Classical Mechanics Virtual Work & d’Alembert’s Principle
Dipan Kumar Ghosh UM-DAE Centre for Excellence in Basic Sciences Kalina, Mumbai 400098 August 15, 2016
1 Constraints
Motion of a system of particles is often constrained, either geometrically or kinematically. Constraints reduce the number of degrees of freedom of a given body. Consider the motion of a single particle in space. For free, unconstrained motion, it has three degrees of freedom which are usually expressed by three coordinates such as x, y, z or r, θ, ϕ etc. If, however, the particle is constrained to move on the surface of a sphere, we must have (taking Cartesian coordinates),
x2 + y2 + z2 = R2 which reduces the number of degrees of freedom by one. Consider two masses connected by a rigid rod, like a dumbbell. Two particles have 6 degrees of freedom, Since the distance between the two bodies remains constant, we have the constraint
2 2 2 2 (x1 − x2) + (y1 − y2) + (z1 − z2) = d which reduces the degree of freedom to 5. These are examples of geometric or holonomic constraints whoch are expressible as algebraic equations involving the coordinates. There are other constraints which restrict the motion of bodies . Some of these are expressible as differential equations which constrain the coordinates and components of velocities. These are called kinematic constraints. Non-integrable kinematic constraints which cannot be reduced to holonomic constraints are called non-holonomic constraints. Thus, if m is the dimension of the configuration space (i.e., the number of generalised coordinates), holonomic constraints are expressible as equations of the form
i f (t, q1, q2, . . . , qm) = 0, 1 ≤ i ≤ k
1 c D. K. Ghosh, IIT Bombay 2 where k is the number of constraints. Holonomic constraints are called scleronomic if they do not explicitly depend on time. Time dependent constraints are called rheonomic. Kinematic constraints are expressed as equations in the phase space
i f (t, q1, q2, . . . , qm;q ˙1, q˙2,..., q˙m) = 0, 1 ≤ i ≤ k
Both the constraints are classified as rheonomic if they explicitly depend on time. Sometimes a constraint may appear to be kinematic but may be in reality holonomic. For instance, a constraint of the type Ax˙ + B = 0 may actually be holonomic if there exists a function f such that A = ∂f/∂x and B = ∂f/∂t. We then have,
∂f df ∂f df = + ∂x dt ∂t which gives a holonomic constraint f = constant.
Example 1:
Consider two masses connected by pulleys, as shown. In general two particles have
6 degrees of freedom. However, m1 can only move along the x direction and m2 along the z direction. Thus y1 = z1 = 0 and x2 = y2 = 0. We are left with two degrees of freedom. However, if m1 moves along the x direction by a distance d, m2 would have to move along z direction by 2d, i.e., we get another holonomic constraint, z2 − 2x1 = 0 which reduces the degree of freedom further by one. The problem is essentially a one dimensional problem. m 1 P 1 z
x
P 2
m 2