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Lecture 7. Dynamics of the particle. General theorems 1
Theoretical Mechanics
9. Dynamics of the particle. General theorems.
The differential equations of the motion
Consider the material point 𝑀 𝑚 moving in the Cartesian frame 𝑂𝑥𝑦𝑧 under
the action of the force 𝐹 = 𝐹 𝑟 , 𝑣 , 𝑡 where 𝐹 𝑋, 𝑌, 𝑍 .
The differential equation of the motion is
(9.1)
along with the initial conditions
(9.2)
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Theoretical Mechanics
In the Cartesian frame 𝑂𝑥 𝑦𝑧 one have
(9.3)
(9.4)
(9.5)
where
A first integral of the differential system of motion (9-3) – (9.4) is a (non-
constant) continuously-differentiable function (class 𝐶1)
which reduces to a constant when the functions (9.6)
satisfy (9.3).
Lecture 7. Dynamics of the particle. General theorems
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
Thus, the problem is reduced to the integration of a system of equations (9.3) with
only 6 − k unknowns (hence, a smaller number of unknowns). If k = 6 , then all the
first integrals are independent, so that the system (9.5) of first integrals determines
all the unknown functions. We notice that for k > 6 the first integrals (9.5) are no
more independent; we may thus set up at the most six independent first integrals.
P.P. Teodorescu, Mechanical Systems, Classical Models. Vol. I: Particle Mechanics, Springer, 2007
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Theoretical Mechanics
(9.8)
(9.7)
From (9.1) we have
Lecture 7. Dynamics of the particle. General theorems
Momentum of a particle
Newton has introduced the notion of momentum (which he called quantity of
motion) representing the product of the mass by the velocity of a point.
Equation (9.8) we have is called the momentum equation. From (9.8) we can
formulate the theorem of momentum:
Another usual notation for momentum is:
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
https://sites.ualberta.ca/~ygu/courses/phys144/notes/momentum1.pdf
Example
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
Example
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
First integrals
• If F = 0 then 𝐻 = 𝑐𝑜𝑛𝑠𝑡, ∀ 𝑡 ≥ 𝑡0. This first integral is the law of momentum
conservation.
Taking into account the initial conditions we obtain:
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Theoretical Mechanics
The motion of the particle P is thus rectilinear and uniform. Besides, this result
corresponds to the principle of inertia, which appears thus as a particular case of
the principle of action of forces (lex secunda).
Lecture 7. Dynamics of the particle. General theorems
• If F ≠ 0, ∀ 𝑡 ≥ 𝑡0 and exist a fixed direction 𝒖 = 𝛼, 𝛽, 𝛾 such that F ∙ 𝑢 = 0,
then using this in (9.1) one have
Thus, (9.9)
is a first integral of the differential equations of motion.
Hence, if the force F is parallel to a fixed plane, then the projection of the velocity of
the free particle P on the normal to this plane is conserved (is constant) in time
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
Example
Moment of momentum (angular momentum)
The moment of the momentum with respect to the pole O of the frame of reference
is called moment of momentum (angular momentum) of the particle, with respect to
this pole, and is given by
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Theoretical Mechanics
(9.11)
(9.10)
Lecture 7. Dynamics of the particle. General theorems
From the equation of motion (9.1) we have
or (9.12)
Momentum of force
F with respect to
the pole O:
𝑀𝑂 𝐹 = 𝑟 × 𝐹
Theorem (theorem of angular momentum) The derivative with respect to time of
the moment of momentum of a particle, with respect to a fixed pole, is equal to the
moment of the force which act upon it, with respect to the same pole.
On projection on the 𝑂𝑥𝑦𝑧 axes:
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
(9.13)
First integrals
• If M𝑂 F = 0 we deduce the law of the angular momentum conservation.
(9.14)
Thus,
(9.15)
Equation (9.15) is equivalent with
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
(9.17)
From (9.15) the following first integrals are obtained
(9.16)
where is the areal velocity of the particle.
In the particular case when 𝐹 ∥ 𝑟 (in this case 𝐹 is called central force) we have
Notice that:
i) the motion is rectilinear if
ii) the motion takes place in a plane if
⇒ 𝑟 ∈ 𝑣 𝑂, 𝑟 𝑂
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
(9.18)
Thus, we obtain the first integral
• If M𝑂 F ≠ 0, ∀ 𝑡 ≥ 𝑡0 and exist a fixed direction 𝒖 = 𝛼, 𝛽, 𝛾 such that
M𝑂 F ∙ 𝑢 = 0, then
Work
Consider the force 𝐹 acting on the point
P (𝑟 = 𝑂𝑃) which effects a real
displacement d𝑟 . The elementary work of
the force 𝐹 is the scalar
(9.19)
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
(9.20)
or
Next, we suppose F depends only on r ( 𝐹 = 𝐹 𝑥, 𝑦, 𝑧 ) and exist a function
U = U 𝑥, 𝑦, 𝑧 with the property:
Theorem. The force 𝐹 is potential (conservative) if and only if
(9.21)
Function 𝑉 ≔ −𝑈 is the potential energy and we say that 𝐹 is conservative.
(9.22)
This is equivalent with the fact that
(9.23)
is an exact differential.
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
A force field F:𝐷 ⊂ 𝑅3 → 𝑅3 is a conservative force field if ∃U: D → R with the
property:
Theorem. Consider 𝐷 ⊂ 𝑅3 a simply connected domain.
A force field 𝐹 : 𝐷 → 𝑅3 is potential (conservative) if and only if one
of the following equivalent conditions takes place:
∀ 𝐶, 𝐶′ ∈ 𝐷 arbitrary curved from A to B.
∀ 𝛾 closed curve
•
•
•
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
• the differential form
is an exact differential, i.e.
Example
is a potential (conservative)
force
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
Force function in the plane
Consider
such that
Thus, such that
We integrate (9.25)1 with respect to x
(9.24)
(9.25)
(9.26)
where 𝜙 𝑦 is determined from (9.25)2:
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
Taking into account (9.24) one obtain
or (9.27)
Using (9.26) and (9.27) we have
(9.28)
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
Next, we suppose that the force field F = 𝐹 𝑟 = 𝐹 𝑥, 𝑦, 𝑧 is conservative
(potential), i,.e ∃U: D → R with the property:
or
Thus, the elementary work is an exact differential
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
Total work along an arc AB of a curve
Consider AB an arc of a curve defined by the equations:
(9.29)
The curvilinear integral
is the total work of the force 𝐹 along the arc AB.
(9.30)
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
If the force is conservative, , , then
(9.31)
and the work is independent of the path from A to B.
If the force is not conservative we use:
(9.32)
and if the force is a function of velocity and time we can use:
(9.33)
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
Kinetic energy
The scalar quantity
(9.34)
Equation (9.35) expresses the following theorem
(9.35)
is the kinetic energy of the particle P ; this quantity depends on the mass and the
velocity of the particle.
From we deduce
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
If the motion of the particle takes place in a conservative field of forces, ,
then exist such that .
The first integral (9.36) is the energy integral, V is the potential energy, while h is
the energy constant and has to be calculated from the initial conditions.
(9.36)
Thus, one have and Eq. (9.35) becomes
total (mechanical) energy
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
https://sites.ualberta.ca/~ygu/courses/phys144/notes/momentum2.pdf
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Theoretical Mechanics
Lecture 7. Dynamics of the particle. General theorems
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