EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6091
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Hamilton-Jacobi Framework for Nonholonomic
Dynamics

K. I. Nawafleh

Physics Department, Mu’tah University, Al-Karak, Jordan

Abstract. We derive generalized Hamilton–Jacobi equations for dynamical systems subject to
nonholonomic constraints. The geometric formulation of Hamilton-Jacobi theory for nonholonomic
constraints is developed, following the ideas of the authors in previous papers. It is shown that the
equations of motion which follow from the principle of d’Alembert are identical to the equations
which follow from the variational action principle. To illustrate the effectiveness of the proposed
framework, we present and analyze two illustrative examples: the motion of a rolling disk on a
horizontal plane and the motion of a knife edge on an inclined plane.

2020 Mathematics Subject Classifications: 70H20, 70F25, 49S05

Key Words and Phrases: Lagrange-d’Alembert Principle, Nonholonomic Constraints

1. Introduction

The theory of systems with nonholonomic constrains goes back to the 19th century [1–
4] Nonholonomic constraints arise naturally in the context of mechanical systems with rigid
bodies rolling without slipping over a surface. Another typical example is the Chaplygin
sleigh. This is a rigid body where one of the contact points with the surface forms a knife
edge. The nonholonomic constraint assumed that there is no motion perpendicular to the
knife edge [5–10] or that the velocity of the contact point remains in the direction of x axis
of the body. The equations of constraints in nonholonomic dynamics, written in terms of
generalized coordinates qi and generalized velocities q̇i.

Typical engineering problems that involve such constraints arise for example in robotics,
where the wheels of a mobile robot are often required to roll without slipping, or if one
interested in guiding the motion of a cutting tool.

Ghori [11]developed a generalization of the classical Hamilton-Jacobi method in order
to cover mechanical systems subject to nonholonomic constraints. The method was based
on a modification of the classical Hamilton-Jacobi equation by supplementing a modifying
function. However, Nzaziev [12] proved that Ghori’s theorem is incorrect. Recently, the
authors [13, 14] introduced a new generalized Hamilton-Jacobi method for such systems.

DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6091

Email address: knawafleh@mutah.edu.jo (K. I. Nawafleh)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 2 of 15

This method used a variational principle for the integral of action in connection with the
introduction of unknown multipliers.

We will start from the actual solutions of the nonholonomic system, and apply a sort of
Hamilton-Jacobi theory to arrive to the generating function, subsequently we will derive
Hamilton’s equations of motion.

This method was governed by a variational principle applied to a certain function. The
resulting variational relation was then treated by introducing some unknown multipliers
in connection with constraint relations. After the elimination of these multipliers the
generalized momenta were found to be certain functions of the partial derivatives of the
Hamilton Jacobi function with respect to the generalized coordinates and the time. Then
the partial differential equation of the classical Hamilton-Jacobi method was modified by
inserting these functions for the generalized momenta in the Hamiltonian of the system.

Unlike previous methods that changed the classical Hamilton-Jacobi equation using
extra assumptions or added functions, this work develops the Hamilton-Jacobi approach
in a clear and organized way, starting directly from a variational principle. It includes
nonholonomic constraints by using Lagrange multipliers and shows that the resulting
equations match those from the well-known Lagrange-d’Alembert principle. The main
contribution of this work is applying this method to find exact solutions and Hamilton-
Jacobi functions for real-world systems, making it easier and more understandable to solve
problems involving constrained motion.

2. Formalism

Hamiltonian approaches to nonholonomic mechanical systems are developed by, for
example [3,4,6,8]. The standard way to derive the equations of motion of a nonholonomic
mechanical systems starts from Lagrange’s equations and the addition of some unknown
force that have to be exerted by the constraint in order to satisfy the nonholonomic
constraint.

Consider a Lagrangian function

L = L(qi, q̇i, t) (1)

along with a set of nonholonomic constraints that are linear functions of the velocities:

fα(qi, q̇i) = 0, α = n+ 1, n+ 2, . . . , n+m (2)

The generalized momenta and the Hamiltonian of the system are defined by the relations:

pi =
∂L

∂q̇i
(3)

H(qi, pi, t) = piq̇i − L(qi, q̇i, t) (4)

When the system is nonholonomic, Hamilton’s variational principle has to be expressed
in the form [14]

δ

∫
Ldt = 0. (5)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 3 of 15

The Lagrangian formulation of these theories requires the configuration space formed by
n generalized coordinates qi and n generalized velocities q̇i.

According to the usual approach of the calculus of variations, it is found that the
generalized Lagrange equations for such nonholonomic systems are given by [15, 16] by
using the Hamilton’s principle:

d

dt

(
∂L

∂q̇i

)
− ∂L

∂qi
= λα

∂fα
∂q̇i

, (6)

these dynamic nonholonomic equations of motion are known as Lagrange-d’Alembert prin-
ciple and are the correct equations to describe the time evolution a mechanical system
under velocity dependent constraints. Where λ is Lagrange multiplier, to be determined
that representing the force of constraint.

This equation represents a system of n +m differential equations that can be solved
for the n+m unknown functions qi(t), λi(t).

We will consider a Lagrangian system with nonholonomic constraints whose motion
obeys the Lagrange-d’Alembert principle in the case of completely integrable constraints.

The standard formulation of the Hamilton–Jacobi problem for a Hamiltonian system
is based on the generating function S(q, t) which is called the Hamilton-Jacobi function:

The set of the Hamilton-Jacobi partial differential equation can be written in compact
form as [17–24]:

∂S

∂t
+H(pi, qi, t) = 0 . (7)

We find the solution of constraint equations by integration whose solution has the following
form:

qi = qi(α1, α2, . . . , αn−m, β1, β2, . . . , βn, t) . (8)

Where n is dimensional configuration space, subject to m nonholonomic constraints. m <
n− 1

We may rewrite the above equations as:

Xi(α1, α2, . . . , αn−m, qi, t) = βi . (9)

Where αi and βi are initial constants of motion respectively related to the initial velocity
and position of the particle.

The central issue in the procedure is to assume the existence of Hamilton’s principal
function from which the relations are obtained [13-14]

We can set:
∂S

∂αi
= βi . (10)

From the above equations we can find the Hamilton-Jacobi function, we will see that the
separation of variables is done in the following simple manner:

S(q1, q2, . . . , qn;α1, α2, . . . , αn, t) = S1(q1, α1) + S2(q2, α2) + · · ·+ Sn(qn, αn). (11)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 4 of 15

The generalized momenta are determined by:

pi =
∂S

∂qi
. (12)

We further have:
α = α(qi, pi, t). (13)

Where we substitute α by their expression in terms of qi and pi.
The Hamiltonian is obtained as:

H = −∂S

∂t
. (14)

Following [7], the Hamilton’s equations of motion are given by [14-16]:

q̇i =
∂H

∂pi
; ṗi = −∂H

∂qi
+ λα

∂fα
∂q̇i

(15)

This represents the equations of motion for nonholonmic constraints. It is worth to mention
that the advantage of this method is that, in spite of the difficulties to solve a partial
differential equation instead of an ordinary differential one, in many cases it works, being
an extremely useful tool, indeed, in these cases the method provides an immediate way to
integrate the equations of motion.

To demonstrate how the proposed scheme works, we will consider two examples.

3. Examples

In order to illustrate the general discussion we will examine two examples subject to
nonholonmic constrains.

Example 1

As a first example of nonholonomic constraints consider the motion of a flat uniform
disk rolls upright without slipping on horizontal plane; we assume that the mass, the
moments of inertia and the radius of the disk are all equal to unity.

The Lagrangian describing the system is

L =
1

2

(
ẋ2 + ẏ2 + θ̇2 + φ̇2

)
(16)

Where θ is the coordinate associated to rolling, φ to pivoting, x and y to point of contact
of the disk with the horizontal plane.

The system has two nonholonomic constraints:

f1 = ẋ− θ̇ cosφ = 0 (17)

f2 = ẏ − θ̇ sinφ = 0 (18)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 5 of 15

The constraint equations (17) and (18) can be rewritten as

ẋ = θ̇ cosφ, (19)

ẏ = θ̇ sinφ. (20)

First we derive the equations of motion as a result of varying the coordinates of the
extended phase space including the multipliers qi and λα according to equation (6), we
obtain

d

dt

(
∂L

∂ẋ

)
− ∂L

∂x
= λ1

∂f1
∂ẋ

, (21)

d

dt

(
∂L

∂ẏ

)
− ∂L

∂y
= λ2

∂f2
∂ẏ

, (22)

d

dt

(
∂L

∂θ̇

)
− ∂L

∂θ
= λ1

∂f1

∂θ̇
+ λ2

∂f2

∂θ̇
, (23)

d

dt

(
∂L

∂φ̇

)
− ∂L

∂φ
= 0 . (24)

These give:

ẍ = λ1 , (25)

ÿ = λ2 , (26)

θ̈ = −λ1 cosφ− λ2 sinφ, (27)

φ̈ = 0. (28)

According to equations (25) and (26), equation (27) can be expressed as:

θ̈ = −ẍ cosφ− ÿ sinφ. (29)

Differentiating constraint functions in equations (19) and (20) with respect to time, we
find

ẍ = θ̈ cosφ− φ̇θ̇ sinφ, (30)

ÿ = θ̈ sinφ+ φ̇θ̇ cosφ, (31)

By inserting equations (30) and (31) in equation (29), we arrive at the following equation

θ̈ = 0. (32)

So we obtain the constrained equations:

θ̈ = 0, φ̈ = 0, ẋ = cosφθ̇, ẏ = sinφθ̇. (33)

This is a mixed set of first and second-order differential equations. From these equations,
we can solve θ(t), φ(t), x(t), y(t) as follows:



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 6 of 15

The first two equations can be easily integrated to give:

θ(t) = α1t+ β1, (34)

φ(t) = α2t+ β2. (35)

Differentiating equations (34) and (35) with respect to time, we get

θ̇ = α1, (36)

φ̇ = α2. (37)

Substituting equations (35) and (36) into equation (19) and integrating the result equation,
we obtain ∫

ẋdt =

∫
α1 cos(α2t+ β2)dt, (38)

and this gives:
x(t) = α3 sinφ+ β3, (39)

where
α3 =

α1

α2
.

Also substituting equations (35), (36) into equation (20) and integrating the result equa-
tion, we have: ∫

ẏdt =

∫
α1 sin(α2t+ β2)dt, (40)

which leads to
y(t) = α4 cosφ+ β4 , (41)

where
α4 = −α1

α2
.

Here β1, β2, β3 and β4 are constants of integration related to the initial values of θ, φ, x, y
respectively while α1, α2, α3 and α4 are the initial values of velocities.

The essential point of our procedure for obtaining the Hamiltonian formalism for non-
holonomic systems is to recognize that the above equations (34), (35), (39), (41) can be
obtained in the Hamilton-Jacobi formalism. We may rewrite these equations respectively
as:

β1 = θ − α1t =
∂S1

∂α1
, (42)

β2 = φ− α2t =
∂S2

∂α2
, (43)

β3 = x− α3 sinφ =
∂S3

∂α3
, (44)

β4 = y − α4 cosφ =
∂S4

∂α4
. (45)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 7 of 15

Solving equations (42-45) simultaneously, we obtain:

S1(θ, α1) = θα1 −
1

2
α2
1t , (46)

S2(φ, α2) = φα2 −
1

2
α2
2t , (47)

S3(x, α3) = xα3 −
1

2
α2
3 sinφ , (48)

S4(y, α4) = yα4 −
1

2
α2
4 cosφ. (49)

We saw that it is possible to separate the variables in the Hamilton-Jacobi equations.
With these results, the Hamilton-Jacobi function S becomes

S = θα1 + φα2 + xα3 + yα4 −
1

2
t(α2

1 + α2
2)−

1

2
α2
3 sinφ− 1

2
α2
4 cosφ. (50)

We interpret S(θ, φ, x, y, α1, α2, α3, α4, t) as the solution of Hamilton-Jacobi equation. The
generalized momenta are derived as

pθ =
∂S

∂θ
= α1 , (51)

pφ =
∂S

∂φ
= α2 −

1

2
α2
3 cosφ+

1

2
α2
4 sinφ, (52)

px =
∂S

∂x
= α3 , (53)

py =
∂S

∂y
= α4 . (54)

From the above equations we can obtain α1, α2, α3, α4 as functions of pi and qi:

α1 = pθ , (55)

α2 = pφ +
1

2
α2
3 cosφ− 1

2
α2
4 sinφ , (56)

α3 = px , (57)

α4 = py . (58)

Following the equation (14), the Hamiltonian is defined as:

H = −∂S

∂t
=

1

2
(α2

1 + α2
2) . (59)

Substituting equation (55) and (56) into equation (59), this leads the following expression
for the Hamiltonian:

H =
1

2

[
P 2
θ +

(
pφ +

1

2
P 2
x cosφ− 1

2
p2y sinφ

)2
]
. (60)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 8 of 15

The Hamilton’s equations of motion can be found from equation (15). So the generalized
velocities corresponding to this Hamiltonian are:

θ̇ =
∂H

∂pθ
= Pθ , (61)

φ̇ =
∂H

∂pφ
=

(
pφ +

1

2
P 2
x cosφ− 1

2
p2y sinφ

)
, (62)

ẋ =
∂H

∂px
=

(
pφ +

1

2
P 2
x cosφ− 1

2
p2y sinφ

)
(Px cosφ) , (63)

ẏ =
∂H

∂py
=

(
pφ +

1

2
P 2
x cosφ− 1

2
p2y sinφ

)
(−py sinφ). (64)

Substituting equations (52–54) into equation (62), we find

φ̇ =

(
α2 −

1

2
α2
3 cosφ+

1

2
α2
4 sinφ+

1

2
α2
3 cosφ− 1

2
α2
4 sinφ

)
,

or,
φ̇ = α2 . (65)

Substituting equations (52–54) into equation (63), and using α2 for the bracket term (from
eq. 65) and Px = α3 (from eq. 57, assuming Px = px):

ẋ = α2α3 cosφ.

Using the definition α3 = α1/α2 (from the context of eq. 39), this simplifies to:

ẋ = α1 cosφ . (66)

According to equation (36), equation (66) can be finally written as follows:

ẋ = θ̇ cosφ. (67)

Substituting equations (52–54) into equation (64), we get:

ẏ = α2(−α4 sinφ) = −α2α4 sinφ,

or, using α4 = −α1/α2 (from the context of eq. 41):

ẏ = α1 sinφ . (68)

According to equation (36), equation (68) can be finally written as follows:

ẏ = θ̇ sinφ. (69)

Using equation (15), this yields

ṗθ = −∂H

∂θ
+ λ1

∂f1

∂θ̇
+ λ2

∂f2

∂θ̇
, (70)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 9 of 15

ṗφ = −∂H

∂φ
+ λ1

∂f1
∂φ̇

+ λ2
∂f2
∂φ̇

, (71)

ṗx = −∂H

∂x
+ λ1

∂f1
∂ẋ

, (72)

ṗy = −∂H

∂y
+ λ2

∂f2
∂ẏ

. (73)

These give

ṗθ = −λ1 cosφ− λ2 sinφ , (74)

ṗφ = 0 , (75)

ṗx = λ1 , (76)

ṗy = λ2 . (77)

These equations are similar to equations (27), (28), (25) and (26) respectively, so that:

ṗθ = θ̈ , (78)

ṗφ = φ̈ , (79)

ṗx = ẍ , (80)

ṗy = ÿ . (81)

We see that the equations of motion that obtained by the Hamilton-Jacobi methods are
equivalent to those which obtained by the Lagrange-d’Alembert principle.

Example 2

As a second example for nonholonomic constraints consider the knife edge on an in-
clined plane. The plane corresponds physically to a blade moving in the xy-plane at an
angle φ to the x-axis.

The Lagrangian of the system is:

L =
1

2
m(ẋ2 + ẏ2) +

1

2
Jφ̇2, (82)

Where J is the moment of inertia of the blade about a vertical axis through the point of
contact. We assume that the mass and the moments of inertia J are equal to unity. Thus
the Lagrangian of the system is given by:

L =
1

2
(ẋ2 + ẏ2 + φ̇2) . (83)

The system has the following nonholonomic constraint:

f = ẋ sinφ− ẏ cosφ = 0 , (84)

This constraint can be rewritten as

ẏ = ẋ tanφ. (85)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 10 of 15

Following equation (2.19) (Note: this refers to an internal numbering, likely eq. 15 in this
paper, or a general formalism section), the equations of motion can be obtained as

d

dt

(
∂L

∂ẋ

)
− ∂L

∂x
= λ

∂f

∂ẋ
, (86)

d

dt

(
∂L

∂ẏ

)
− ∂L

∂y
= λ

∂f

∂ẏ
, (87)

d

dt

(
∂L

∂φ̇

)
− ∂L

∂φ
= λ

∂f

∂φ̇
. (88)

These give:

ẍ = λ sinφ, (89)

ÿ = −λ cosφ, (90)

φ̈ = 0. (91)

From equations (89) and (90), we obtain

ÿ = − ẍ

tanφ
, (92)

Differentiating constraint equation (85) with respect to time to eliminate λ, we find

ÿ = ẍ tanφ+ φ̇ẋ sec2 φ , (93)

Inserting equation (92) in (93) and multiplying the result by (− tanφ) leads to:

ẍ = −ẍ tan2 φ− φ̇ẋ sec2 φ tanφ, (94)

By using the identity 1 + tan2 φ = sec2 φ, equation (94) reduces to:

ẍ = −φ̇ẋ tanφ. (95)

From which we obtain the constrained equations:

φ̈ = 0, ẍ = −φ̇ẋ tanφ, ẏ = ẋ tanφ. (96)

Now we can solve φ(t), x(t), y(t) as follows:
The first equation can be easily integrated to give:

φ(t) = α1t+ β1. (97)

Differentiating equation (97) with respect to time, we obtain

φ̇ = α1. (98)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 11 of 15

Now integrate equation (95), we get∫
ẍ

ẋ
dt =

∫
(− tanφ)dφ, φ̇ =

dφ

dt
. (99)

This gives:
ln ẋ = ln cosφ ,

or,
ẋ = cosφ. (100)

Substituting equation (97) into equation (100) and integrating the result equation:∫
ẋdt =

∫
cos(α1t+ β1)dt , (101)

Leads to
x(t) = α2 sinφ+ β2 , (102)

where

α2 =
1

α1
.

Now substituting equation (100) in constraint equation (85) we get:

ẏ = sinφ, (103)

Inserting equation (97) into equation (103) and integrating the result equation∫
ẏdt =

∫
sin(α1t+ β1)dt , (104)

we find
y(t) = α3 cosφ+ β3, (105)

where

α3 = − 1

α1
.

Again β1, β2 and β3 are constant of integration related to the initial values of φ, x, y while
α1, α2 and α3 are the initial values of velocities.

Following the same steps as in the preceding example we may rewrite equations (97),
(102) and (105) respectively as:

β1 = φ− α1t =
∂S1

∂α1
, (106)

β2 = x− α2 sinφ =
∂S2

∂α2
, (107)

β3 = y − α3 cosφ =
∂S3

∂α3
. (108)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 12 of 15

Solving equations (106–108) simultaneously, we obtain:

S1(φ, α1) = φα1 −
1

2
α2
1t, (109)

S2(x, α2) = xα2 −
1

2
α2
2 sinφ, (110)

S3(y, α3) = yα3 −
1

2
α2
3 cosφ. (111)

Following equation (11), we finally collect expressions from equations (109–111), which
give the Hamilton-Jacobi function S(φ, x, y, α1, α2, α3, t):

S = φα1 + xα2 + yα3 −
1

2
α2
1t−

1

2
α2
2 sinφ− 1

2
α2
3 cosφ . (112)

The generalized momenta can be derived as

pφ =
∂S

∂φ
= α1 −

1

2
α2
2 cosφ+

1

2
α2
3 sinφ, (113)

px =
∂S

∂x
= α2 , (114)

py =
∂S

∂y
= α3. (115)

From the above equations we can obtain α1, α2, α3 as functions of pi and qi.

α1 = pφ +
1

2
α2
2 cosφ− 1

2
α2
3 sinφ , (116)

α2 = px , (117)

α3 = py . (118)

Making use the definition of equation (14), the Hamiltonian is defined as:

H = −∂S

∂t
=

1

2
α2
1 , (119)

Inserting equation (116) into equation (119) we get the following expression for the Hamil-
tonian:

H =
1

2

[
pφ +

1

2
(P 2

x cosφ− p2y sinφ)

]2
. (120)

The corresponding Hamilton’s equations of motion can be found from equation (15). So
the generalized velocities corresponding to this Hamiltonian are:

φ̇ =
∂H

∂pφ
=

[
pφ +

1

2
(P 2

x cosφ− p2y sinφ)

]
, (121)

ẋ =
∂H

∂px
=

[
pφ +

1

2
(P 2

x cosφ− p2y sinφ)

]
(Px cosφ), (122)



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 13 of 15

ẏ =
∂H

∂py
=

[
pφ +

1

2
(P 2

x cosφ− p2y sinφ)

]
(−py sinφ) . (123)

Substituting equations (113–115) into equation (121), we find

φ̇ =

[
α1 −

1

2
α2
2 cosφ+

1

2
α2
3 sinφ+

1

2
α2
2 cosφ− 1

2
α2
3 sinφ

]
,

or,
φ̇ = α1 . (124)

Inserting equations (113–115) into equation (122), and using Px = px = α2:

ẋ = α1α2 cosφ

or, using α2 = 1/α1 (from context of eq. 102):

ẋ = cosφ. (125)

Substituting equations (113–115) into equation (123), and using py = α3:

ẏ = α1(−α3 sinφ) = −α1α3 sinφ,

or, using α3 = −1/α1 (from context of eq. 105):

ẏ = sinφ. (126)

Using equation (15), this yields:

ṗφ = −∂H

∂φ
+ λ

∂f

∂φ̇
, (127)

ṗx = −∂H

∂x
+ λ

∂f

∂ẋ
, (128)

ṗy = −∂H

∂y
+ λ

∂f

∂ẏ
. (129)

These give

ṗφ = 0, (130)

ṗx = λ sinφ, (131)

ṗy = −λ cosφ. (132)

These equations similar to equations (91), (89) and (90) respectively so:

ṗφ = φ̈ , (133)

ṗx = ẍ , (134)

ṗy = ÿ . (135)

According to the previous discussion we see that the equations of motion that obtained by
the Hamilton-Jacobi methods are equivalent to those obtained by the Lagrange-d’Alembert
principle.



K. I. Nawafleh / Eur. J. Pure Appl. Math, 18 (3) (2025), 6091 14 of 15

4. Conclusion

This work shedding further light on constrained systems of type nonholonomic con-
straints, especially on determining the Hamilton-Jacobi function S. Finding S enables us
to get the solutions of the equations of motion. These solutions are obtained in terms of
the time and the coordinates that correspond to dependent momenta; these are treated as
independent variables, just like the time t. This is followed by determining a new method
to integrate the dynamical equations in particular, nonholonomic systems. We obtain the
equations of motion for a Lagrangian system with nonholonomic constraints making use
of the d’Alembert principle. It is shown that the results obtained using the HamiltonJa-
cobi method are equivalent to those obtained by the Lagrange-d’Alembert principle. The
method is applied for two examples: the first one is the motion of a flat uniform disk rolls
upright without slipping on horizontal plane, and the second one is the motion of a knife
edge on an inclined plane.

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