In this paper, we develop a Hamilton–Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems, and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton–Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton–Jacobi problem for a Čaplygin system to the Hamilton–Jacobi problem for a forced Lagrangian system.

## I. INTRODUCTION

The classical formulation^{1–3} of the Hamilton–Jacobi problem for a Hamiltonian system on *T*^{*}*Q* consists in looking for a function *S* on $Q\xd7R$, called the principal function (also known as the generating function), such that

where $H:T*Q\u2192R$ is the Hamiltonian function. With the *ansatz**S*(*q*^{i}, *t*) = *W*(*q*^{i}) − *tE*, where *E* is a constant, the equation above reduces to

where $W:Q\u2192R$ is the so-called characteristic function. Both Eqs. (1) and (2) are known as the Hamilton–Jacobi equation.

Despite the difficulties to solve a partial differential equation instead of a system of ordinary differential equations, i.e., to solve the Hamilton–Jacobi equation instead of Hamilton equations, the Hamilton–Jacobi theory provides a remarkably powerful method to integrate the dynamics of many Hamiltonian systems. In particular, for a completely integrable system, if one knows as much constants of the motion in involution as degrees of freedom of the system, one can obtain a complete solution of the Hamilton–Jacobi problem and completely solve the Hamiltonian system or, in other words, reduce it to quadratures.^{4–6}

Geometrically, the Hamilton–Jacobi equation (2) can be written as

where d*W* is a 1-form on *Q*. This 1-form transforms the integral curves of a vector field $XHdW$ on *Q* into integral curves of the dynamical vector field *X*_{H} on *TQ* (the latter satisfying Hamilton equations). This geometric procedure^{2,7} has been extended to many other different contexts, such as nonholonomic systems,^{7–11} singular Lagrangian systems,^{12–14} higher-order systems,^{15} field theories,^{16–22} or contact systems.^{23} An unifying Hamilton–Jacobi theory for almost-Poisson manifolds was developed in Ref. 24. The Hamilton–Jacobi theory has also been generalized to Hamiltonian systems with non-canonical symplectic structures,^{25} non-Hamiltonian systems,^{26} locally conformally symplectic manifolds,^{27} Nambu–Poisson and Nambu–Jacobi manifolds,^{28,29} Lie algebroids,^{30} and implicit differential systems.^{31} The applications of Hamilton–Jacobi theory include the relation between classical and quantum mechanics,^{32–34} information geometry,^{35,36} control theory,^{37} and the study of phase transitions.^{38}

In the same fashion, in this paper, we develop a Hamilton–Jacobi theory for systems with external forces. This paper is the natural continuation of our previous paper about symmetries and constants of the motion of systems with external forces^{39} (see also Ref. 40). Mechanical systems with external forces (so-called forced systems) appear commonly in engineering and describe certain physical systems with dissipation.^{39–41} Moreover, they emerge after a process of reduction in a nonholonomic Čaplygin system.^{9,10,42–45} A particular type of external forces is the so-called Rayleigh forces,^{39,40,46} i.e., forces that can be written as the derivative of a “potential” with respect to the velocities. Forced systems on a Lie group have been studied in Ref. 46.

This paper is organized as follows: In Sec. II, we recall the geometric concepts we will make use of. In Sec. III, we develop a Hamilton–Jacobi theory for Hamiltonian systems with external forces. We consider the complete solutions, particularized for Rayleigh forces, and discuss some examples. The analogous theory for Lagrangian systems with external forces is described in Sec. IV. In Sec. V, we present a method for the reduction and reconstruction of solutions of the Hamilton–Jacobi problem for forced Hamiltonian systems with symmetry. Finally, Sec. VI is devoted to the reduction of Čaplygin systems to forced Lagrangian systems in order to obtain solutions of the forced Hamilton–Jacobi problem and reconstruct solutions of the nonholonomic Hamilton–Jacobi problem.

## II. PRELIMINARIES

### A. Semibasic forms and fibered morphims

Consider a fiber bundle *π*: *E* → *M*. Let us recall^{2,47,48} that a 1-form *β* on *E* is called *semibasic* if

for all vertical vector fields *Z* on *E*. If (*x*^{i}, *y*^{a}) are fibered (bundle) coordinates, then the vertical vector fields are locally generated by {*∂*/*∂y*^{a}}. Hence, *β* is a semibasic 1-form if it is locally written as

We shall particularize this definition for the cases of tangent and cotangent bundles. In what follows, let *Q* be an *n*-dimensional differentiable manifold. Given a morphism of fiber bundles,

one can define an associated semibasic 1-form^{47,48}*β*_{D} on *TQ* by

where $vq\u2208TqQ,uvq\u2208Tvq(TQ)$.

If locally *D* is given by

then

Conversely, given a semibasic 1-form *β* on *TQ*, one can define the following morphism of fiber bundles:

for every $vq,wq\u2208TqQ,uwq\u2208Twq(TQ)$, with $T\tau Q(uwq)=wq$. In local coordinates, if

then

Here, $(qi,q\u0307i)$ are bundle coordinates in *TQ*.

Hence, there exists a one-to-one correspondence between semibasic 1-forms on *TQ* and fibered morphisms from *TQ* to *T*^{*}*Q*.

### B. Hamiltonian mechanics

An external force is geometrically interpreted as a semibasic 1-form on *T*^{*}*Q*. A Hamiltonian system with external forces (the so-called *forced Hamiltonian system*) (*H*, *β*) is given by a Hamiltonian function $H:T*Q\u2192R$ and a semibasic 1-form *β* on *T*^{*}*Q*. Let *ω*_{Q} = −d*θ*_{Q} be the canonical symplectic form of *T*^{*}*Q*. Locally these objects can be written as

where (*q*^{i}, *p*_{i}) are bundle coordinates in *T*^{*}*Q*.

The dynamics of the system is given by the vector field *X*_{H,β}, defined by

If *X*_{H} is the Hamiltonian vector field for *H*, that is,

and *Z*_{β} is the vector field defined by

then we have

Locally, the above equations can be written as

Then, a curve *q*^{i}(*t*), *p*_{i}(*t*) in *T*^{*}*Q* is an integral curve of *X*_{H,β} if and only if it satisfies the *forced Hamilton equations*

Let us recall that the *Poisson bracket* is the bilinear operation,

with *X*_{f}, *X*_{g} being the Hamiltonian vector fields associated with *f* and *g*, respectively.

*H*,

*β*) be a forced Hamiltonian system on

*T*

^{*}

*Q*. A function

*f*on

*T*

^{*}

*Q*is called a

*constant of the motion*(or a

*conserved quantity*) if

*f*is constant along the solutions of the forced Hamilton equations (5).

### C. Lagrangian systems with external forces

The *Poincaré*–*Cartan* 1-form on *TQ* associated with the Lagrangian function $L:TQ\u2192R$ is

where *S*^{*} is the adjoint operator of the vertical endomorphism on *TQ*, which is locally

The Poincaré–Cartan 2-form is *ω*_{L} = −d*θ*_{L}, so locally

One can easily verify that *ω*_{L} is symplectic if and only if *L* is regular, that is, if the Hessian matrix

is invertible. The *energy* of the system is given by

where Δ is the Liouville vector field,

Similarly to the Hamiltonian framework, an external force is represented by a semibasic 1-form *α* on *TQ*. Locally,

The dynamics of the forced Lagrangian system (*L*, *α*) is given by

that is, the integral curves of the *forced Euler*–*Lagrange vector field**ξ*_{L,α} satisfy the forced Euler–Lagrange equations,

We can write *ξ*_{L,α} = *ξ*_{L} + *ξ*_{α}, where

This vector field is a *second order differential equation* (SODE), that is,

*L*,

*α*) be a forced Lagrangian system on

*TQ*. A function

*f*on

*TQ*is called a

*constant of the motion*(or a

*conserved quantity*) if

*f*is constant along the solutions of the forced Euler–Lagrange equations (12).

### D. Rayleigh forces

An external force $R\u0304$ is said to be *of Rayleigh type* (or simply *Rayleigh* for short)^{39,40} if there exists a function $R$ on *TQ* such that

which can be locally written as

This function $R$ is called the *Rayleigh dissipation function* (or the *Rayleigh potential*). In other words, the fibered morphism $DR\u0304:TQ\u2192T*Q$ associated with $R\u0304$ is given by the fiber derivative of $R$ (see Ref. 46), namely,

A *Rayleigh system*$(L,R)$ is a forced Lagrangian system with Lagrangian function *L* and with external force $R\u0304$ generated by the Rayleigh potential $R$. For a Rayleigh system $(L,R)$ with Rayleigh force $R\u0304$, the forced Euler–Lagrange vector field is denoted by $\xi L,R\u0304$, given by

This vector field can be written as $\xi L,R\u0304=\xi L+\xi R\u0304$, where

and the forced Euler–Lagrange equations (12) can be written as

If a Rayleigh potential $R$ on *TQ* defines a Rayleigh force $R\u0304$ on *TQ*, $R+f$ also defines $R\u0304$ for any function *f* on *Q*. In other words, given a Rayleigh force $R\u0304$, its associated Rayleigh potential $R$ is defined up to the addition of a basic function on *TQ*.

The *vertical differentiation*^{47} d_{S} on *T*^{*}*Q* is given by

where *ι*_{S} denotes the *vertical derivation*, given by

for any function *f*, any *p*-form *ω*, and any vector fields *X*_{1}, …, *X*_{p} on *TQ*. In particular,

for any function *f* on *TQ*. We can then write a Rayleigh force as

A *linear Rayleigh force* $R\u0304$ is a Rayleigh force for which $R$ is a quadratic form in the velocities, namely,

where *R*_{ij} is symmetric and non-degenerate, and hence, the Rayleigh force is

In such a case, one can define an associated *Rayleigh tensor**R* ∈ *T*^{*}*Q* × *T*^{*}*Q*, given by

A *linear Rayleigh system* (*L*, *R*) is a Rayleigh system such that $R\u0304$ is a linear Rayleigh force with Rayleigh tensor *R*.

The *Legendre transformation* is a mapping Leg: *TQ* → *T*^{*}*Q* such that the diagram

commutes. Here, *τ*_{q} and *π*_{Q} are the canonical projections on *Q*. Locally,

with $pi=\u2202L/\u2202q\u0307i$. In what follows, let us assume that the Lagrangian *L* is *hyper-regular*, i.e., that Leg is a (global) diffeomorphism.

### E. Dissipative bracket

*dissipative bracket*is a bilinear map [·, ·]:

*C*

^{∞}(

*TQ*) ×

*C*

^{∞}(

*TQ*) →

*C*

^{∞}(

*TQ*) given by

*S*is the vertical endomorphism and

*X*

_{f}is the Hamiltonian vector field associated with

*f*on (

*TQ*,

*ω*

_{L}), namely,

*The dissipative bracket* [·, ·] *on* (*TQ*, *ω*_{L}) *verifies the following properties:*

[

*f*,*g*] = [*g*,*f*]*(it is symmetric) and*[

*f*,*gh*] = [*f*,*h*]*g*+ [*f*,*g*]*h**(“Leibniz rule”)*

*for all functions* *f*, *g* *on* *TQ**.*

Since the dissipative bracket is bilinear and verifies the Leibniz rule, it is a derivation or a so-called Leibniz bracket.^{49}

*Consider a Rayleigh system*$(L,R)$

*on*(

*TQ*,

*ω*

_{L})

*. A function*

*f*

*on*

*TQ*

*is a constant of the motion of*$(L,R)$

*if and only if*

*where*$\u22c5,\u22c5$

*is the Poisson bracket*

*(6)*

*defined by*

*ω*

_{L}

*.*

*f*is a constant of the motion.□

Other types of dissipative systems, particularly thermodynamical systems, exhibit a “double bracket” dissipation, i.e., their dynamics are described in terms of two brackets [in our case, the Poisson bracket (6) and the dissipative bracket (17)]. As a matter of fact, the dissipative bracket we defined above has certain similarities with other types of brackets.

The dissipative bracket [·, ·] defined above resembles the dissipative bracket (·, ·) appearing in the metriplectic framework.^{50,51} Both brackets are symmetric and bilinear. However, the latter requires the additional assumption that (*E*_{L}, *f*) vanishes identically for every function *f* on *TQ*. Clearly, this requirement does not hold for our dissipative bracket.

_{GL}.

^{52}This bracket, together with symmetry and bilinearity, satisfies the positivity condition, i.e., [

*f*,

*f*]

_{GL}≥ 0 holds pointwisely for all

*f*on

*TQ*. As a matter of fact, this holds for our bracket in the case of many relevant Lagrangians. For instance, consider a Lagrangian of the form

*m*

_{i}. Then,

### F. Natural Lagrangians and Hamiltonian Rayleigh forces

Consider a *natural Lagrangian* *L* on *TQ*, i.e., a Lagrangian function of the form

where

is a (pseudo)Riemannian metric on *Q*. Clearly, *L* is regular if and only if *g* is non-degenerate. As it is well-known, these are the usual Lagrangians in classical mechanics. The Legendre transformation is now linear,

where $(gij)=(gij)\u22121$. In other words, the Legendre transformation consists simply in the “raising and lowering of indices” defined by the metric *g*.

Consider a linear Rayleigh system $(L,R)$, where *L* is natural. The associated *Hamiltonian Rayleigh potential* $R\u0303$ on *T*^{*}*Q* is given by

Similarly, the *Hamiltonian Rayleigh force*$R\u0303$ on *T*^{*}*Q* is given by

When the Lagrangian is regular, the Legendre transformation is well-defined, so we can define a tensor field $S\u0303\u2208T(T*Q)\u2297T*Q$ given by

In particular, if the Lagrangian is natural, then we have

Hence, the Hamiltonian Rayleigh force $R\u0303$ can be expressed in terms of the Rayleigh potential $R\u0303$ as

We shall omit the adjective Hamiltonian and refer to $R\u0303$ and $R\u0303$ as the Rayleigh potential and the Rayleigh force, respectively, if there is no danger of confusion.

Let us introduce the *vertical differentiation*^{47} $dS\u0303$ on *T*^{*}*Q* as

where $\iota S\u0303$ is defined analogously to *ι*_{S} by replacing *S* with $S\u0303$. In particular,

for any function *f* on *T*^{*}*Q*. We can then write a Hamiltonian Rayleigh force as

Consider a linear Rayleigh system (*L*, *R*), where *L* is natural. The associated Hamiltonian Rayleigh potential $R\u0303$ on *T*^{*}*Q* is given by

where

Similarly, the Hamiltonian Rayleigh force $R\u0303$ on *T*^{*}*Q* is given by

where $Rji=Rkjgik$. The associated Hamiltonian Rayleigh tensor $R\u0302\u2208TQ\u2297T*Q$ is given by

The linear Hamiltonian Rayleigh force $R\u0303$ can thus be written as

This motivates the next definition of linear Rayleigh forces in the Hamiltonian framework, without the need of considering natural Lagrangians, as follows:

*linear Hamiltonian Rayleigh force*if it can be written as

*Hamiltonian Rayleigh tensor*. A

*linear Hamiltonian Rayleigh system*$(H,R\u0302)$ is a forced Hamiltonian system whose external force is a linear Hamiltonian Rayleigh force. When there is no ambiguity, the adjective Hamiltonian will be omitted.

Since *T*^{*}*Q*, unlike *TQ*, has not a canonical vertical endomorphism, there is not a natural way to define Hamiltonian non-linear Rayleigh forces, besides Legendre-transforming Lagrangian Rayleigh forces.

## III. HAMILTON–JACOBI THEORY FOR SYSTEMS WITH EXTERNAL FORCES

Let (*H*, *β*) be a forced Hamiltonian system on *T*^{*}*Q*. Its dynamical vector field *X*_{H,β} is given by Eq. (3). Given a 1-form *γ* on *Q* (i.e., a section of *π*_{Q}: *T*^{*}*Q* → *Q*), it is possible to project *X*_{H,β} along *γ*(*Q*), obtaining the vector field

on *Q*, so that the following diagram commutes:

*The vector fields* *X*_{H,β} *and* $XH,\beta \gamma $ *are* *γ**-related if and only if* *X*_{H,β} *is tangent to* *γ*(*Q*)*.*

*X*

_{H,β}and $XH,\beta \gamma $ are

*γ*-related if

Therefore, the integral curves of $XH,\beta \gamma $ are mapped to integral curves of *X*_{H,β} [which satisfy the forced Hamilton Eqs. (5)] via *γ*. Indeed, if *σ* is an integral curve of $XH,\beta \gamma $, then

so *γ*◦*σ* is an integral curve of *X*_{H,β}. Conversely, if *γ*◦*σ* is an integral curve of *X*_{H,β} for every integral curve *σ* of *Y* = *Tπ*_{Q}◦*X*_{H,β}◦*γ*, then

for every integral curve *σ*, and hence, *X*_{H,β} and *Y* are *γ*-related.

From Eq. (4), locally we have that

Then, Eq. (21) yields

In other words,

that is,

If *γ* is closed, we have

that is,

Let us recall that a *Lagrangian submanifold*$L\u2282T*Q$ is a maximal isotropic submanifold, i.e., a submanifold such that *ω*_{Q}|_{S} = 0 and dim *S* = 1/2 dim *T*^{*}*Q* = dim *Q*. Clearly, a 1-form *γ* on *Q* is closed if and only if Im *γ* is a Lagrangian submanifold.

A 1-form *γ* on *Q* is called a *solution of the Hamilton*–*Jacobi problem* for (*H*, *β*) if

it is closed and

it satisfies Eq. (22).

This equation is known as the *Hamilton*–*Jacobi equation*.

The results above can be summarized in the following theorem:

*Let* *γ* *be a closed 1-form on* *Q**. Then, the following conditions are equivalent:*

*γ**is a solution of the Hamilton–Jacobi problem for*(*H*,*β*)*.**For every curve*$\sigma :R\u2192Q$*such that*$\sigma \u0307(t)=T\pi Q\u25e6XH,\beta \u25e6\gamma \u25e6\sigma (t)$*for all**t**, then**γ*◦*σ**is an integral curve of**X*_{H,β}.Im

*γ**is a Lagrangian submanifold of**T*^{*}*Q**and**X*_{H,β}*is tangent to it.*

*H*+

*β*vanishes on Im

*γ*, and hence,

*β*is non-degenerate, it is a symplectic form and Im

*γ*is a Lagrangian submanifold on (

*T*

^{*}

*Q*, d

*β*). In general, it is not easy to see whether d

*β*is non-degenerate or not. However, there is a type of external forces for which this is simple: the Rayleigh forces linear in the momenta.

Let $R\u0302$ be a Hamiltonian Rayleigh tensor on *T*^{*}*Q*, and let $R\u0303$ be the associated Hamiltonian Rayleigh force on *T*^{*}*Q*. If $R\u0302$ is non-degenerate, then $dR\u0303$ is a symplectic form on *T*^{*}*Q*.

*X*=

*X*

^{i}

*∂*/

*∂q*

^{i}+

*Y*

_{i}

*∂*/

*∂p*

_{i}on

*T*

^{*}

*Q*. Then,

*X*∈ ker Ω

_{Q,γ}if and only if

*R*is positive-definite), then $X\u2208kerdR\u0303$ if and only if

*X*= 0, so $dR\u0303$ is symplectic.□

When $Rji$ does not depend on (*q*^{i}), we can make the change of bundle coordinates,

so that $dR\u0303=\omega Q$ and $(qi,p\u0303i)$ are Darboux coordinates.

Consider a linear Rayleigh system $(H,R\u0302)$. Suppose that $R\u0302$ is non-degenerate. Then, a closed 1-form *γ* on *Q* is a solution of the Hamilton–Jacobi problem for $(H,R\u0303)$ if and only if Im *γ* is a Lagrangian submanifold of $(T*Q,dR\u0303)$.

For a Rayleigh system $(H,R\u0303)$, the Hamilton–Jacobi equation can also be written as

We will also refer to the Hamilton–Jacobi problem for $(H,R\u0304)$ as the Hamilton–Jacobi problem for $(H,R)$. In the case of a linear Rayleigh system $(H,R\u0302)$, we have

so the Hamilton–Jacobi equation (23) can be written as

We will also refer to the Hamilton–Jacobi problem for $(H,R\u0304)$ as the Hamilton–Jacobi problem for $(H,R\u0302)$.

One can consider a more general problem by relaxing the hypothesis of *γ* being closed.

A *weak solution of the Hamilton*–*Jacobi problem* for (*H*, *β*) is a 1-form *γ* on *Q* such that *X*_{H,β} and $XH,\beta \gamma $ are *γ*-related. Here, $XH,\beta \gamma $ is the vector field defined by (20).

*Consider a* 1*-form* *γ* *on* *Q**. Then, the following statements are equivalent:*

*γ**is a weak solution of the Hamilton–Jacobi problem for*(*H*,*β*)*.**γ**satisfies the equation*(25)$\iota XH,\beta \gamma d\gamma =\u2212d(H\u25e6\gamma )\u2212\gamma *\beta .$*X*_{H,β}*is tangent to the submanifold*Im*γ*⊂*T*^{*}*Q**.**If*$\sigma :R\u2192Q$*satisfies*$\sigma \u0307(t)=T\pi Q\u25e6XH,\beta \u25e6\gamma \u25e6\sigma (t),$*then**γ*◦*σ**is an integral curve of**X*_{H,β}.

Let

Observe that Eq. (25) holds if and only if

(local expressions). Let *γ* be a 1-form *γ* on *Q*. Let *H* be a Hamiltonian function on *T*^{*}*Q*, and let $\beta ,R\u0303$, and $R\u0302$ be an external force, a Rayleigh potential, and a Rayleigh tensor on *T*^{*}*Q*, respectively. If *γ* is closed, then

*γ*is a solution of the Hamilton–Jacobi problem for (*H*,*β*) if and only if$\u2202H\u2202qi+\u2202H\u2202pj\u2202\gamma j\u2202qi+\beta i\u25e6\gamma =0,i=1,\u2026,n.$*γ*is a solution of the Hamilton–Jacobi problem for $(H,R\u0303)$ if and only if$\u2202H\u2202qi+\u2202H\u2202pj\u2202\gamma j\u2202qi+\u2202R\u0303\u2202pi\u25e6\gamma =0,i=1,\u2026,n.$*γ*is a solution of the Hamilton–Jacobi problem for $(H,R\u0302)$ if and only if$\u2202H\u2202qi+\u2202H\u2202pj\u2202\gamma j\u2202qi+Rji\gamma j=0,i=1,\u2026,n.$Moreover, if*γ*is not necessarily closed, it is a weak solution of the Hamilton–Jacobi problem for (*H*,*β*) if and only ifSimilar expressions can be easily found for Rayleigh forces or linear Rayleigh forces.$\u2202H\u2202qi+\u2202H\u2202pj\u2202\gamma j\u2202qi+\beta i\u25e6\gamma +\u2202H\u2202pj\u2202\gamma i\u2202qj\u2212\u2202\gamma j\u2202qi=0.$

### A. Complete solutions

The main interest in the standard Hamilton–Jacobi theory lies in finding a complete family of solutions to the problem.^{7,24} As it is explained below, knowing a complete solution of the Hamilton–Jacobi problem for a forced Hamiltonian system is tantamount to completely integrating the system, namely, there is a constant of the motion for each degree of freedom of the system, and these constants of the motion are in mutual involution.

Consider a forced Hamiltonian system (*H*, *β*) on *T*^{*}*Q* and assume that dim *Q* = *n*.

Let $U\u2286Rn$ be an open set. A map Φ: *Q* × *U* → *T*^{*}*Q* is called a *complete solution of the Hamilton–Jacobi problem* for (*H*, *β*) if

Φ is a local diffeomorphism and

- for any
*λ*= (*λ*_{1}, …,*λ*_{n}) ∈*U*, the mapis a solution of the Hamilton–Jacobi problem for ($\Phi \lambda :Q\u2192T*Qq\u21a6\Phi \lambda (q)=\Phi (q,\lambda 1,\u2026,\lambda n)$*H*,*β*).

For the sake of simplicity, we shall assume Φ to be a global diffeomorphism. Consider the functions given by

where *π*_{a} denotes the projection over the *a*th component of $Rn$.

*The functions* *f*_{a} *are constants of the motion. Moreover, they are in involution, i.e.,* $fa,fb=0$*, where* $\u22c5,\u22c5$ *is the Poisson bracket defined by* *ω*_{Q}*.*

*p*∈

*T*

^{*}

*Q*, suppose that

*f*

_{a}(

*p*) =

*λ*

_{a}for each

*a*= 1, …,

*n*. Observe that

*X*

_{H,β}is tangent to Im Φ

_{λ}, and hence,

*a*= 1, …,

*n*, that is,

*f*

_{1}, …,

*f*

_{n}are constants of the motion. In addition,

*H*,

*β*), with

*a*,

*b*= 1, …,

*n*, so the constants of the motion are in involution. Consider the 1-form

*γ*on

*Q*given by

*γ*is closed. In fact, it is exact,

*γ*is a complete solution of the Hamilton–Jacobi problem. When

*κ*

_{i}= 0 for every

*i*= 1, …,

*n*,

*H*. See Example 4 for the Lagrangian counterpart of this example.

*g*does not depend on

*q*(for instance,

*g*

_{ij}=

*m*

_{i}

*δ*

_{ij}, with

*m*

_{i}being the mass of the

*i*th particle), and

*q*. Then, the Hamilton–Jacobi equation (24) can be locally written as

*γ*

_{λ}is closed; in fact, it is exact,

## IV. HAMILTON–JACOBI THEORY FOR LAGRANGIAN SYSTEMS WITH EXTERNAL FORCES

As it has been seen in Sec. III, the natural framework for the Hamilton–Jacobi theory is the Hamiltonian formalism on the cotangent bundle. Following the work of Cariñena, Gràcia, Marmo, Martínez, Muñoz-Lecanda, and Román-Roy,^{7,8,32,33,53} we introduce an analogous problem in the Lagrangian formalism on the tangent bundle as follows:

A vector field *X* on *Q* is called a *solution of the Lagrangian Hamilton–Jacobi problem* for (*L*, *α*) if

Leg◦

*X*is a closed 1-form and*X*satisfies the equation(26)$d(EL\u25e6X)=\u2212X*\alpha .$This equation is known as the

*Lagrangian Hamilton*–*Jacobi equation*. When there is no risk of ambiguity, we shall refer to the Lagrangian Hamilton–Jacobi problem (respectively, equation) as simply the Hamilton–Jacobi problem (respectively, equation).

If *γ* = Leg◦*X* is a closed 1-form, then Im *γ* is a Lagrangian submanifold of (*T*^{*}*Q*, *ω*_{Q}). Therefore, Im *X* is a Lagrangian submanifold of (*TQ*, *ω*_{L}). In other words, Leg◦*X* is closed if and only if *X*^{*}*ω*_{L} = 0. Moreover, it is easy to see that *X* and *ξ*_{L,α} are *X*-related, that is,

*Let* *X* *be a vector field on* *Q* *that satisfies* *X*^{*}*ω*_{L} = 0*. Then, the following assertions are equivalent:*

*X is a solution of the Hamilton–Jacobi problem for*(*L*,*α*)*.*Im

*X**is a Lagrangian submanifold of**TQ**invariant by**ξ*_{L,α}*.**For every curve*$\sigma :R\u2192Q$*such that**σ**is an integral curve of**X**, then*$X\u25e6\sigma :R\u2192TQ$*is an integral curve of**ξ*_{L,α}*.*

As in the Hamiltonian case, one can consider a more general problem by relaxing the hypothesis of Leg◦*X* being closed.

A *weak solution of the Hamilton–Jacobi problem* for (*L*, *α*) is a vector field *X* on *Q* such that *X* and *ξ*_{L,α} are *X*-related.

Clearly, a weak solution of the Hamilton–Jacobi problem for (*L*, *α*) is a solution of the Hamilton–Jacobi problem for (*L*, *α*) if and only if *X*^{*}*ω*_{L} = 0.

*Let* *X* *be a vector field on* *Q**. Then, the following statements are equivalent:*

*X is a solution of the generalized Hamilton–Jacobi problem for*(*L*,*α*)*.**X**satisfies the equation*(27)$\iota X(X*\omega L)=d(EL\u25e6X)+X*\alpha .$*The submanifold*Im*X*⊂*TQ**is invariant by**ξ*_{L,α}.*If*$\sigma :R\u2192Q$*is an integral curve of**X**, then**X*◦*σ**is an integral curve of**ξ*_{L,α}*.*

*X*and

*ξ*

_{L,α}are

*X*-related, we can write

The proof of the converse is completely analogous to the one of Theorem 1 in Ref. 7.□

### A. Equivalence between Lagrangian and Hamiltonian Hamilton–Jacobi problems

Given a forced Lagrangian system (*L*, *α*) on *TQ* (with *L* hyper-regular), one can obtain an associated forced Hamiltonian system (*H*, *β*) on *T*^{*}*Q*, where

Moreover, the dynamical vector fields *ξ*_{L,α} and *X*_{H,β} [given by Eqs. (3) and (11), respectively] are Leg-related, i.e.,

*Consider a hyper-regular forced Lagrangian system* (*L*, *α*) *on* *TQ**, with the associated forced Hamiltonian system* (*H*, *β*) *on* *T*^{*}*Q**. Then,* *X* *is a (weak) solution of the Hamilton–Jacobi problem for* (*L*, *α*) *if and only if* *γ* = Leg◦*X* *is a (weak) solution of the Hamilton–Jacobi problem for* (*H*, *β*)*.*

*X*be a weak solution of the Hamilton–Jacobi problem for (

*L*,

*α*). Then,

*X*and

*ξ*

_{L,α}are

*X*-related. Composing the left- and right-hand sides with

*Tπ*

_{Q}from the left, we obtain

*γ*is a weak solution of the Hamilton–Jacobi problem for (

*H*,

*β*).

*γ*is a solution of the Hamilton–Jacobi problem for (

*H*,

*β*),

*X*is

*γ*-related to

*X*

_{H,β}. Moreover,

*X*is a weak solution of the Lagrangian Hamilton–Jacobi problem.

Obviously, the Lagrangian weak solution is a solution if and only if the associated Hamiltonian solution is a closed 1-form.□

This result could be extended for regular but not hyper-regular Lagrangians (i.e., Leg is a local diffeomorphism), where it only holds in the open sets where Leg is a diffeomorphism.

### B. Complete solutions

Complete solutions for the Hamilton–Jacobi problem are defined analogously to the ones in *T*^{*}*Q* (see Definition 7).

Let $U\u2286Rn$ be an open set. A map Φ: *Q* × *U* → *TQ* is called a *complete solution of the Hamilton*–*Jacobi problem* for (*L*, *α*) if

Φ is a local diffeomorphism and

- for any
*λ*= (*λ*_{1}, …,*λ*_{n}) ∈*U*, the mapis a solution of the Hamilton–Jacobi problem for ($\Phi \lambda :Q\u2192TQq\u21a6\Phi \lambda (q)=\Phi (q,\lambda 1,\u2026,\lambda n)$*L*,*α*).

^{39,40}with

*X*is a Lagrangian submanifold. As a matter of fact,

*q*(0) =

*q*

_{0}and $q\u0307(0)=q\u03070$. Similarly, the integral curves of

*X*are given by

*n*dimensions, namely,

*f*

_{a}are constants of the motion. In addition,

*a*,

*b*= 1, …,

*n*, so they are in involution. Then,

*g*does not depend on

*q*(for instance,

*g*

_{ij}=

*m*

_{i}

*δ*

_{ij}, with

*m*

_{i}being the mass of the

*i*th particle), and

*R*

_{ij}does not depend on

*q*. Then, the Hamilton–Jacobi equation can be locally written as

## V. REDUCTION AND RECONSTRUCTION OF THE HAMILTON–JACOBI PROBLEM

Let *G* be a connected Lie group acting freely and properly on *Q* by a left action Φ, namely,

As usual, we denote by $g$ the Lie algebra of *G* and denote the dual of $g$ by $g*$. For each *g* ∈ *G*, we can define a diffeomorphism

Under these conditions, the quotient space *Q*/*G* is a differentiable manifold and *π*_{G}: *Q* → *Q*/*G* is a *G*-principal bundle. The action Φ induces a lifted action $\Phi T*$ on *T*^{*}*Q* given by

for every $\alpha q\u2208Tq*Q$. Since Φ is a diffeomorphism, its lift to *T*^{*}*Q* leaves *θ*_{Q} invariant;^{2} in other words, *θ*_{Q} is *G*-invariant.

The *natural momentum map* $J:T*Q\u2192g$ is given by

for each $\xi \u2208g$. Here, *ξ*_{Q} is the infinitesimal generator of the action of $\xi \u2208g$ on *Q*, and $\xi Qc$ is the generator of the lifted action on *T*^{*}*Q*. The natural momentum map is *G*-equivariant for the lifted action on *T*^{*}*Q*. For each $\xi \u2208g$, we have a function $J\xi :T*Q\u2192R$ given by

that is,

A vector field *Z* on *Q* defines an associated function *ιZ* on *T*^{*}*Q*. Locally, if

then

Given a vector field *X* on *Q*, its *complete lift*^{54} is a vector field *X*^{c} on *T*^{*}*Q* such that

for any vector field *Z* on *Q*. Locally, if *X* has the form above, then

*G*-invariant forced Hamiltonian system (

*H*,

*β*) is a forced Hamiltonian system (

*H*,

*β*) such that

*H*and

*β*are both

*G*-invariant, namely,

If *H* is known to be *G*-invariant, the subgroup *G*_{β} such that (*H*, *β*) is *G*_{β}-invariant can be found through the following lemma:

*Consider a forced Hamiltonian system* (*H*, *β*)*. Suppose that* *H* *is* *G**-invariant. Let* $\xi \u2208g$*. Then, the following statements hold:*

*J*^{ξ}*is a constant of the motion if and only if*$\beta (\xi Qc)=0.$*If the previous equation holds, then**ξ**leaves**β**invariant if and only if*$\iota \xi Qcd\beta =0.$*Moreover, the vector subspace*(28)$g\beta =\xi \u2208g\u2223\beta (\xi Qc)=0,\iota \xi Qcd\beta =0$*is a Lie subalgebra of*$g$*.*

*θ*

_{Q}is $g$-invariant. Contracting with the dynamical vector field

*X*

_{H,β}[given by Eq. (3)] yields

*H*is $g$-invariant, so, by Eq. (7),

*J*

^{ξ}is a constant of the motion if and only if $\beta (\xi Qc)$ vanishes. The proofs of the other statements are completely analogous to the ones of the Lagrangian counterpart of this lemma (see Refs. 39 and 40 and also Ref. 46).□

In the following paragraphs, we shall briefly recall some results we will make use of (see Refs. 55 and 56 and references therein for more details).

A *G**-invariant Lagrangian submanifold* $L\u2282T*Q$ is a Lagrangian submanifold in *T*^{*}*Q* such that $\Phi gT*(L)=L$ for all *g* ∈ *G*. Given the momentum map *J* defined above and a Lagrangian submanifold $L\u2282T*Q$, it can be shown that *J* is constant along $L$ if and only if $L$ is *G*-invariant.

The quotient space *T*^{*}*Q*/*G* has a Poisson structure induced by the canonical symplectic structure on *T*^{*}*Q* such that *π*: *T*^{*}*Q* → *T*^{*}*Q*/*G* is a Poisson morphism.

Denote by *G*_{β} ⊂ *G* the Lie subgroup whose Lie algebra is $g\beta $, defined by (28). Let $\mu \u2208g*$. Let us assume that $(G\beta )\mu =G$. Then, *J*^{−1}(*μ*)/*G* is a symplectic leaf of *T*^{*}*Q*/*G*. Moreover, *J*^{−1}(*μ*) is a coisotropic submanifold. Assume that $L\u2282J\u22121(\mu )$ is a Lagrangian submanifold. Then, by the coisotropic reduction theorem,^{2} $\pi (L)$ is a Lagrangian submanifold to *J*^{−1}(*μ*)/*G*.

Furthermore, it can be shown that *J*^{−1}(*μ*)/*G* is diffeomorphic to the cotangent bundle *T*^{*}(*Q*/*G*). In addition, if *J*^{−1}(*μ*)/*G* is endowed with the symplectic structure *ω*_{μ} given by the Marsden–Weinstein reduction procedure, it is symplectomorphic to *T*^{*}(*Q*/*G*) endowed with a modified symplectic structure $\omega \u0303Q/G$. This modified symplectic form is given by the canonical symplectic form plus a magnetic term, namely,

where *ω*_{Q/G} is the canonical symplectic form on *T*^{*}(*Q*/*G*).

Combining the previous paragraphs, $\pi (L)$ can be seen as a Lagrangian submanifold of a cotangent bundle with a modified symplectic structure. Let $g\u0303*$ denote the adjoint bundle to *π*_{G}: *Q* → *Q*/*G* via the coadjoint representation

A connection *A* on *π*_{G}: *Q* → *Q*/*G* induces a splitting

This identification is given by

where $horq:T\pi G(q)(Q/G)\u2192Hq$ denotes the horizontal lift of the connection *A*. Here, $Hq$ denotes the horizontal space of *A* at *q* ∈ *Q*, namely,

If *α*_{q} ∈ *J*^{−1}(*μ*), then *J*(*α*_{q}) = *μ* and Ψ([*α*_{q}]) = (*α*_{q}◦hor_{q}, *μ*), so

Consider a *G*-invariant forced Hamiltonian system (*H*, *β*) on *T*^{*}*Q*. Then, *H* = *H*_{G}◦*π*, where $HG:T*Q/G\u2192R$ is the reduced Hamiltonian on *T*^{*}*Q*/*G*. Similarly, *β* = *π*^{*}*β*_{G}, where *β*_{G} is the reduced external force on *T*^{*}*Q*/*G*. Moreover, we can define the reduced Hamiltonian $H\u0303\mu $ on *T*^{*}(*Q*/*G*) by

for each $(\alpha \u0303q\u0303)\u2208Tq\u0303*(Q/G)$, where $q\u0303=[q]\u2208Q/G$ and $H\u0303=HG\u25e6\Psi \u22121$. Similarly, let $\beta \u0303=(\psi \u22121)*\beta G$ and

for each $(\alpha \u0303q\u0303)\u2208Tq\u0303*(Q/G)$.

Let *γ* be a *G*-invariant solution of the Hamilton–Jacobi problem for (*H*, *β*). Then, the following diagram commutes:

(reduction). *Let* *γ* *be a* *G**-invariant solution of the Hamilton–Jacobi problem for* (*H*, *β*)*. Let* $L=Im\gamma $ *and* $L\u0303=\Psi \u25e6\pi (L)$*. Then,* *γ* *induces a mapping* $\gamma \u0303\mu $ *such that* $Im\gamma \u0303\mu =L\u0303$ *and* $\gamma \u0303\mu $ *is a solution the Hamilton–Jacobi problem for* $(H\u0303\mu ,\beta \u0303\mu )$*.*

*H*+

*β*vanishes along $L$, clearly d

*H*

_{G}+

*β*

_{G}vanishes along $\pi (L)$. If $\alpha \u0303q\u0303\u2208L\u0303$, then $\psi \u22121(\alpha \u0303q\u0303)\u2208\pi (L)$,

*γ*is

*G*-invariant, it induces a mapping $\gamma \u0303:Q\u2192T*(Q/G)$, which is also

*G*-invariant. This mapping, in turn, induces a reduced solution $\gamma \u0303\mu :Q/G\u2192T*(Q/G)$ such that $\gamma \u0303=\pi G*\gamma \u0303\mu $ and $Im\gamma \u0303\mu =L\u0303$.□

*Let*$L\u0303$

*be a Lagrangian submanifold of*(

*T*

^{*}(

*Q*/

*G*),

*ω*

_{Q/G}+

*B*

_{μ})

*for some*$\mu \u2208g*$,

*which is a fixed point of the coadjoint action. Assume that*$L\u0303=Im\gamma \u0303\mu $

*, where*$\gamma \u0303\mu $

*is a solution of the Hamilton–Jacobi problem for*$(H\u0303\mu ,\beta \u0303\mu )$

*. Let*

*and take*

*Then,*

$L$

*is a**G**-invariant Lagrangian submanifold of**T*^{*}*Q**and*$L=Im\gamma $

*, where**γ**is a solution of the Hamilton–Jacobi problem for*(*H*,*β*)*.*

*G*-invariant mapping

*G*-invariant mapping

*γ*:

*Q*→

*T*

^{*}

*Q*. Let $\alpha \u0303q\u0303\u2208L\u0303$, so $(\alpha \u0303q\u0303,[\mu ]q\u0303)\u2208T*(Q/G)\xd7Q/Gg\u0303*$ and

*T*

^{*}

*Q*. The momentum map is

*q*,

*p*) and the natural projection

*h*,

*r*) if and only if

*γ*

_{λ}of the Hamilton–Jacobi for $(H,R\u0303)$, given by

## VI. ČAPLYGIN SYSTEMS

A nonholonomic mechanical system is given by a Lagrangian function $L=LqA,q\u0307A$ subject to a family of constraint functions,

For the sake of simplicity, we shall assume that the constraints Φ^{i} are linear in the velocities, i.e., $\Phi iqA,q\u0307A=\Phi Ai(q)q\u0307A$. Then, the nonholonomic equations of motion are

where $\lambda i=\lambda iqA,q\u0307A,1\u2264i\u2264m$, are Lagrange multipliers to be determined.

Geometrically, the constraints are given by a vector sub-bundle *M* of *TQ* locally defined by Φ^{i} = 0. The dynamical equations can then be rewritten intrinsically as

Under certain compatibility conditions, the vector field *X* is unique and it is denoted by *X*_{nh}.

A *Čaplygin system* (also spelled as Chaplygin) is a nonholonomic mechanical system such that we have the following:

The configuration manifold

*Q*is a fibered manifold, say,*ρ*:*Q*→*N*, over a manifold*N*.The constraints are provided by the horizontal distribution of an Ehresmann connection Γ in

*ρ*.The Lagrangian $L:TQ\u2192R$ is Γ-invariant.

A particular case is when *ρ*: *Q* → *N* = *Q*/*G* is a principal *G*-bundle and Γ is a principal connection. As a matter of fact, some Refs. 42, 44, 45, and 57 restrict their definition of the *Čaplygin system* to this particular case. Our more general definition is also considered in Refs. 9 and 58.

Let us recall that the connection Γ induces a Whitney decomposition $TQ=H\u2295V\rho $, where $H$ is the horizontal distribution and *Vρ* = ker *Tρ* is the vertical distribution. Take fibered coordinates $qA=qa,qi$ such that $\rho qa,qi=qa$.

With a slight abuse of notation, let $hor:TQ\u2192H$ denote the horizontal projector hereinafter and let $\Gamma ai=\Gamma aiqA$ denote the Christoffel components of the connection Γ. Let us recall that Γ may be considered as a (1, 1)-type tensor field on *Q* with Γ^{2} = id, so hor = hor^{2} = (1/2)(id + Γ). The curvature of Γ is the (1, 2)-tensor field $R=12[hor,hor]$, where [hor, hor] is the Nijenhuis tensor of hor,^{47} that is,

for each pair of vector fields *X* and *Y* on *N*. Locally,

where

The constraints are given by

that is, the solutions are horizontal curves with respect to Γ.

Since the Lagrangian is Γ-invariant,

for all *Y* ∈ *T*_{y}*N*, *y* = *ρ*(*q*_{1}) = *ρ*(*q*_{2}), where $YH$ denotes the horizontal lift of *Y* to *Q*. We can then introduce a function *ℓ* on *TN* such that

so locally we have

Now let *ρ*(*q*) = *y* and $x\u2208H$ with *τ*_{Q}(*x*) = *q*; let *u* ∈ *T*_{y}*N*, *U* ∈ *T*_{u}(*TN*) and *X* ∈ *T*_{x}(*TQ*) such that *X* projects onto

We can then introduce a 1-form α on *TN* such that

where *θ*_{L} is the Poincaré-Cartan 1-form associated with *L*, given by Eq. (8). In other words, α is locally given by

It can be shown that *ℓ* is a regular Lagrangian and that the Čaplygin system is equivalent to the forced Lagrangian system (*ℓ*, α).

Assume that *ℓ* is hyper-regular. Then, the Čaplygin system has an associated forced Hamiltonian system (*h*, β), with

and

In particular, if *L* is natural, we have

The Hamilton–Jacobi equation for (*h*, β) is thus locally

In particular, if *L* is purely kinetical,

Let *D* denote a distribution on *Q* whose annihilator is

Then, we can form the algebraic ideal $I(D0)$ in the algebra Λ^{*}(*Q*), namely, if a *k*-form $\nu \u2208I(D0)$, then

where β_{i} ∈ Λ^{k−1}(*q*) and 1 ≤ *i* ≤ *m*.

[Ref. 9 (Theorem 4.3)]. *Let* $H$ *denote the horizontal distribution defined by the connection* Γ *in* *ρ*: *Q* → *N**. Let* *X* *be vector field on* *Q* *such that* $X(Q)\u2282H$ *and* $d(Leg\u25e6X)\u2208I(H0)$*. Then, the following conditions are equivalent:*

*For every curve*$\sigma :R\u2192Q$*such that*$\sigma \u0307(t)=T\tau Q\u25e6Xnh\u25e6X\u25e6\sigma (t)$*for all**t**, then**X*◦*σ**is an integral curve of**X*_{nh}*.*$dEL\u25e6X\u2208H0$

*.**A vector field**X**satisfying these conditions is called a solution of the nonholonomic Hamilton–Jacobi problem for*(*L*, Γ)*.*

[Ref. 9 (Theorem 4.5)]. *Assume that a vector field* *X* *on* *Q* *is a solution for the nonholonomic Hamilton–Jacobi problem for* (*L*, Γ)*. If* *X* *is* *ρ**-projectable to a vector field* *Y* *on* *N* *and* $\gamma =Leg\u2113*Y$ *is closed, then* *Y* *is a solution of the Lagrangian Hamilton–Jacobi problem for* (*ℓ*, α) *and* *γ* *is a solution of the Hamilton–Jacobi problem for* (*h*, β)*.*

*Conversely, let*

*γ*

*be a solution of the Hamilton–Jacobi problem for*(

*h*, β)

*. Then,*

*Y*= Leg

^{−1}◦

*γ*

*is a solution of the Lagrangian Hamilton–Jacobi problem for*(

*ℓ*, α)

*. If*

*then the horizontal lift*$YH$

*is a solution for the nonholonomic Hamilton–Jacobi problem for*(

*L*, Γ)

*.*

(mobile robot with fixed orientation). Consider the motion of a robot whose body maintains a fixed orientation with respect to the environment. The robot has three wheels with radius *R*, which turn simultaneously about independent axes and perform a rolling without sliding over a horizontal floor (see Refs. 9, 42, and 59 for more details).

*x*,

*y*) denote the position of the center of mass, and let

*θ*and

*ψ*denote the steering and rotation angles of the wheels, respectively. Hence, the configuration manifold is $Q=S1\xd7S1\xd7R2$, and the Lagrangian of the system is

*m*is the mass,

*J*is the moment of inertia, and

*J*

_{ω}is the axial moment of inertia of the robot.

*Q*by translations, namely,

*G*-bundle

*ρ*:

*Q*⟶

*N*=

*Q*/

*G*with a principal connection given by the $g$-valued 1-form,

*G*and $e1,e2$ is the canonical basis of $R2$ (identified with $g$).

*ℓ*, α) on

*TN*is given by

*ℓ*, α) is given by

*L*, Γ).

## VII. CONCLUSIONS AND OUTLOOK

In this paper, we have obtained a Hamilton–Jacobi theory for Hamiltonian and Lagrangian systems with external forces. We have discussed the complete solutions of the Hamilton–Jacobi problem. Our results have been particularized for forces of Rayleigh type. We have presented a dissipative bracket for Rayleigh systems. Furthermore, we have studied the reduction and reconstruction of the Hamilton–Jacobi problem for forced Hamiltonian systems with symmetry. Additionally, we have shown how the Hamilton–Jacobi problem for a Čaplygin system can be reduced to the Hamilton–Jacobi problem for a forced Lagrangain system in order to obtain solutions of the latter and reconstruct solutions of the former.

In a previous paper, we studied the symmetries, conserved quantities, and reduction of forced mechanical systems (see Ref. 39, see also Ref. 40). Making use of results from this paper, one can obtain the constants of the motion in involution of a forced system and relate them with complete solutions of the Hamilton–Jacobi problem for that system (see Example 3). Furthermore, Lemma 15 from Ref. 39 has been translated to the Hamiltonian formalism (see Lemma 12) in order to extend the method of reduction of the Hamilton–Jacobi problem^{55} for forced Hamiltonian systems.

In another paper,^{60} we develop a Hamilton–Jacobi theory for forced discrete Hamiltonian systems. Our approach is based on the construction of a discrete flow on *Q* × *Q* (unlike the case without external forces,^{61} where the discrete flow is defined on *Q*). We define a discrete Rayleigh potential. Additionally, we present some simulations and analyze their numerical accuracy.

An additional open problem is the particularization of the results from this paper when the configuration space *Q* is a Lie group *G* with the Lie algebra $g$. If (*L*, *α*) is a *G*-invariant forced Lagrangian system on *TG*, the forced Euler–Lagrange equations for (*L*, *α*) are reduced to the Euler–Poincaré equations with forcing on $g$ (see Ref. 46). We also plan to extend our results on forced systems to the Lie algebroid framework in order to use Atiyah algebroids when the system enjoys symmetries. Furthermore, we plan to extend the results from this paper for time-dependent forced Lagrangian systems in the framework of cosymplectic geometry (see Ref. 62). Additionally, the applications of the dissipative bracket (17) will be studied elsewhere.

## ACKNOWLEDGMENTS

We thank the referee for his/her constructive comments. The authors acknowledge financial support from the Spanish Ministry of Science and Innovation (MICINN) under Grant No. PID2019-106715GB-C21 and “Severo Ochoa Programme for Centres of Excellence in R&D” (Grant No. CEX2019-000904-S). Manuel Lainz wishes to thank MICINN and the Institute of Mathematical Sciences (ICMAT) for the FPI-Severo Ochoa predoctoral contract (No. PRE2018-083203). Asier López-Gordón would like to thank MICINN and ICMAT for the predoctoral contract (No. PRE2020-093814). He is also grateful for enlightening discussions on fiber bundles with his friend and colleague Alejandro Pérez-González.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.