Solutions of DEs and PDEs as Potential Maps Using First Order Lagrangians

Constantin Udrişte

Abstract

Recently we have solved a problem rised for the first time by Poincaré (find a suitable geometric structure that describes the trajectories of a given vector field like geodesics), showing that the trajectories of a given vector field are pregeodesics in a suitable Riemann-Jacobi, Riemann-Jacobi-Lagrange or Finsler-Jacobi structure [6]-[10]. Continuing to develop similar ideas, the present paper and [11] show that solutions of DEs or PDEs are potentials maps via first order Lagrangians or via generalized Lorentz world-force laws.

This paper is organized as follows. Section 1 and Section 3 review the notion of (single-time respectively multi-time) jet bundle of order one, and establish second-order derivative operator along a local section (suitable decomposition of the Laplacian), adapted dual bases, a Sasaki-like metric, a generalized Lorentz World-Force Law, a suitable second-order prolongation of a first-order (DEs respectively PDEs) system, first-order Lagrangians producing the prolongation, and Lorentz-Udrişte World-Force Law. Section 2 and Section 4 give the Hamiltonian and the non-degenerate distinguished symplectic relative 2-form which permit the changing of the Lagrangian dynamics into covariant Hamilton equations.

Mathematics Subject Classification: 34C40, 31C12, 53C43, 58E20 Key words: jet bundle of order one, DEs, PDEs, potential maps, Lagrangians of order one, covariant Hamilton equations

1 Solutions of DEs as potential maps

Unless specifically denied, all manifolds, all objects on them, and all maps from one manifold into another will be C^{∞}; however, we sometimes redundantly write ”a C^{∞} manifold”, and so on, for emphasis.

Let (T = R, h) and (M, g) be semi-Riemann manifolds of dimensions 1 and n. Hereafter we shall assume that the manifold T is oriented. Latin letters will be used for indexing the components of geometrical objects attached to the manifold M.

Local coordinates will be written

Balkan Journal of Geometry and Its Applications, Vol.6, No.1, 2001, pp. 93-108 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

t = t^{1}, x = (x^{i}), i = 1, …, n,

and the components of the corresponding metric tensors and Christoffel symbols will be denoted by h_{11}, g_{ij}, H_{11}^{1}, G_{jk}^{i}. Indices of distinguished objects will be rised and lowered in the usual fashion.

Let C^{∞}(T, M) = {φ : T → M | φ of class C^{∞}}. For any φ, ψ ∈ C^{∞}(T, M), we define the equivalence relation φ ∼ ψ at (t_{0}, x_{0}) ∈ T × M by

x^{i}(t_{0}) = y^{i}(t_{0}) = x_{0}^{i}, \frac{dx^{i}}{dt} (t_{0}) = \frac{dy^{i}}{dt} (t_{0}).

Using the factorization

J_{(t_{0},x_{0})}^{1}(T, M) = C^{∞}(T, M)/ ∼

we introduce the jet bundle of order one

J^{1}(T, M) = ⋃_{(t_{0},x_{0})∈T ×M} J_{(t_{0},x_{0})}^{1}(T, M).

Denoting by [φ]_{(t_{0},x_{0})} the equivalence class of the map φ, we define the projection

π : J^{1}(T, M) → T × M, π[φ]_{(t_{0},x_{0})} = (t_{0}, φ(t_{0})).

Suppose that the base T × M is covered by a system of coordinate neighborhoods (U × V, t^{α}, x^{i}). Then we can define the diffeomorphism

F_{U×V} : π^{−1}(U × V) → U × V × R^{1·n}

F_{UV} [φ]_{(t_{0},x_{0})} = (t_{0}, x_{0}^{i}, \frac{dx^{i}}{dt} (t_{0})) .

Consequently J^{1}(T, M) is a differentiable manifold of dimension 1 + n + 1 · n = 2n + 1. The coordinates on π^{−1}(U × V) ⊂ J^{1}(T, M) will be

(t^{1} = t, x^{i}, y^{i} = \frac{dx^{i}}{dt} ) ,

where

t^{1}([φ]_{(t_{0},x_{0})}) = t^{1}(t_{0}), x^{i}([φ]_{(t_{0},x_{0})}) = x^{i}(x_{0}), y^{i}([φ]_{(t_{0},x_{0})}) = \frac{dx^{i}}{dt} (t_{0}).

A local changing of coordinates (t, x^{i}, y^{i}) → (t̄, x̄^{i}, ȳ^{i}) is given by

(1)  t̄= t̄(t), x̄^{i} = x̄^{i}(x^{j}), ȳ^{i} = \frac{∂x̄^{i}}{∂x^{j}} \frac{dt}{dt̄} y^{j},

where

\frac{dt̄}{dt} > 0, det (\frac{∂x̄^{i}}{∂x^{j}} ) ≠ 0.

Solutions of DEs and PDEs as Potential Maps Using First Order Lagrangians 95

The expression of the Jacobian matrix of the local diffeomorphism (1) shows that the jet bundle of order one J^{1}(T, M) is always orientable.

Let

H_{11}^{1} = \frac{1}{2}h^{11} \frac{dh_{11}}{dt^{1}} = \frac{1}{2}h_{11}^{−1} \frac{dh_{11}}{dt^{1}} = \frac{1}{2} \frac{d}{dt^{1}} \sqrt{|h_{11}|},

G_{jk}^{i} be the components of the connections induced by h and g respectively. If

(t = t^{1}, x^{i}, y^{i} = \frac{dx^{i}}{dt} ) are the coordinates of a point in J^{1}(T, M), then

\frac{δ}{dt} \frac{dx^{i}}{dt} = \frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt}

are the components of a distinguished tensor on T × M. Also

(\frac{δ}{δt} = \frac{d}{dt} + H_{11}^{1}y^{i} \frac{∂}{∂y^{i}} , \frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − G_{ik}^{h}y^{k} \frac{∂}{∂y^{h}} , \frac{∂}{∂y^{i}} )

(dt, dx^{j}, δy^{j} = dy^{j} − H_{11}^{1}y^{j}dt + G_{hk}^{j}y^{h}dx^{k})

are dual frames on J^{1}(T, M), i.e.,

dt (\frac{δ}{δt}) = 1, dt (\frac{δ}{δx^{i}} ) = 0, dt (\frac{∂}{∂y^{i}} ) = 0

dx^{j} (\frac{δ}{δt}) = 0, dx^{j} (\frac{δ}{δx^{i}} ) = δ_{i}^{j}, dx^{j} (\frac{∂}{∂y^{i}} ) = 0

δy^{j} (\frac{δ}{δt}) = 0, δy^{j} (\frac{δ}{δx^{i}} ) = 0, δy^{j} (\frac{∂}{∂y^{i}} ) = δ_{i}^{j}.

Using these frames, we define on J^{1}(T, M) the induced Sasaki-like metric

S_{1} = h_{11}dt ⊗ dt + g_{ij}dx^{i} ⊗ dx^{j} + h^{11}g_{ij}δy^{i} ⊗ δy^{j}.

The semi-Riemann geometry of the manifold (J^{1}(T, M), S_{1}) was developed recently in [4].

Now we shall generalize the Lorentz world-force law which was initially stated [5] for particles in nonquantum relativity. Definition. Let F = (F_{j}^{i}) and U = (U^{i}) be C^{∞} distinguished tensors on T × M, where ω_{ji} = g_{hi}F_{j}^{h} is skew-symmetric with respect to j and i. Let c(t, x) be a C^{∞} real function on T × M. A map φ : T → M obeys a Generalized Lorentz World-Force Law with respect to F, U, c iff

h^{11} \frac{δ}{dt} \frac{dx^{i}}{dt} = h^{11} (g^{ij} \frac{∂c}{∂x^{j}} + F_{j}^{i} \frac{dx^{j}}{dt} + U^{i}) .

Now we remark that a C^{∞} distinguished tensor field X^{i}(t, x), i = 1, …, n on T × M defines a family of trajectories as solutions of DEs system of order one

(2)  \frac{dx^{i}}{dt} = X^{i}(t, x(t)).

The distinguished tensor field X^{i}(t, x) and semi-Riemann metrics h and g determine the potential energy density

f : T × M → R, f = \frac{1}{2}h^{11}g_{ij}X^{i}X^{j}.

The distinguished tensor field (family of trajectories) X^{i} on (T ×M, h_{11} +g) is called: 1) timelike, if f < 0;

2) nonspacelike or causal, if f ≤ 0;

3) null or lightlike, if f = 0;

4) spacelike, if f > 0.

Let X^{i} be a distinguished tensor field of everywhere constant energy. If X^{i} (the system (2)) has no critical point on M, then upon rescaling, it may be supposed that f ∈ {−1, 0, 1}. Generally, E = {x_{0} ∈ M|X^{i}(t, x_{0}) = 0, ∀t ∈ T } is the set of critical points of the distinguished tensor field, and this rescaling is possible only on T × (M \ E).

Using the operator (derivative along a solution of (2) via the decomposition \frac{δ}{dt} · \frac{d}{dt})

\frac{δ}{dt} \frac{dx^{i}}{dt} = \frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt} ,

the Levy-Civita connection D of (R, h) and the Levy-Civita connection ∇ of (M, g), we obtain the prolongation (DEs system of order two)

(3)  \frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt} = DX^{i} + (∇_{j}X^{i})\frac{dx^{j}}{dt} ,

where

∇_{j}X^{i} = \frac{∂X^{i}}{∂x^{j}} + G_{jk}^{i}X^{k}, DX^{i} = \frac{∂X^{i}}{∂t} − H_{11}^{1}X^{i}.

The distinguished tensor field X^{i}, the metric g, and the connection ∇ determine the external distinguished tensor field

F_{j}^{i} = ∇_{j}X^{i} − g^{ih}g_{kj}∇_{h}X^{k},

which characterizes the helicity of the distinguished tensor field X^{i}.

The DEs system (3) can be written in the equivalent form

(4)  \frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt} = g^{ih}g_{kj}(∇_{h}X^{k})\frac{dx^{j}}{dt} + F_{j}^{i} \frac{dx^{j}}{dt} + DX^{i}.

Now we modify this DEs system into

(5)  \frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt} = g^{ih}g_{kj}(∇_{h}X^{k})X^{j} + F_{j}^{i} \frac{dx^{j}}{dt} + DX^{i}.

The system (5) is still a prolongation of the DEs system (2). Theorem. The kinematic system (2) can be prolonged to the second order dynamical system (5).

Corollary. Choosing the metrics h and g such that f ∈ {−1, 0, 1}, then the kinematic system (2) can be prolonged to the second order dynamical system

\frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt} = F_{j}^{i} \frac{dx^{j}}{dt} + DX^{i}.

We shall show that the dynamical system (5) has a variational structure, being in fact an Euler-Lagrange system. We identify J^{1}(T × M) with its dual via the semi-Riemann metrics h and g. Theorem. 1) The solutions of the DEs system (5) are the extremals of the Lagrangian

L = \frac{1}{2}h^{11}g_{ij} (\frac{dx^{i}}{dt} − X^{i}) (\frac{dx^{j}}{dt} − X^{j}) \sqrt{|h_{11}|} =

= (\frac{1}{2}h^{11}g_{ij} \frac{dx^{i}}{dt} \frac{dx^{j}}{dt} − h^{11}g_{ij} \frac{dx^{i}}{dt} X^{j} + f) \sqrt{|h_{11}|}.

2) If F_{j}^{i} = 0, then the solutions of the DEs system (5) are the extremals of the Lagrangian

L = (\frac{1}{2}h^{11}g_{ij} \frac{dx^{i}}{dt} \frac{dx^{j}}{dt} + f) \sqrt{|h_{11}|}.

3) Both Lagrangians produce the same Hamiltonian

H = (\frac{1}{2}h^{11}g_{ij} \frac{dx^{i}}{dt} \frac{dx^{j}}{dt} − f) \sqrt{|h_{11}|}.

Theorem (Lorentz-Udrişte World-Force Law) see also [6]-[10].

1) Every solution of DEs system

\frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt} = g^{ih}g_{kj}(∇_{h}X^{k})X^{j} + DX^{i}

is a potential map on the semi-Riemann manifold (T × M, h + g).

2) Every solution of DEs system (5) is a horizontal potential map on the semi-Riemann-Lagrange manifold

(T × M, h + g, N(_{1}^{i})_{j} = G_{jk}^{i}y^{k} − F_{j}^{i}, M(_{1}^{i})_{1} = −H_{11}^{1}y^{i}).

Corollary. Every DE generates a Lagrangian of order one via the associated first order DEs system and suitable metrics on the manifold of independent variable and on the manifold of functions. In this sense the solutions of the initial DE are potential maps produced by a suitable Lagrangian. Proof. Let t ∈ R denote a real variable, usually referred to as the time. It may be pointed out that the DE

(6)  \frac{d^{n}x}{dt^{n}} = f (t, x, \frac{dx}{dt}, …, \frac{d^{n−1}x}{dt^{n−1}}) ,

where x is the unknown function, is equivalent to a system (2). For if we set x = x^{1}, then (6) is equivalent to

\frac{dx^{1}}{dt} = x^{2}, \frac{dx^{2}}{dt} = x^{3}, …, \frac{dx^{n−1}}{dt} = x^{n}

\frac{dx^{n}}{dt} = f(t, x^{1}, x^{2}, …, x^{n}),

which is of type (2). Therefore, the preceding theory applies.

2 Hamiltonian approach

Let (Q, Ω) be a symplectic manifold (of even dimension). The Hamiltonian vector field X_{H} of the function H ∈ F(Q) is defined by

X_{H} Ω = dH.

We generalize this relation as

X_{H}^{1} Ω_{1} = \sqrt{|h_{11}|}dH,

using the distinguished objects

X_{H}^{1}, Ω_{1}, H

and the manifold J^{1}(T, M). For another point of view, see also [11] and compare with [2], [3]. Theorem. The DEs system

\frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt} = g^{ih}g_{kj}(∇_{h}X^{k})X^{j}

transfers in J^{1}(T, M) as a Hamilton DEs system with respect to the Hamiltonian

H = \frac{1}{2}h^{11}g_{ij}y^{i}y^{j} − f

and the non-degenerate distinguished symplectic relative 2-form

Ω = Ω_{1} ⊗ dt^{1}, Ω_{1} = g_{ij}dx^{i} ∧ δy^{j}\sqrt{|h_{11}|}.

Proof. Let

θ = θ_{1} ⊗ dt^{1}, θ_{1} = g_{ij}y^{i}dx^{j}\sqrt{|h_{11}|}

be the distinguished Liouville relative 1-form on J^{1}(T, M). We find

Ω_{1} = −dθ_{1}.

We introduce

X_{H} = X_{H}^{1} \frac{δ}{δt}, X_{H}^{1} = u^{1l} \frac{δ}{δx^{l}} + \frac{δu^{1l}}{dt} \frac{∂}{∂y^{l}}

as the distinguished Hamiltonian object associated to the function H.

The relation

X_{H}^{1} Ω_{1} = \sqrt{|h_{11}|}dH,

Solutions of DEs and PDEs as Potential Maps Using First Order Lagrangians 99 where

dH = h^{11}g_{ij}y^{j}δy^{i} − h^{11}g_{ij}(DX^{i})X^{j}dt − h^{11}g_{ij}X^{j}∇_{k}X^{i}dx^{k},

implies

g_{ij}u^{1i}δy^{j} − g_{ij} \frac{δu^{1j}}{dt} dx^{i} = dH.

Consequently, it appears the PDEs system of Hamilton type

\{ \begin{matrix}u^{1i} = h^{11}y^{i} \\ \frac{δu^{1i}}{dt} = g^{hi}h^{11}g_{jk}X^{j}(∇_{h}X^{k}) \end{matrix}

together the condition

h^{11}g_{ij}(DX^{i})X^{j} = 0.

Theorem. The DEs system

\frac{d^{2}x^{i}}{dt^{2}} − H_{11}^{1} \frac{dx^{i}}{dt} + G_{jk}^{i} \frac{dx^{j}}{dt} \frac{dx^{k}}{dt} = g^{ih}g_{kj}(∇_{h}X^{k})X^{j} + F_{j}^{i} \frac{dx^{j}}{dt} + DX^{i}

transfers in J^{1}(T, M) as a Hamilton DEs system with respect to the Hamiltonian

H = \frac{1}{2}h^{11}g_{ij}y^{i}y^{j} − f

and the non-degenerate distinguished symplectic relative 2-form

Ω = Ω_{1} ⊗ dt, Ω_{1} = (g_{ij}dx^{i} ∧ δy^{j} + ω_{ij}dx^{i} ∧ dx^{j} + g_{ij}(DX^{i})dt ∧ dx^{j})\sqrt{|h_{11}|}, where

ω_{ji} = g_{hi}F_{j}^{h}.

Proof. Let

θ = θ_{1} ⊗ dt^{1}, θ_{1} = (g_{ij}y^{i}dx^{j} − g_{ij}X^{i}dx^{j})\sqrt{|h_{11}|}

be the distinguished Liouville relative 1-form on J^{1}(R, M). We find

Ω_{1} = −dθ_{1}.

We denote

X_{H} = X_{H}^{1} \frac{δ}{δt}, X_{H}^{1} = h^{11} \frac{δ}{δt} + u^{1l} \frac{δ}{δx^{l}} + \frac{δu^{1l}}{dt} \frac{∂}{∂y^{l}}

the distinguished Hamiltonian object of the function H. The relation

X_{H}^{1} Ω_{1} = \sqrt{|h_{11}|}dH

can be written

g_{ij}u^{1i}δy^{j} − g_{ij} \frac{δu^{1j}}{dt} dx^{i} + 2ω_{ij}u^{1i}dx^{j} − g_{ij}(DX^{i})u^{1j}dt + h^{11}g_{ij}(DX^{i})dx^{j} = dH,

where

dH = −h^{11}g_{ij}(DX^{i})X^{j}dt + h^{11}g_{ij}y^{j}δy^{i} − h^{11}g_{ij}X^{j}(∇_{k}X^{i})dx^{k}.

Via these relations we identify a PDEs system of Hamilton type,

\{ \begin{matrix}u^{1i} = h^{11}y^{i} \\ \frac{δu^{1i}}{dt} = g^{hi}h^{11}g_{jk}X^{j}(∇_{h}X^{k}) + 2g^{hi}ω_{jh}u^{1j} + h^{11}DX^{i} \end{matrix}

together the condition

g_{ij}(DX^{i})(u^{1j} − h^{11}X^{j}) = 0.

3 Solutions of PDEs as Potential Maps

All manifolds and maps are C^{∞}, unless otherwise stated.

Let (T, h) and (M, g) be semi-Riemann manifolds of dimensions p and n. Hereafter we shall assume that the manifold T is oriented. Greek (Latin) letters will be used for indexing the components of geometrical objects attached to the manifold T (manifold M).

Local coordinates will be written

t = (t^{α}), α = 1, …, p
x = (x^{i}), i = 1, …, n,

and the components of the corresponding metric tensors and Christoffel symbols will be denoted by h_{αβ}, g_{ij}, H_{βγ}^{α}, G_{jk}^{i}. Indices of tensors or distinguished tensors will be rised and lowered in the usual fashion.

Let C^{∞}(T, M) = {φ : T → M| φ of class C^{∞}}. For any φ, ψ ∈ C^{∞}(T, M) we define the equivalence relation φ ∼ ψ at (t_{0}, x_{0}) ∈ T × M, by

x^{i}(t_{0}) = y^{i}(t_{0}) = x_{0}^{i}, \frac{∂x^{i}}{∂t^{α}} (t_{0}) = \frac{∂y^{i}}{∂t^{α}} (t_{0}).

Using the factorization

J_{(t_{0},x_{0})}^{1}(T, M) = C^{∞}(T, M)/_{∼},

we introduce the jet bundle of order one

J^{1}(T, M) = ⋃_{(t_{0},x_{0})∈T ×M} J_{t_{0},x_{0}}^{1}(T, M).

Denoting by [φ]_{(t_{0},x_{0})} the equivalence class of the map φ, we define the projection

π : J^{1}(T, M) → T × M, π[φ]_{(t_{0},x_{0})} = (t_{0}, φ(t_{0})).

Suppose that the base T × M is covered by a systems of coordinate neighborhood (U × V, t^{α}, x^{i}). Then we can define the diffeomorphism

F_{U×V} : π^{−1}(U × V) → U × V × R^{pn},

F_{UV} [φ]_{(t_{0},x_{0})} = (t_{0}^{α}, x_{0}^{i}, \frac{∂x^{i}}{∂t^{α}} (t_{0})) .

Consequently J^{1}(T, M) is a differentiable manifold of dimension p + n + pn. The coordinates on π^{−1}(U × V) ⊂ J^{1}(T, M) will be

(t^{α}, x^{i}, x_{α}^{i}),

where

t^{α} ([φ]_{(t_{0},x_{0})}) = t^{α}(t_{0}), x^{i} ([φ]_{(t_{0},x_{0})}) = x^{i}(x_{0}), x_{α}^{i} ([φ]_{(t_{0},x_{0})}) = \frac{∂x^{i}}{∂t^{α}} (t_{0}).

A local changing of coordinates (t^{α}, x^{i}, x_{α}^{i}) → (t̄^{α}, x̄^{i}, x̄_{α}^{i}) is given by

(7)  t̄^{α} = t̄^{α}(t^{β}), x̄^{i} = x̄^{i}(x^{j}), x̄_{α}^{i} = \frac{∂x̄^{i}}{∂x^{j}} \frac{∂t^{β}}{∂t̄^{α}} x_{β}^{j},

where

det (\frac{∂t̄^{α}}{∂t^{β}} ) > 0, det (\frac{∂x̄^{i}}{∂x^{j}} ) ≠ 0.

The expression of the Jacobian matrix of the local diffeomorphism (7) shows that the jet bundle of order one J^{1}(T, M) is always orientable.

Let H_{βγ}^{α}, G_{jk}^{i} be the components of the connections induced by h and g respectively. If (t^{α}, x^{i}, x_{α}^{i}) are the coordinates of a point in J^{1}(T, M), then

x_{αβ}^{i} = \frac{∂^{2}x^{i}}{∂t^{α}∂t^{β}} − H_{αβ}^{γ}x_{γ}^{i} + G_{jk}^{i}x_{α}^{j}x_{β}^{k}

are the components of a distinguished tensor on T × M. Also

(\frac{δ}{δt^{α}} = \frac{∂}{∂t^{α}} + H_{αβ}^{γ}x_{γ}^{i} \frac{∂}{∂x_{β}^{i}} , \frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − G_{ik}^{h}x_{α}^{k} \frac{∂}{∂x_{α}^{h}} , \frac{∂}{∂x_{α}^{i}} ) ,

(dt^{β}, dx^{j}, δx_{β}^{j} = dx_{β}^{j} − H_{βλ}^{γ}x_{γ}^{j}dt^{λ} + G_{hk}^{j}x_{β}^{h}dx^{k})

are dual frames on J^{1}(T, M), i.e.,

dt^{β} (\frac{δ}{δt^{α}} ) = δ_{α}^{β}, dt^{β} (\frac{δ}{δx^{i}} ) = 0, dt^{β} (\frac{∂}{∂x_{α}^{i}} ) = 0

dx^{j} (\frac{δ}{δt^{α}} ) = 0, dx^{j} (\frac{δ}{δx^{i}} ) = δ_{i}^{j}, dx^{j} (\frac{∂}{∂x_{α}^{i}} ) = 0

δx_{β}^{j} (\frac{δ}{δt^{α}} ) = 0, δx_{β}^{j} (\frac{δ}{δx^{i}} ) = 0, δx_{β}^{j} (\frac{∂}{∂x_{α}^{i}} ) = δ_{i}^{j}δ_{β}^{α}.

Using these frames, we define on J^{1}(T, M) the induced Sasaki-like metric

S_{1} = h_{αβ}dt^{α} ⊗ dt^{β} + g_{ij}dx^{i} ⊗ dx^{j} + h^{αβ}g_{ij}δx_{α}^{i} ⊗ δx_{β}^{j}.

The semi-Riemann geometry of the manifold J^{1}(T, M) was developed recently in [4].

The Lorentz world-force law formulated usually for particles [5] can be generalized as follows: Definition. Let F_{α} = (F_{j}^{i}_{α}) and U_{αβ} = (U_{αβ}^{i}) be C^{∞} distinguished tensors on T × M, where ω_{jiα} = g_{hi}F_{j}^{h}_{α} is skew-symmetric with respect to j and i. Let c(t, x) be a C^{∞} real function on T × M. Suppose (T, h) is a Riemannian manifold. A C^{∞} map φ : T → M obeys a Generalized Lorentz World-Force Law with respect to F_{α}, U_{αβ}, c iff

h^{αβ}x_{αβ}^{i} = g^{ij} \frac{∂c}{∂x^{j}} + h^{αβ}F_{j}^{i}_{α}x_{β}^{j} + h^{αβ}U_{αβ}^{i},

i.e., iff it is a potential map of a suitable geometrical structure.

Let us show that the solutions of a system of PDEs of order one are potential maps in a suitable geometrical structure of the jet bundle of order one. For that we remark that any C^{∞} distinguished tensor field X_{α}^{i}(t, x) on T × M defines a family of p-dimensional sheets as solutions of the PDEs system of order one

(8)  x_{α}^{i} = X_{α}^{i}(t, x(t)),

if the complete integrability conditions

\frac{∂X_{α}^{i}}{∂t^{β}} + \frac{∂X_{α}^{i}}{∂x^{j}} X_{β}^{j} = \frac{∂X_{β}^{i}}{∂t^{α}} + \frac{∂X_{β}^{i}}{∂x^{j}} X_{α}^{j}

are satisfied.

To any distinguished tensor field X_{α}^{i}(t, x) and semi-Riemann metrics h and g we associate the potential energy density

f : T × M → R, f = \frac{1}{2}h^{αβ}g_{ij}X_{α}^{i}X_{β}^{j}.

The distinguished tensor field X_{α}^{i} (family of p-dimensional sheets) on (T × M, h + g) is called:

1) timelike, if f < 0;

2) nonspacelike or causal, if f ≤ 0;

3) null or lightlike, if f = 0;

4) spacelike, if f > 0.

Let E = {x_{0} ∈ M| X_{α}^{i}(t, x_{0}) = 0, ∀t ∈ T } be the set of critical points of the system (8). If f = constant, upon rescaling on T × (M \ E), it may be supposed that f ∈ {−1, 0, 1}.

The derivative along a solution of (8),

\frac{δ}{∂t^{β}} x_{α}^{i} = x_{αβ}^{i} = \frac{∂^{2}x^{i}}{∂t^{α}∂t^{β}} − H_{αβ}^{γ}x_{γ}^{i} + G_{jk}^{i}x_{α}^{j}x_{β}^{k},

produces the decomposition \frac{δ}{∂t^{β}} · \frac{∂}{∂t^{α}}. This operator, the Levy-Civita connection D of (T, h) and the Levy-Civita connection ∇ of (M, g) produce the prolongation (PDEs system of order two)

(9)  x_{αβ}^{i} = D_{β}X_{α}^{i} + (∇_{j}X_{α}^{i})x_{β}^{j},

which can be converted into the prolongation

(10)  h^{αβ}x_{αβ}^{i} = g^{ih}h^{αβ}g_{ij}(∇_{h}X_{α}^{k})X_{β}^{j} + h^{αβ}F_{j}^{i}_{α}x_{β}^{j} + h^{αβ}D_{β}X_{α}^{i},

where

F_{j}^{i}_{α} = ∇_{j}X_{α}^{i} − g^{ih}g_{kj}∇_{h}X_{α}^{k}

is the external distinguished tensor field which characterizes the helicity of the distinguished tensor field X_{α}^{i}. Theorem. Any solution of PDEs system (8) is a solution of the PDEs system (10).

The first term in the second hand member of the PDEs system (10) is the force (gradf)^{i}. Therefore, choosing the metrics h and g such that f ∈ {−1, 0, 1}, the system (10) reduces to

(10′)  h^{αβ}x_{αβ}^{i} = g^{ih}F_{j}^{i}_{α}x_{β}^{j} + h^{αβ}D_{β}X_{α}^{i}.

Now, let us describe the variational structure of the PDEs system (10). Theorem. The solutions of PDEs system (10) are the extremals of the Lagrangian

\begin{aligned}L & = & \frac{1}{2}h^{αβ}g_{ij}(x_{α}^{i} − X_{α}^{i})(x_{β}^{j} − X_{β}^{j})\sqrt{|h|} = \\ & = & (\frac{1}{2}h^{αβ}g_{ij}x_{α}^{i}x_{β}^{j} − h^{αβ}g_{ij}x_{α}^{i}X_{β}^{j} + f) \sqrt{|h|}. \end{aligned}

If F_{jα}^{i} = 0, then this Lagrangian can be replaced by

L = (\frac{1}{2}h^{αβ}g_{ij}x_{α}^{i}x_{β}^{j} + f) \sqrt{|h|}.

2) Both Lagrangians produce the same Hamiltonian

H = (\frac{1}{2}h^{αβ}g_{ij}x_{α}^{i}x_{β}^{j} − f) \sqrt{|h|}.

Theorem (Lorentz-Udrişte World-Force Law). Suppose (T, h) is a Riemannian manifold. Every solution of the PDEs system (8) is a horizontal potential map of the semi-Riemann-Lagrange manifold

(T × M, h + g, N(_{α}^{i})_{j} = G_{jk}^{i}x_{α}^{k} − F_{j}^{i}_{α}, M(_{α}^{i})_{β} = −H_{αβ}^{γ}x_{γ}^{i}).

Corollary. Every PDE generates a Lagrangian of order one via the associated first order PDEs system and suitable metrics on the manifold of independent variables and on the manifold of functions. In this sense the solutions of the initial PDE are potential maps produced by a suitable Lagrangian. Proof. Let

\frac{∂^{r}x}{∂(t^{p})^{r}} = F (t^{α}, x, x̄^{(r)})

be a PDE of order r, where x̄^{(r)} represent the partial derivatives of x with respect to

t^{α}, till the order r inclusively, excepting the partial derivative \frac{∂^{r}x}{∂(t^{p})^{r}}. This equation

is equivalent to a system (8).

For the sake of simplicity, we take r = 2. We denote \frac{∂x}{∂t^{α}} = x_{α} = u^{α} and we find the partial derivatives of the functions (x, u^{α}) using the system

x_{α} = u^{α}
u_{β}^{α} = u_{α}^{β}, α ≠ β
u_{2}^{2} = f(t^{α}, x, u_{μ}^{λ}), excepting λ = μ = 2.

We shall find a PDEs system of order one with p(1 + p) equations, which is of type (8). Therefore, the preceding theory applies. Example. Consider the Monge-Ampère equation det(Hess u) = F, where u : T → R is the unknown function, Hess means the Hessian with respect to the semi-Riemann structure of the manifold T, and F is a given function of t, u, du. This equation is clasely related to the study of the curvature of a manifold [14]. The Monge-Ampère equation is equivalent to the first-order system

\frac{∂u}{∂t^{α}} = ω_{α}, D_{β}ω_{α} = η_{βα}

with the restriction

det(η_{αβ}) = F (t, u, ω).

It appears the Lagrangian

L = \frac{1}{2}h^{αβ} (\frac{∂u}{∂t^{α}} − ω_{α}) (\frac{∂u}{∂t^{β}} − ω_{β}) + \frac{1}{2}h^{αβ}h^{γδ}(D_{γ}ω_{α} − η_{γα})(D_{δ}ω_{β} − η_{δβ})

subject to

det(η_{αβ}) = F (t, u, ω),

and the preceding theory applies.

Particularly, the Monge-Amperè equation

u_{t^{1}t^{1}}u_{t^{2}t^{2}} − u_{t^{1}t^{2}}^{2} = F (t^{1}, t^{2}, u, u_{t^{1}}, u_{t^{2}})

is equivalent to the first order PDEs system

u_{t^{1}} = v
u_{t^{2}} = w
v_{t^{1}} = \sqrt{F} cos hζ
v_{t^{2}} = \sqrt{F} sin hζ
w_{t^{1}} = \sqrt{F} sin hζ
w_{t^{2}} = \sqrt{F} cos hζ

where ζ is an arbitrary function of (t^{1}, t^{2}, u, v, w). Let (M, g_{ij} = δ_{ij}) be the Riemannian manifold of coordinates (u, v, w), (T, h_{αβ} = δ_{αβ}) be the Riemannian manifold of coordinates (t^{1}, t^{2}) and J^{1}(T × M) be the jet bundle of order one. In this sense, the solutions of the previous system are extremals (potential maps) of the Lagrangian

L = (u_{t^{1}} − v)^{2} + (u_{t^{2}} − w)^{2} + (v_{t^{1}} − \sqrt{F} cos hζ)^{2}+
+ (v_{t^{2}} − \sqrt{F} sin hζ)^{2} + (w_{t^{1}} − \sqrt{F} sin hζ)^{2} + (w_{t^{2}} − \sqrt{F} cos hζ)^{2}

4 Covariant Hamilton Equations

Recall that on a symplectic manifold (Q, Ω) of even dimension q, the Hamiltonian vector field X_{H} of a function H ∈ F(Q) is defined by

X_{H} Ω = dH.

This relation can be generalized as

X_{H}^{α} Ω_{α} = \sqrt{|h|}dH,

using the distinguished objects X_{H}, Ω, H on J^{1}(T, M). For another point of view, see also [11] and compare with [2], [3]. Theorem. The PDEs system

h^{αβ}x_{αβ}^{i} = g^{ih}h^{αβ}g_{jk}(∇_{h}X_{α}^{j})X_{β}^{k}

transfers in J^{1}(T, M) as a covariant Hamilton PDEs system with respect to the Hamiltonian

H = \frac{1}{2}h^{αβ}g_{ij}x_{α}^{i}x_{β}^{j} − f

and the non-degenerate distinguished polysymplectic relative 2–form

Ω = Ω_{α} ⊗ dt^{α}, Ω_{α} = g_{ij}dx^{i} ∧ δx_{α}^{j}\sqrt{|h|}.

Proof. Let

θ = θ_{α} ⊗ dt^{α}, θ_{α} = g_{ij}x_{α}^{i}dx^{j}\sqrt{|h|}

be the distinguished Liouville relative 1–form on J^{1}(T, M). It follows

Ω_{α} = −dθ_{α}.

We denote by

X_{H} = X_{H}^{β} \frac{δ}{δt^{β}} , X_{H}^{β} = u^{βl} \frac{δ}{δx^{l}} + \frac{δu^{βl}}{∂t^{α}} \frac{∂}{∂x_{α}^{l}}

the distinguished Hamiltonian object of the function H. Imposing

X_{H}^{α} Ω_{α} = \sqrt{|h|}dH,

where

dH = h^{αβ}g_{ij}x_{β}^{j}δx_{α}^{i} − h^{αβ}g_{ij}(D_{γ}X_{α}^{i})X_{β}^{j}dt^{γ} − h^{αβ}g_{ij}X_{β}^{j}∇_{k}X_{α}^{i}dx^{k}

we find

g_{ij}u^{αi}δx_{α}^{j} − g_{ij} \frac{δu^{αj}}{∂t^{α}} dx^{i} = dH.

Consequently, it appears the Hamilton PDEs system

\{ \begin{matrix}u^{αi} = h^{αβ}x_{β}^{i} \\ \frac{δu^{αi}}{∂t^{α}} = g^{hi}h^{αβ}g_{jk}X_{β}^{j}(∇_{h}X_{α}^{k}) \end{matrix}

together the condition

h^{αβ}g_{ij}(D_{γ}X_{α}^{i})X_{β}^{j} = 0.

Theorem. The PDEs system

h^{αβ}x_{αβ}^{i} = g^{ih}h^{αβ}g_{kj}(∇_{h}X_{α}^{k})X_{β}^{j} + h^{αβ}F_{j}^{i}_{α}x_{β}^{j} + h^{αβ}D_{β}X_{α}^{i}

transfers in J^{1}(T, M) as a covariant Hamilton PDEs system with respect to the Hamiltonian

H = \frac{1}{2}h^{αβ}g_{ij}x_{α}^{i}x_{β}^{j} − f

and the non-degenerate distinguished polysymplectic relative 2–form

\begin{matrix}Ω = Ω_{α} ⊗ dt^{α}, \\ Ω_{α} = (g_{ij}dx^{i} ∧ δx_{α}^{j} + ω_{ijα}dx^{i} ∧ dx^{j} + g_{ij}(D_{β}X_{α}^{i})dt^{β} ∧ dx^{j})\sqrt{|h|}. \end{matrix}

Proof. Let

\begin{matrix}θ = θ_{α} ⊗ dt^{α}, \\ θ_{α} = (g_{ij}x_{α}^{i}dx^{j} − g_{ij}X_{α}^{i}dx^{j})\sqrt{|h|} \end{matrix}

be the distinguished Liouville relative 1–form on J^{1}(T, M). It follows

Ω_{α} = −dθ_{α}.

We denote by

\begin{matrix}X_{H} = X_{H}^{β} \frac{δ}{δt^{β}} , \\ X_{H}^{β} = h^{βγ} \frac{δ}{δt^{γ}} + u^{βl} \frac{δ}{δx^{l}} + \frac{δu^{βl}}{∂t^{α}} \frac{∂}{∂x_{α}^{l}} \end{matrix}

the distinguished Hamiltonian object of the function H. Imposing

X_{H}^{α} Ω_{α} = \sqrt{|h|}dH,

where

dH = −h^{αβ}g_{ij}(D_{γ}X_{α}^{i})X_{β}^{j}dt^{γ} + h^{αβ}g_{ij}x_{β}^{j}δx_{α}^{i} − h^{αβ}g_{ij}X_{β}^{j}(∇_{k}X_{α}^{i})dx^{k},

we find

\begin{aligned}g_{ij}u^{αi}δx_{α}^{j} & − & g_{ij} \frac{δu^{αj}}{∂t^{α}} dx^{i} + 2ω_{ijα}u^{αi}dx^{j}− \\ & − & g_{ij}(D_{β}X_{α}^{i})u^{αj}dt^{β} + h^{αβ}g_{ij}(D_{β}X_{α}^{i})dx^{j} = dH. \end{aligned}

Consequently, we obtain the Hamilton PDEs system

\{ \begin{matrix}u^{αi} = h^{αβ}x_{β}^{i} \\ \frac{δu^{αi}}{∂t^{α}} = g^{hi}h^{αβ}g_{jk}X_{β}^{j}(∇_{h}X_{α}^{k}) + 2g^{hi}ω_{jhα}u^{αj} + h^{αβ}D_{β}X_{α}^{i} \end{matrix}

together the condition

g_{ij}(D_{γ}X_{α}^{i})(u^{αj} − h^{αβ}X_{β}^{j}) = 0.

Acknowledgements. A version of the paper was presented at the Centennial Vrănceanu, Romanian Academy, University of Bucharest, June 30 - July 4, 2000, being an homage for the greatest Romanian geometer whose Work [12], [13] includes the geometrical theory of PDEs.

References

[1] J.Eells, L.Lemair, Harmonic maps of Riemannian manifolds, Amer. J. Math. 86(1964), 109-160.

[2] G.Giachetta, L.Mangiarotti, G.Sardonashwily, Covariant Hamiltonian field theory, http://xxx.lanl.gov/hep-th/9904062.

[3] M.J.Gotay, J.Isenberg, J.E. Marsden, R.Montgomery, J.Sniatycki, P.B.Yasskin, Momentum maps and Classical relativistic fields, Part.I: Covariant field theory, http://xxx.lanl.gov/physical/9801019.

[4] M.Neagu, C.Udrişte, Multi-Time Dependent Sprays and Harmonic Maps on J^{1}(T, M), http://xxx.lanl.gov/math/DG/0009049.

[5] R.K.Sachs and H.Wu, General relativity for mathematicians, Springer-Verlag, New-York, 1977.

[6] C.Udrişte, Geometric dynamics, Southeast Asian Bulletin of Mathematics 24, 1(2000), 313-322.

[7] C.Udrişte, Geometric dynamics, Kluwer Academic Publishers, 2000.

[8] C.Udrişte, Dynamics induced by second-order objects, BSG Proceedings 4, Global Analysis, Differential Geometry, Lie Algebras, Editor Grigorios Tsagas, pp.161-168, Geometry Balkan Press, 2000.

[9] C.Udrişte, M.Neagu, Geometrical interpretation of solutions of certain PDEs, Balkan Journal of Geometry and Its Applications, 4,1 (1999), 138-145.

[10] C.Udrişte, A.Udrişte, From flows and metrics to dynamics, Proceedings of International Symposium on Mathematics and Mathematical Sciences, Calcutta Mathematical Society, India, Jan 22-24, 2000.

[11] C.Udrişte, Nonclassical Lagrangian dynamics and potential maps, Proceedings of the Conference on Mathematics in Honour of Professor Radu Roşca at the Occasion of his Ninetieth Birthday, Katholieke University Brussel, Katholieke University Leuven, Belgium, Dec. 11-16, 1999, http://xxx.lanl.gov/math.DS/0007060, 2000.

[12] G.Vrănceanu, Lecţii de de geometrie diferenţială, vol.2, Editura Didactică şi Pedagogică, Bucureşti, 1964.

[13] G.Vrănceanu, Opera Matematică, vol.1-3, Editura Academiei Romane, Bucureşti, 1969, 1971, 1973.

[14] S.T.Yau, Survey on partial differential equations in differential geometry, Seminar on Differential Geometry, Ed.S.T.Yau, Annals of Mathematics Studies, Princeton, University Press, 1982, pp.3-71.

University Politehnica of Bucharest Department of Mathematics I Splaiul Independenţei 313 77206 Bucharest, Romania email:udriste@mathem.pub.ro