1 Introduction

The use of geometric methods in classical mechanics [1–5] has been very useful to get a better understanding of different problems offering us new related questions and answers. Constraints force us to replace affine spaces by differentiable manifolds, coordinates becoming then a local concept. Differential equations are replaced by vector fields, a global concept, in such a way that the integral curves of such vector fields are the solutions of a system of differential equations in a coordinate system.

The explicit determination of such integral curves is based on the existence of symmetries and constants of the motion. Note, however, that two conformally related vector fields have the same constants of motion, and therefore, the integral curves of one are, up to a reparametrization, integral curves of the other. Even in the simplest case of an affine space as configuration space, where global coordinates do exist, we can consider nonlinear systems. Note that if we start with a harmonic oscillator system, the dynamics is linear, but a deformation of the system produces nonlinear systems, the periodic motion having a period depending on the energy. This was the motivation for the appearance of q-oscillators, with applications in quantum problems [6, 7] and more recently f-oscillators [8]. For instance, we can consider a smooth function of a Hamiltonian H and define the corresponding Hamilton function f(H).

Recall that geometric structures that are compatible with the vector field responsible of the dynamics are playing a relevant role. In particular, we can fix next our attention on symplectic structures, \((M,\omega )\)—where \(\omega \) is a non-degenerate closed 2-form in the manifold M—which provide a common geometric framework to deal with both Hamiltonian and Lagrangian mechanics (in the regular case). A differentiable function H determines a uniquely defined vector field \(X_H\) as solution of the equation \(i(X_H)\omega =dH\), which is called the Hamiltonian vector field with Hamiltonian H. Then the new Hamiltonian f(H) produces a new Hamiltonian vector field, \(f'(H)\, X_H\), conformally equivalent to the initial one, \(X_H\). These f-oscillators, for instance, have been used in [9–11].

On the other side, the Lagrangian approach to conservative systems was originally devised to deal with holonomic constraints in a way invariant under changes of coordinates, the Hamilton variational principle leading to the Euler–Lagrange equations of motion determining the stationary solutions of the action. In the regular case, the Lagrangian approach can also be seen as a particular case of Hamiltonian dynamical system, but now the Lagrangian function L determines both the symplectic form \(\omega _L\) and the Hamiltonian (i.e. the energy \(E_L\)) and, as a sub-product, the dynamical vector field (i.e. the equations of motion). In this respect, the inverse problem of mechanics has been playing an important role, the main result being the one given by Helmholtz [12, 13]. This new geometric approach allows us to answer questions not only on the existence, but also on the non-uniqueness, of such a Lagrangian function (i.e. the existence of alternative Lagrangians), which has been shown to provide constants of the motion, as proved in [14] for the one-dimensional case and generalized in [15] for the multidimensional case (see also [16] for a geometric approach).

This geometric approach to Lagrangian mechanics suggested the use of more general Lagrangian functions than those of a mechanical type, sometimes called non-standard Lagrangians, for instance for second-order Riccati and Abel equations [17, 18], and this question has received much attention during the last years, and many applications have been developed [19–24].

In a recent paper of this journal [25], El-Nabulsi presents some of the implications of non-standard Lagrangians in non-inertial dynamics. The paper forms part of a long series of papers [25–30] where the author has been dealing with many applications of such Lagrangians and in particular he analysed the relation of equations of motion corresponding to a Lagrangian L with those of the deformed Lagrangians \(e^{\lambda L}\) or \(L^{1+\gamma }\). The theory as developed by El-Nabulsi is worth of a deeper analysis in order to extend its applicability to cases in which the configuration space is not \({\mathbb {R}}^n\), but there exist holonomic constraints and globally defined coordinates do not exist. The theory here developed is much more general, because it is an intrinsic approach to the problem, with no explicit choice of coordinates and offering a new geometric perspective which allows us to answer different questions as, for instance, under what conditions the equations of motion for \(\phi (L)\) are equivalent to those of L? Of course, not only Lagrangians of mechanical type have to be considered and therefore the associated conserved quantity called energy has no the usual form of kinetic term plus potential energy.

2 Notation and basic definitions: deformation of Lagrangians

In this section, we will shortly describe the geometrical approach to classical mechanics making use of the theory of symplectic manifolds. For more details we refer the reader to [1, 4, 5].

In the non autonomous case, the configuration space is an n-dimensional differentiable manifold Q and its tangent bundle \(\tau :TQ\rightarrow Q\) is the velocity phase space. A function \(L\in C^\infty (TQ)\), called Lagrangian function, allows us to construct a Hamiltonian dynamical system \((TQ,\omega _L,E_L)\) by means of two intrinsic tensor fields in TQ: a (1, 1)-tensor field called vertical endomorphism S and the Liouville vector field \(\varDelta \in {\mathfrak {X}}(TQ)\) generator of dilations along fibres (see e.g. [2–5] and references therein). The Liouville 1-form is defined as \(\theta _L={\text {d}}L\circ S\) and the energy function \(E_L\) as \(E_L=\varDelta (L)-L\). The Lagrangian function L is said to be regular when the exact 2-form \(\omega _L=-d\theta _L\) is regular, i.e. the only vector field \(X\in {{\mathfrak {X}}}(TQ)\) such that \(i(X)\omega _L=0\) is the zero vector field. In this case, \((TQ,\omega _L,E_L)\) is a Hamiltonian dynamical system, the dynamical vector field \(\varGamma _L\) being determined by the equation \(i(\varGamma _L)\omega _L={\text {d}}E_L\). One can prove that then \(\varGamma _L\) is a second-order differential equation vector field (hereafter shortened as SODE), i.e. \(S(\varGamma _L)=\varDelta \). Moreover, for the SODE vector field \(\varGamma _L\) the condition \(i(\varGamma _L)\omega _L={\text {d}}E_L\) is equivalent to \({\mathcal {L}}_{\varGamma _L}\theta _L-{\text {d}}L=0\).

We also remark that the 1-form \({\mathcal {L}}_{\varGamma _L}\theta _L-{\text {d}}L\) is semibasic (i.e. its contractions with vertical vector fields vanish). Consider local coordinates for TQ induced from local coordinates in Q: given a local chart for the configuration space Q, \((U,\varphi )\), providing n coordinates \(q^i=\text {pr}^i\circ \varphi \), where \(\text {pr}^i:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\), for \(i=1,\ldots ,n\), denotes the projection on the i-th coordinate in \({\mathbb {R}}^n\), we have an associated basis of the \(C^\infty (U)\)-module \({\mathfrak {X}}(U)\) of local vector fields, usually denoted \(\{\partial /\partial q^j\mid j=1,\ldots ,n\}\), with a dual basis \(\{dq^j\mid j=1,\ldots ,n\}\) of 1-forms. A vector in a point \(q\in U\) is \(v=v^j\,(\partial /\partial q^j)|_q\), with \(v^j=\langle dq^j,v\rangle \). This provides a local chart in \(\tau ^{-1}(U)\).

In terms of these local coordinates, the expressions for the Liouville vector field \(\varDelta \) and the vertical endomorphism S are:

$$\begin{aligned} \varDelta =v^i{\partial \over \partial {v^i}},\qquad S={\partial \over \partial {v^i}}\otimes dq^i, \end{aligned}$$
(1)

and the expressions of the Cartan forms, \( \theta _L \) and \( \omega _L \), and the energy function \(E_L\) are

$$\begin{aligned} \theta _L= & {} {\displaystyle {\partial L \over \partial v^i}}\,dq^i \nonumber \\ \omega _L= & {} {\displaystyle {\partial ^2L \over \partial q^j\,\partial v^i}}\,dq^i \wedge dq^j + {\displaystyle {\partial ^2L \over \partial v^j\,\partial v^i}}\,dq^i \wedge dv^j \\ E_L= & {} v^i\,{\displaystyle {\partial L \over \partial v^i} }- L \, ,\nonumber \end{aligned}$$
(2)

and the dynamical vector field \(\varGamma _L\) is

$$\begin{aligned} \varGamma _{L}= v^i\,{\displaystyle {\partial \over \partial q^i}} + f^i(q,v)\,{\displaystyle {\partial \over \partial v^i}}\,, \end{aligned}$$
(3)

where

$$\begin{aligned} f^i(q,v) =W^{ij}\,\left[ \,{\displaystyle {\partial L \over \partial q^j}}\ -\ {\displaystyle {\partial ^2 L \over \partial q^k \partial v^j}}\,v^k\,\right] \, , \end{aligned}$$
(4)

with \(W^{ij}\,W_{jk}=\delta ^i_k\) and

$$\begin{aligned} W_{ij}={\partial ^2L\over \partial v^i\partial v^j}, \end{aligned}$$

while the semibasic 1-form \({\mathcal {L}}_{\varGamma _L}\theta _L-{\text {d}}L\) is given by:

$$\begin{aligned} {\mathcal {L}}_{\varGamma _L}\theta _L-{\text {d}}L= \left( W_{ij} f^j+v^j{\partial ^2L\over \partial v^i\partial q^j}-{\partial L\over \partial q^i}\right) dq^i, \end{aligned}$$

which clearly shows that the Euler–Lagrange equations are but a local expression for the vanishing of such semibasic 1-form.

The relation between the equations of motion for a Lagrangian L and for its deformation \(\phi (L)\) is given by:

Theorem 1

Let \(\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a differentiable function and L a regular Lagrangian for a given dynamics, i.e. a SODE vector field \(\varGamma _L\) such that

$$\begin{aligned} {\mathcal {L}}_{\varGamma _L}\theta _L-{\text {d}}L=0. \end{aligned}$$
(5)

Then, the equations of motion for \(\phi (L)\), \({\mathcal {L}}_{\varGamma _{\phi (L)}}( \theta _{\phi (L)})-d(\phi (L))=0\) are equivalent to

$$\begin{aligned} {\varGamma _{\phi (L)}} (\phi '(L))\,\theta _L+\phi '(L)({\mathcal {L}}_{\varGamma _{\phi (L)}} \theta _L-{\text {d}}L)=0. \end{aligned}$$
(6)

Proof

Recall that by definition \(\theta _L={\text {d}}L\circ S\) and then

$$\begin{aligned} \theta _{\phi (L)}=d(\phi (L))\circ S=\phi '(L)\,{\text {d}}L\circ S=\phi '(L)\,\theta _L, \end{aligned}$$

where we have used that \(d(\phi (L))=\phi '(L)\,{\text {d}}L\). Then,

$$\begin{aligned} {\mathcal {L}}_{\varGamma } ( \theta _{\phi (L)})-d(\phi (L))= & {} {\varGamma } (\phi '(L))\,\theta _L+\phi '(L)\,{\mathcal {L}}_{\varGamma } \theta _L\\&-\phi '(L) {\text {d}}L \end{aligned}$$

for every vector field \(\varGamma \); putting \(\varGamma =\varGamma _{\phi (L)}\), we finally deduce that the relation between Euler–Lagrange equations of L and \(\phi (L)\) is given by (6). \(\square \)

Observe also that \({\varGamma _{\phi (L)}} (\phi '(L))=\phi ''(L)\,\varGamma _{\phi (L)}(L)\), i.e. relation (6) can be rewritten as

$$\begin{aligned} \phi ''(L)\,{\varGamma _{\phi (L)}} (L)\,\theta _L+\phi '(L)({\mathcal {L}}_{\varGamma _{\phi (L)}} \theta _L-{\text {d}}L)=0. \end{aligned}$$
(7)

We can study the modified Euler–Lagrange Eq. (6) obtained from the deformed Lagrangian \(\phi (L)\) for specific choices of the function \(\phi \).

Case \(\phi (L)=e^{\lambda \,L}\):

As now \(\phi '(L)=\lambda \, \phi (L)\) is a non-vanishing function and, moreover, \({\varGamma _{\phi (L)}} (\phi '(L))=\phi ''(L)\, \varGamma _{\phi (L)}(L)=\lambda ^2\phi (L)\, \varGamma _{\phi (L)}(L)\), the modified equations of motion (6) obtained from \(\phi (L)\) are

$$\begin{aligned} {\mathcal {L}}_{\varGamma _{\phi (L)}} \theta _L-{\text {d}}L+\lambda \, \varGamma _{\phi (L)}(L)\,\theta _L=0, \end{aligned}$$
(8)

which is the intrinsic expression of (4) of Reference [26] in the autonomous case and \(\lambda =1\), because (remember that \(\ddot{q}^i=f^i(q,\dot{q})\) and for a function \(F(q,\dot{q}\)), the total time derivative \({\text {d}}F/{\text {d}}t\) means \(\dot{q}^i\partial /\partial q^i+f^i\partial /\partial \dot{q}^i\)) in this case the right-hand side of (8) is:

$$\begin{aligned} {\partial ^2L\over \partial v^i\partial v^j} f^j+v^j{\partial ^2L\over \partial v^i\partial q^j}\!-\!{\partial L\over \partial q^i}\!+\!\lambda \, {\varGamma _{\phi (L)}}(L)\,{\partial L\over \partial v^i}=0. \end{aligned}$$

The term \(\lambda \, \varGamma _{\phi (L)}\,\theta _L\) in (8) can be interpreted as a deformation, depending on the parameter \(\lambda \), of the dynamical Eq. (5).

Case \(\phi (L)=L^{\gamma +1}\):

In this case, \(\phi '(L)=(\gamma +1)\,L^\gamma \), while \(\phi ''(L)=\gamma (\gamma +1)\,L^{\gamma -1}\), and we also have \(\varGamma _{\phi (L)}(\phi '(L))=\gamma (\gamma +1)L^{\gamma -1}\, \varGamma _{\phi (L)}(L)\); therefore, the modified Euler–Lagrange equations obtained from \(\phi (L)\) are

$$\begin{aligned}&\gamma (\gamma +1)L^{\gamma -1}\,\varGamma _{\phi (L)}(L)\,\theta _L+(\gamma +1)L^{\gamma }\\&\quad ({\mathcal {L}}_{\varGamma _{\phi (L)}} \theta _L-{\text {d}}L)=0; \end{aligned}$$

if \((\gamma +1)L\ne 0\), the preceding expression becomes:

$$\begin{aligned} {\mathcal {L}}_{\varGamma _{\phi (L)}} \theta _L-{\text {d}}L+\frac{\gamma }{L}\,\varGamma _{\phi (L)}(L)\,\theta _L=0, \end{aligned}$$

which, as in the previous case, is the intrinsic expression (30) of [26]. The term \(( \gamma / L)\varGamma _{\phi (L)}(L)\,\theta _L\) represents the deformation of the Euler–Lagrange equations of the corresponding dynamical equation (5).

3 Dynamically equivalent deformations

As indicated before, different Lagrange functions can lead to the same vector field \(\varGamma _L\), and then, they are called dynamically equivalent [16]. For instance, for each pair of real values a and b, the function \(\bar{L}=a\, L+b\) is dynamically equivalent to L. Is there any other type of functions \(\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) such that \(\phi (L)\) is dynamically equivalent to L (i.e. \(\varGamma _{\phi (L)}=\varGamma _L\))? The answer is given in the following theorem:

Theorem 2

Let \(\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a differentiable function and L a regular Lagrangian for a given dynamics, i.e. a SODE vector field \(\varGamma _L\) satisfying (5). Then, \(\phi (L)\) is dynamically equivalent to L if either there exist real numbers a and b such that \(\phi (L)=a\, L+b\), or L is a constant of the motion.

Proof

Recall the fundamental relation (6). Consequently, as L is an admissible Lagrangian for \(\varGamma _L\) if and only if \({\mathcal {L}}_{\varGamma _L}\theta _L={\text {d}}L\), then \(\phi (L)\) is admissible too if and only if

$$\begin{aligned} \varGamma _L(\phi '(L))=0. \end{aligned}$$
(9)

Now, using that \(\varGamma _L(\phi '(L))=\phi ''(L)\,\varGamma _L(L)\), we can conclude that if \(\phi (L)\) and L produce equivalent sets of equations, then either there exist real numbers \(a,b\in {\mathbb {R}}\) such that \(\phi (L)=a\,L+b\), i.e. \(\phi ''(L)=0\), or L is a constant of the motion. In the last case \(\phi (L)\) is equivalent to L for any differentiable function \(\phi \). \(\square \)

A typical example is the free dynamics on surfaces, i.e. motion under the only action of constraint functions, which is usually described by a Lagrangian defined by the induced Riemannian metric (see e.g. [31]). In this case, the energy coincides with the Lagrangian, which therefore is conserved. This implies that any function of the kinetic energy is an alternative Lagrangian. The case \(\phi (x)=\sqrt{x}\) gives rise to a singular Lagrangian and was studied in [32].

Note also that as L does not depend on the time, the energy \(E_L=\varDelta (L)-L\) is conserved, \(\varGamma _L(E_L)=0\), and then if the function \(\phi (L)\) produces an equivalent set of Euler–Lagrange equations, \(\varGamma _L(\varDelta (L))=0\) is a necessary and sufficient condition for \(\phi (L)\) to be equivalent to L, that is,

$$\begin{aligned} \varGamma _L(\varDelta (L))=0\Longleftrightarrow \varGamma _L(\phi (L))=0, \end{aligned}$$

assuming that \(\phi ''(L)\ne 0\). This shows that the assertion of [26] concerning Lagrangians such that \(v^i \partial L/\partial v^i=K\ne 1\) is also true when \(v^i \partial L/\partial v^i\) is a constant of motion.

The two particular cases we have considered are depending on a parameter and can be considered as a deformation of the original Lagrangian equations which correspond to the zero value of the parameter. Such deformation of the Lagrangian amounts to a deformation of both the symplectic structure \(\omega _L\) and the energy which lead to the deformed dynamical vector field. In the general case of a parameterized family of functions \(\phi _\lambda (L)\), the modified equations of motion include a term depending on the parameter which turns out to be \((\phi ''_\lambda (L )/\phi '_\lambda (L))\,{\varGamma _L} (L)\,(\partial L/\partial v^i)\).

This geometric approach to the problem is valid for the case of systems with a nontrivial configuration space, and it is simpler, intrinsic, i.e. independent of the choice of coordinates, and has allowed us to answer the question on the existence of dynamically equivalent deformations of the Lagrangian.

We have to remark that the importance of having alternative but not gauge equivalent Lagrangians in the search for non-Noether constants of motion [14–16] compels us to admit non-standard Lagrangians; otherwise the freedom for finding alternative Lagrangians is actually very small. Then, in cases for which a standard Lagrangian does not exist, we may find a non-standard Lagrangian and therefore an associated constant of the motion [17], very useful in the reduction process to simpler cases, the corresponding non-standard energy. Moreover, there are cases where alternative Lagrangians exist [18], and then we can obtain more constants of motion related to the traces of powers of the associated recursion operator [15, 16]. This justifies the relevance of the existence of alternative Lagrangians. As a final comment, the quantization of classical systems described by alternative Lagrangians, whatever the quantization prescription be used, will lead to inequivalent quantum systems. However, in most cases, the geometric formulation of the classical theory is a previous necessary ingredient for the quantization process, which enforces the convenience of using geometric approaches.