" Lagrange’s Equations with Undetermined Multipliers Marion, Section 7.5 • Holonomic Constraints are defined as those
which can be expressed as algebraic relations among the coordinates.
– If a system has only holonomic constraints ⇒ We can always find a proper set of generalized coordinates in which the equations of motion do not explicitly depend on constraints. • Constraints which depend on both the coordinates & the velocities are non-holonomic unless the constraint equations can be integrated to give relations among the coordinates.
• A constraint which depends on both the coordinates & the velocities is of the form: F(xαi,xαi,t) = 0
– It is non-holonomic unless it can be integrated to give another function G(xαi,t) = 0
• As an example of this, consider a constraint relation of the form: ∑i Ai xi + B = 0 (i = 1,2,3) (1)
(1) is non-integrable (& so the constraint is nonholonomic) unless Ai & B have the special forms [f = f(xi,t)]:
Ai = (∂f/∂xi), B = (∂f/∂t)
– In that case, (1) becomes: ∑i(∂f/∂xi)(dxi/dt) + (∂f/∂t) = 0 • Or, (1) becomes: (df/dt) = 0, which can be integrated to find: f(xi,t) = const, or f(xi,t) - const = 0, a holonomic constraint!
• This example shows that ⇒ Constraints which can be written in the differential form: ∑j(∂f/∂qj)dqj + (∂f/∂t)dt = 0 (2) are holonomic constraints. • Often, in practical physical situations, the constraints can be written in the differential form (2). If this is the case, the constraints can be explicitly incorporated into Lagrange’s Equations using the method of Lagrange’s undetermined multipliers (Sect. 6.6).
⇒ The formalism of Sect. 6.6 (changing to the notation of Ch. 7) shows that if the holonomic constraints can be written in the differential form:
∑j(∂fk/∂qj)dqj = 0
j =1,2, ..,s; k =1,2,..m
(A)
(also can be shown explicitly using the variational form of Hamilton’s Principle) Lagrange’s Equations become:
(∂L/∂qj) - (d/dt)[∂L/∂qj] + ∑kλk(t)(∂fk/∂qj) = 0
(B) λk(t) ≡ Lagrange’s undetermined multipliers
We could also have added a term (∂fk/∂t)dt to (A) & still have gotten (B).
Summary • Lagrange’s Equations with constraints: (∂L/∂qj) - (d/dt)[∂L/∂qj] + ∑kλk(t)(∂fk/∂qj) = 0 λk(t) ≡ Lagrange’s undetermined multipliers
– The λk(t) are determined as part of solution to the problem! – The physical interpretation of the λk(t) will be discussed next.
Lagrange’s Equations
(∂L/∂qj) - (d/dt)[∂L/∂qj] + ∑kλk(t)(∂fk/∂qj) = 0 (C)
• The major advantage to Lagrangian Mechanics: Explicit inclusion of forces is not necessary. Emphasis is placed on the dynamics of the system rather than on the calculation of forces. • Often, however, we might want or need to know the forces of constraint. It is explicitly shown in graduate texts (& we’ll see in some examples) that the Lagrange multipliers λk(t) can be used to calculate the (generalized) forces of constraint!
Generalized Forces
• Specifically, it can be shown that the Generalized Forces of Constraint Qj (corresponding to the generalized coordinates qj) are given by:
Qj ≡ ∑kλk(t)(∂fk/∂qj)
The last term in (C)!
• Just as the generalized coordinates qj do not necessarily have units of length, the corresponding generalized forces Qj do not necessarily have units of force! We’ll see this in the examples!
Example 7.9
• A disk, radius R, rolls without slipping down an inclined plane (total length ) as shown. Find the equations of motion, the force of constraint, & the angular acceleration. • Note: The generalized coordinates y & θ are not independent. They are related by y = Rθ. ⇒ The constraint equation is
f(y,θ) = y - Rθ = 0
This is equivalent to the differential versions:
(∂f/∂y) = 1; (∂f/∂θ) = -R
Worked on the board with 2 separate methods: without & with Lagrange multipliers.
Example 7.10
• A particle of mass m starts from rest on top of smooth hemisphere of radius a (figure). Find the force of constraint & determine the angle at which m leaves hemisphere.
Summary • Lagrange’s Equations with constraints:
(∂L/∂qj) - (d/dt)[∂L/∂qj] + ∑kλk(t)(∂fk/∂qj) = 0
(C) • The usefulness of Lagrange Eqtns with undetermined multipliers:
– The Lagrange multipliers λk(t) can be used to obtain the forces of constraint. These are often needed. Get λk(t) as part of the solution to the equations of motion. – When a proper set of generalized coordinates is not wanted or is too difficult to get, we can use this method to increase the number of generalized coordinates by including the constraints explicitly in the equations of motion.
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