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Physics 325: Fall 2018
Lecture 25
Lagrangian mechanics
The principle of least action (Hamilton’s Principle): The path taken by a mechanical system (subject to no non-conservative forces) with coordinates qi between the fixed endpoints {t 1 , ~q(t 1 )} and {t 2 , ~q(t 2 )} minimizes the action
S[~q(t)] =
∫ (^) t 2
t 1
L(qi, q˙i; t)dt (1)
where
L = T − U (2)
is the Lagrangian, T is the kinetic energy, and U is the potential energy (in an inertial reference frame). The equations of motion satisfied by the path ~q(t) satisfy the equations of motion
d dt
∂L
∂ q˙i
∂L
∂qi
Example:
Mass m on a spring of spring constant k. Let’s do this using both the Newtonian methods with which we are familiar, and the new Lagrangian method. Here, we call the deviation of the spring from its equilibrium position x.
Newtonian method: Here we use Newton’s second law
Fx = max ⇒ −kx = mx.¨ (4)
Lagrangian method: In this case the kinetic and potential energies are
T =
m x˙^2 , U =
kx^2 (5)
and so we construct the Lagrangian function
L = T − U =
m x˙^2 −
kx^2 (6)
The equation of motion is the Euler-Lagrange equation
∂L ∂x
= −kx = d dt
∂L
∂ x˙
= mx¨ (7)
which agrees with the Newtonian approach.
Example:
Unconstrained particle of mass m in 3D subject to a conservative force F~ = −∇~U. We work in cartesian coordinates qi = {x, y, z}, and write
T =
m 2
( ˙x^2 + ˙y^2 + ˙z^2 ) =
~r˙ · ~r˙ (8)
the Lagrangian is then
L = T − U = m 2
( ˙x^2 + ˙y^2 + ˙z^2 ) − U (9)
and the Euler-Lagrange equations are
m¨x = −
∂U
∂x
my¨ = −
∂U
∂y
mz¨ = −
∂U
∂z
which can be written
m~¨r = −∇~U (13)
which is the same as Newton’s second law.
Notes:
a) We will use generalized coordinates qi(t) instead of the more familiar Cartesian or Polar coordinates. i = 1, 2 , ....s where s is the number of degrees of freedom. The qi can be any set of numbers that we find useful and simple for describing the configuration of our system. We usually choose a set of generalized coordinates q that automatically satisfy constraints, for example
- The angle of a plane pendulum rather than the x and y coordinates of the tip.
- the distance of a bead along a wire rather than the xyz coordinates of the bead.
We sometimes find it convenient to choose generalized coordinates that are subject to further constraints, in which case the number (n) of qi is greater than the number of degrees of freedom n > s. Of this, more later.
b) The Lagrangian L, depends on the generalized coordinates qi, and on their time derivatives, the generalized velocities dqi/dt, but not higher derivatives, e.g. d^2 qi/dt^2. L will sometimes also depend explicitly on time. What is key about L is not its value so much as how it depends on all these variables: L = L(qi, dqi/dt; t). Be sure you recognize the distinction: L can depend on time explicitly in a known way, and/or through its dependence on q(t). The latter is implicit and is not known until after we have solved for the motion.
Generalized coordinates & constrained motion
Configuration space
In D-spatial dimensions, the configuration space of N free particles is described by N D independent coordinates, or degrees of freedom. For example, in 3-dimensions, there are 3N degrees of freedom. Equivalently, we can think of the configuration as a single point in
RDN^ = R ︸ D^ ⊗ RD^ ⊗ RD︷︷^ ⊗ RD^ ⊗ · · · RD︸ N −copies
This N D-dimensional Euclidean space is called the configuration space of the particles, C.
Trajectories in real space (left panel) and configuration space (right panel).
State space
The motion of the system gives rise to particle trajectories, {~r 1 (t), ~r 2 (t), ~r 3 (t),... , ~rN (t)} in RD^ (D- dimensional Euclidean space) and a corresponding trajectory in C. However, in order to completely specify the state of the system, it is also necessary to specify the particle velocities
~v 1 (t) = ~r˙ 1 (t), ~v 2 (t) = ~r˙ 2 (t), · · · , ~vN (t) = ~r˙N (t). (21)
Note: We consider only systems which satisfy the principle of determinacy (Newton): The state of the system at any instant t uniquely determines the state at any subsequent instant t′.
- This does not exclude deterministic chaos.
- It does exclude equations of motion with higher derivatives,
~r = fi.
- It does exclude non-causal solutions of general relativity, e.g. G¨odel’s rotating universe which has closed time-like loops. The positions and velocities of the N particles together define a single point in a 2N D- dimensional state-space or phase-space, S.
R ︸ D^ ⊗ RD^ ⊗ RD︷︷^ ⊗ RD^ ⊗ · · · RD︸ N −configurations
⊗ R ︸ D^ ⊗ RD^ ⊗ RD︷︷^ ⊗ RD^ ⊗ · · · RD︸
N −velocities
Constraints
When a mechanical system consists of N mass points which are not all free, then the available configurations space, C , is not all of RDN^ but a subset. Similarly for the state space S.
Holonomic constraints
Suppose that we have k constraints that may be written in the form
f 1 (~r 1 , ~r 2 ,... , ~rN , t) = f 2 (~r 1 , ~r 2 ,... , ~rN , t) = .. . fk(~r 1 , ~r 2 ,... , ~rN , t) =0. (23)
Such constraints are said to be holonomic and the system has s = 3N − k degrees of freedom in D = 3 dimensions, or s = DN − k degrees of freedom in general. For example, for the simple pendulum, s = 2 − 1 = 1 and f 1 = x^2 + y^2 − l^2 = 0. The constraints may be satisfied by choosing s independent variables q 1 , q 2 ,... , qs, such that the original N coordinates may be expressed as
~r 1 =~r 1 (q 1 , q 2 ,... , qs, t) ~r 2 =~r 1 (q 1 , q 2 ,... , qs, t) .. . ~rN =~rN (q 1 , q 2 ,... , qs, t). (24)
The q 1 , q 2 ,... , qs are called generalized coordinates.
Non-holonomic constraints
Non-holonomic constraints are those which cannot be put in the form of Eqn. (23). There are many varieties of non-holonomic constraint, for example
- Inequalities: Particle in a box { 0 ≤ x ≤ a, 0 ≤ y ≤ a, 0 ≤ z ≤ a}; particle outside a sphere {x^2 + y^2 + z^2 = l^2 }.
- Differential (non-integrable) constraints: A wheel rolling on a horizontal plane, always verti- cal, no slipping. In this case, the constraint is that the velocity of the centre of the wheel is constrained to be equal to the velocity of the point of contact. In equations, v = a φ˙. This constraint can be re-written as
x˙ =v cos θ = a φ˙ cos θ (25) y˙ =v sin θ = a φ˙ sin θ (26)
or
dx − a cos θdφ =0 (27) dy − a sin θdφ =0 (28)
These relations cannot be integrated without solving the problem. Physically, there is no functional relation between the angle and the other coordinates x, y, θ.
Recall the EL eqn is
d dt
∂L
∂ θ˙
∂L
∂θ
or
d dt
[mL^2 θ˙ + mL cos θ x˙] −
∂θ
[mL cos θ θ˙ x˙ + mgL cos θ] = 0 (38)
or, (again using the chain rule, but be careful; both x and θ are functions of time!)
mL^2 θ¨ + mLcosθx¨ − mL sin θ θ˙ x˙ + mL sin θ θ˙ x˙ + mgL sin θ = 0 (39)
You are advised to confirm that you get the same expression. We are fortunate to find that the middle terms cancel each other. We are left with
mL^2 θ¨ + mgL sin θ = −mL cos θ¨x (40)
The left side may be recognized as our familiar nonlinear pendulum equation. The right side is an effective force that looks like torque due to a fictitious sideways gravity.
There is no Euler-Lagrange equation associated with x. x is not a dynamical coordinate, because it is apriori prescribed; we do not need a governing differential equation for it. We do NOT write d dt
∂L
∂ x˙
∂L
∂x
Example: Bead on a spinning hoop
A hoop of radius R spins on its vertical axis at a fixed rate Ω as illustrated, with a bead of mass m free to slide along the hoop.
This case is a particle (the bead) in three spatial dimensions, so its configuration space has dimension
- There are two constraints, the bead must be on the hoop, which amounts to the constraint
x^2 + y^2 + z^2 = R^2 (42)
and, the beads angle in the x-y plane is specified
φ = arctan
( (^) y x
= Ωt (43)
Thus we expect one degree of freedom. Now, we note that if we work in spherical-polar coordinates centered on the center of the hoop, such that
x =R sin θ cos Ωt, x˙ = R θ˙ cos θ cos Ωt − RΩ sin θ sin Ωt (44) y =R sin θ sin Ωt y˙ = R θ˙ cos θ sin Ωt + RΩ sin θ cos Ωt (45) z = − R cos θ z˙ = R sin θ θ˙ (46)
Then, we compute
T = m 2
( ˙x^2 + ˙y^2 + ˙z^2 ) (47)
= m 2
R θ˙ cos θ cos Ωt − RΩ sin θ sin Ωt
R θ˙ cos θ sin Ωt + RΩ sin θ cos Ωt
m 2
R^2 θ˙^2 + m 2
R^2 Ω^2 sin^2 θ (49)
The bead has potential energy
Ubead = mgz = −Rmg cos θ (50)
The Lagrangian L = T − U is therefore
L =
mR^2 ( θ˙)^2 +
m(R sin θ)^2 Ω^2 + Rmg cos θ (51)
The EL equation is: (there is only one because there is only one dynamical coordinate)
d dt
[
∂L
∂ θ˙
= mR^2 θ˙
]
[
∂L
∂θ = −Rmg sin θ + mR^2 sin θ cos θΩ^2
]
Again using the chain rule to evaluate the d/dt
mR^2 θ¨ + Rmg sin θ − mR^2 sin θ cos θΩ^2 = 0 (53)
This is the second order nonlinear equation that governs θ(t). It could have been derived in other ways. In the first third of the course, it was derived using F = ma in spherical coordinates and by introducing constraint forces (normal forces) between the hoop and the bead. Another way would be to make a free body diagram in the rotating frame that includes centrifugal force. (Coriolis force is out of the paper and compensated by a normal force between the wire and the bead.) Resolving the forces along the wire, and using Newton, one gets the same governing equation for θ.
