5.S: Calculus of Variations (Summary)
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Euler’s differential equation
The calculus of variations has been introduced and Euler’s differential equation was derived. The calculus of variations reduces to varying the functions yi(x), where i=1,2,3,...n, such that the integral
F=∫x2x1f[yi(x),y′i(x);x]dx
is an extremum, that is, it is a maximum or minimum. Here x is the independent variable, yi(x) are the dependent variables plus their first derivatives y′i≡dyidx. The quantity f[y(x),y′(x);x] has some given dependence on yi,y′i and x. The calculus of variations involves varying the functions yi(x) until a stationary value of F is found which is presumed to be an extremum. It was shown that if the yi(x) are independent, then the extremum value of F leads to n independent Euler equations
∂f∂yi−ddx∂f∂y′i=0
where i=1,2,3..n. This can be used to determine the functional form yi(x) that ensures that the integral F=∫x2x1f[y(x),y′(x);x]dx is a stationary value, that is, presumably a maximum or minimum value.
Note that Euler’s equation involves partial derivatives for the dependent variables yi,y′i, and the total derivative for the independent variable x.
Euler’s integral equation
It was shown that if the function ∫x2x1f[yi(x),y′i(x);x] does not depend on the independent variable, then Euler’s differential equation can be written in an integral form. This integral form of Euler’s equation is especially useful when ∂f∂x=0, that is, when f does not depend explicitly on x, then the first integral of the Euler equation is a constant f−y′∂f∂y′=constant
Constrained variational systems
Most applications involve constraints on the motion. The equations of constraint can be classified according to whether the constraints are holonomic or non-holonomic, the time dependence of the constraints, and whether the constraint forces are conservative.
Generalized coordinates in variational calculus
Independent generalized coordinates can be chosen that are perpendicular to the rigid constraint forces and therefore the constraint does not contribute to the functional being minimized. That is, the constraints are embedded into the generalized coordinates and thus the constraints can be ignored when deriving the variational solution.
Minimal set of generalized coordinates
If the constraints are holonomic then the m holonomic equations of constraint can be used to transform the n coupled generalized coordinates to s=n−m independent generalized variables qi,q′i. The generalized coordinate method then uses Euler’s equations to determine these s=n−m independent generalized coordinates. ∂f∂qi−ddx∂f∂q′i=0
Lagrange multipliers for holonomic constraints
The Lagrange multipliers approach for n variables, plus m holonomic equations of constraint, determines all N+m unknowns for the system. The holonomic forces of constraint acting on the N variables, are related to the Lagrange multiplier terms λk(x)∂gk∂yi) that are introduced into the Euler equations.
That is,
∂f∂yi−dfdx∂f∂y′i+m∑kλk(x)∂gk∂yi
where the holonomic equations of constraint are given by
gk(yi;x)=0
The advantage of using the Lagrange multiplier approach is that the variational procedure simultaneously determines both the equations of motion for the N variables plus the m constraint forces acting on the system.