Inverse problem for Lagrangian mechanics
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s can arise as the Euler–Lagrange equations for some Lagrangian function. There has been a great deal of activity in the study of this problem since the early 20th century. A notable advance in this field was a 1941 paper by the American
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Jesse Douglas Jesse Douglas (July 3, 1897 – September 7, 1965) was an American mathematician and Fields Medalist known for his general solution to Plateau's problem. Life and career He was born to a Jewish family in New York City, the son of Sarah ...
, in which he provided
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conditions for the problem to have a solution; these conditions are now known as the Helmholtz conditions, after the
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physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...
.


Background and statement of the problem

The usual set-up of
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...
on ''n''-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
R''n'' is as follows. Consider a
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
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''u'' :  , ''T''nbsp;→ R''n''. The action of the path ''u'', denoted ''S''(''u''), is given by :S(u) = \int_^ L(t, u(t), \dot(t)) \, \mathrm t, where ''L'' is a function of time, position and
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
known as the Lagrangian. The
principle of least action Action principles lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called a Lagrangian describing the physical sy ...
states that, given an initial state ''x''0 and a final state ''x''1 in R''n'', the trajectory that the system determined by ''L'' will actually follow must be a minimizer of the action functional ''S'' satisfying the boundary conditions ''u''(0) = ''x''0, ''u''(T) = ''x''1. Furthermore, the critical points (and hence minimizers) of ''S'' must satisfy the Euler–Lagrange equations for ''S'': :\frac \frac - \frac = 0 \quad \text 1 \leq i \leq n, where the upper indices ''i'' denote the components of ''u'' = (''u''1, ..., ''u''''n''). In the classical case :T(\dot) = \frac m , \dot , ^, :V :
, T The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
\times \mathbb^ \to \mathbb, :L(t, u, \dot) = T(\dot) - V(t, u), the Euler–Lagrange equations are the second-order ordinary differential equations better known as
Newton's laws of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
: :m \ddot^ = - \frac \quad \text 1 \leq i \leq n, :\mboxm \ddot = - \nabla_ V(t, u). The inverse problem of Lagrangian mechanics is as follows: given a system of second-order ordinary differential equations :\ddot^ = f^ (u^, \dot^) \quad \text 1 \leq i, j \leq n, \quad \mbox that holds for times 0 ≤ ''t'' ≤ ''T'', does there exist a Lagrangian ''L'' :  , ''T''nbsp;× R''n'' × R''n'' → R for which these ordinary differential equations (E) are the Euler–Lagrange equations? In general, this problem is posed not on Euclidean space R''n'', but on an ''n''-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
''M'', and the Lagrangian is a function ''L'' :  , ''T''nbsp;× T''M'' → R, where T''M'' denotes the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
of ''M''.


Douglas' theorem and the Helmholtz conditions

To simplify the notation, let :v^ = \dot^ and define a collection of ''n''2 functions Φ''j''''i'' by :\Phi_^ = \frac \frac \frac - \frac - \frac \frac \frac. Theorem. (Douglas 1941) There exists a Lagrangian ''L'' :  , ''T''nbsp;× T''M'' → R such that the equations (E) are its Euler–Lagrange equations
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
there exists a
non-singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singular ...
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
''g'' with entries ''g''''ij'' depending on both ''u'' and ''v'' satisfying the following three Helmholtz conditions: :g \Phi = (g \Phi)^, \quad \mbox :\frac + \frac \frac g_ + \frac \frac g_ = 0 \mbox 1 \leq i, j \leq n, \quad \mbox :\frac = \frac \mbox 1 \leq i, j, k \leq n. \quad \mbox (The
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is in use for the repeated indices.)


Applying Douglas' theorem

At first glance, solving the Helmholtz equations (H1)–(H3) seems to be an extremely difficult task. Condition (H1) is the easiest to solve: it is always possible to find a ''g'' that satisfies (H1), and it alone will not imply that the Lagrangian is singular. Equation (H2) is a system of ordinary differential equations: the usual theorems on the existence and uniqueness of solutions to ordinary differential equations imply that it is, ''in principle'', possible to solve (H2). Integration does not yield additional constants but instead first integrals of the system (E), so this step becomes difficult ''in practice'' unless (E) has enough explicit first integrals. In certain well-behaved cases (e.g. the
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for the
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connection on a
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), this condition is satisfied. The final and most difficult step is to solve equation (H3), called the ''closure conditions'' since (H3) is the condition that the differential 1-form ''g''''i'' is a closed form for each ''i''. The reason why this is so daunting is that (H3) constitutes a large system of coupled partial differential equations: for ''n'' degrees of freedom, (H3) constitutes a system of :2 \left( \begin n + 1 \\ 3 \end \right) partial differential equations in the 2''n'' independent variables that are the components ''g''''ij'' of ''g'', where :\left( \begin n \\ k \end \right) denotes the
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
. In order to construct the most general possible Lagrangian, one must solve this huge system! Fortunately, there are some auxiliary conditions that can be imposed in order to help in solving the Helmholtz conditions. First, (H1) is a purely algebraic condition on the unknown matrix ''g''. Auxiliary algebraic conditions on ''g'' can be given as follows: define functions :Ψ''jk''''i'' by :\Psi_^ = \frac \left( \frac - \frac \right). The auxiliary condition on ''g'' is then :g_ \Psi_^ + g_ \Psi_^ + g_ \Psi_^ = 0 \mbox 1 \leq i, j \leq n. \quad \mbox In fact, the equations (H2) and (A) are just the first in an infinite hierarchy of similar algebraic conditions. In the case of a
parallel connection Two-terminal components and electrical networks can be connected in series or parallel. The resulting electrical network will have two terminals, and itself can participate in a series or parallel topology. Whether a two-terminal "object" is ...
(such as the canonical connection on a Lie group), the higher order conditions are always satisfied, so only (H2) and (A) are of interest. Note that (A) comprises :\left( \begin n \\ 3 \end \right) conditions whereas (H1) comprises :\left( \begin n \\ 2 \end \right) conditions. Thus, it is possible that (H1) and (A) together imply that the Lagrangian function is singular. As of 2006, there is no general theorem to circumvent this difficulty in arbitrary dimension, although certain special cases have been resolved. A second avenue of attack is to see whether the system (E) admits a submersion onto a lower-dimensional system and to try to "lift" a Lagrangian for the lower-dimensional system up to the higher-dimensional one. This is not really an attempt to solve the Helmholtz conditions so much as it is an attempt to construct a Lagrangian and then show that its Euler–Lagrange equations are indeed the system (E).


References

* * {{cite journal , author=Rawashdeh, M., & Thompson, G. , title=The inverse problem for six-dimensional codimension two nilradical Lie algebras , journal=Journal of Mathematical Physics , volume=47 , issue=11 , year=2006 , issn=0022-2488 , doi=10.1063/1.2378620 , pages=112901 , bibcode=2006JMP....47k2901R Calculus of variations Lagrangian mechanics Inverse problems