ε ∈ \{−1, 1\}, \end{matrix}

is not variational in classical sense. For ε = −1, (18) represents the Whittaker equations and for ε = 1 we obtain the Urrutia-Hojmann equations. Both systems can be studied with the same Lagrangian spray if (18) is expressed like that

(19)  \{ \begin{matrix}ẍ^{1} + x^{1} = (1 − ε)x^{1}, \\ ẍ^{2} + x^{2} = x^{2} − εẋ^{1}. \end{matrix}

The spray defined by the system

(20)  \{ \begin{matrix}ẍ^{1} + x^{1} = 0, \\ ẍ^{2} + x^{2} = 0, \end{matrix}

is variational (S_{0} = y^{1} \frac{∂}{∂x^{1}} + y^{2} \frac{∂}{∂x^{2}} − x^{1} \frac{∂}{∂y^{1}} − x^{2} \frac{∂}{∂y^{2}} ) and the coresponding La-

grangian of (20) is L_{0} = y^{1}y^{2} − x^{1}x^{2}. For example if ε = −1 the metric of T M_{2}

has the nonzero components G_{14} = G_{41} = G_{23} = G_{32} = \frac{1}{2}. The first order system

associated to (19) is ẏ^{1} − x^{1} = 0, ẏ^{2} − y^{1} = 0, ẋ^{1} − y^{1} = 0, ẋ^{2} − y^{2} = 0 and the corresponding Lagrangian is L = (ẋ^{2}−y^{2})ẏ^{1}+(ẋ^{1}−y^{1})ẏ^{2}+x^{1}y^{2}+(y^{1})^{2}−x^{1}ẋ^{2}−y^{1}ẋ^{1}. Aknowledgements. A version of this paper was presented at the Third Conference of the Balkan Society of Geometers, Workshop on Electromagnetic Flows and Dynamics, July 31 - August 3, 2000, University POLITEHNICA of Bucharest, Romania.

Supported by MEN Grant No 21815/28.09.1998, CNCSU-31.

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West University of Timişoara Department of Algebra Bulevardul V. Pîrvan 4, 1900 Timişoara, Romania E-mail: obadeanu@hilbert.math.uvt.ro