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Lecture # 4
Variational Principles and Lagrange’s Equations
1. Hamilton’s Principle
2. Calculus of Variations
3. Derivation of Lagrange’s Equations from Hamilton’s Principle
4. Hamilton’s Principle for Nonholonomic Systems
5. Conservation Theorems and Symmetry Properties
6. Energy Function and the Conservation of Energy
1. Hamilton’s Principle
1.1 Action Integral and Hamilton’s principle
The entire motion of the system between times t1 and t2, and small virtual variations of this motion
from the actual motion. This motion is such that the line integral (called the action or the action
integral),
• where L = T − V, has a stationary value for the actual path of the motion.
• The “stationary value means that the integral along the given path has the same value to within
first-order infinitesimals as that along all neighboring paths (i.e., those that differ from it by
infinitesimal displacements).
• We can summarize Hamilton’s principle by saying that the motion is such that the variation of
the line integral I for fixed t1 and t2 is zero:
1.2 Motion of a System and Configuration Space
• The instantaneous configuration of a system is described by the values of the n generalized coordinates q1,
. . . , qn, and corresponds to a particular point in a Cartesian hyperspace.
• q’s form the n coordinate axes. This n-dimensional space is therefore known as configuration space.
• As time goes on, the state of the system changes and the system point moves in configuration space
tracing out a curve, described as “the path of motion of the system.”
• The “motion of the system,” refers to the motion of the system point along this path in configuration space.
• The configuration space has no necessary connection with the physical three-dimensional space, just as
the generalized coordinates are not necessarily position coordinates.
• The path of motion in configuration space has no resemblance to the path in space of any actual particle;
each point on the path represents the entire system configuration at some given instant of time.
2. Calculus of Variations
To find the curve for which some given line integral has a stationary value:
Consider the problem in an essentially one-dimensional form,
• have a function f (y, 𝑦, x) defined on a path y = y(x) between two values x1 and x2,
where 𝑦 is the derivative of y with respect to x.
• We wish to find a particular path y(x) such that the line integral J of the function ‘f’ between x1 and x2, has a
stationary value relative to paths differing infinitesimally from the correct function y(x).
Consider only such varied paths for which y(x1) = y1, y(x2) = y2.
• Since J must have a stationary value for the correct path relative to any neighboring path.
• The variation must be zero relative to some particular set of neighboring paths labeled by an infinitesimal parameter α.
• Such a set of paths might be denoted by y(x, α), with y(x, 0) representing the correct path.
• If we select any function η(x) that vanishes at x = x1 and x = x2, then a possible set of varied paths is given by;
Calculus of Variations ………………
For simplicity, it is assumed that both the correct path y(x) and the auxiliary function η(x) are well-behaved functions—
continuous between x1 and x2. For any such parametric family of curves, J in Eq. (2.3) is also a function of α:
The condition for obtaining a stationary point is;
and
• Since J must have a stationary value for the correct path relative to any neighboring path.
• The variation must be zero relative to some particular set of neighboring paths labeled by an infinitesimal parameter α.
• Such a set of paths might be denoted by y(x, α), with y(x, 0) representing the correct path.
• If we select any function η(x) that vanishes at x = x1 and x = x2, then a possible set of varied paths is given by;
Calculus of Variations ………………
For simplicity, it is assumed that both the correct path y(x) and the auxiliary function η(x) are well-behaved functions—
continuous and nonsingular between x1 and x2, with continuous first and second derivatives in the same interval. For
any such parametric family of curves, J in Eq. (2.3) is also a function of α:
The condition for obtaining a stationary point is;
and
Calculus of Variations ………………
Consider the second term of these integrals: Integrating by parts, the integral becomes;
All varied curves pass through the points (x1, y1), (x2, y2), and hence the partial derivative of y with respect to α at x1 and x2
must vanish.
Calculus of Variations ………………
Integrating by parts, the integral becomes;
All varied curves pass through the points (x1, y1), (x2, y2), and hence the partial derivative of y with respect to α at x1 and x2
must vanish.
Consider the second term of these integrals:
Calculus of Variations ………………
Integrating by parts, the integral becomes;
All varied curves pass through the points (x1, y1), (x2, y2), and hence the partial derivative of y with respect to α at x1 and x2
must vanish.
Consider the second term of these integrals:
Calculus of Variations ………………
Integrating by parts, the integral becomes;
All varied curves pass through the points (x1, y1), (x2, y2), and hence the partial derivative of y with respect to α at x1 and x2
must vanish. Therefore, the first term of (2.8) vanishes and Eq. (2.7) reduces to;
The condition for a stationary value, Eq. , is therefore equivalent to the equation
• The partial derivative of y with respect to α is a function of x that is arbitrary except for continuity and end point conditions.
• For example, for the particular parametric family of varied paths given by Eq. (2.4), it is the arbitrary function η(x).
• We can imagine constructing a function η that is positive in the immediate vicinity of any chosen point in the interval and zero
everywhere else. From Eq. (2.9), it therefore follows that J can have a stationary value only if;
Consider the second term of these integrals:
Calculus of Variations ………………
The differential quantity, represents the infinitesimal departure of the varied path from the correct path y(x) at the
point x and thus corresponds to the virtual displacement δy;
This concluded that J is stationary for the correct path and Eq. 2.9 can be
written;
The infinitesimal variation of J about the correct path is;
⸫y(x) satisfy the differential equation (2.11).
1. Shortest distance between two points in a plane.
2. Minimum surface of revolution.
3. The brachistochrone problem.
3. Derivation of Lagrange’s Equations from Hamilton’s Principle
The fundamental problem of the calculus of variations is easily generalized to the case where f is a function of many
independent variables yi , and their derivatives 𝑦i . Then a variation of the integral J ,
Consider J as a function of parameter α that labels a possible set of curves y1(x, α)
. . . . . .
where y1(x, 0), y2(x, 0), etc., are the solutions of the extremum problem (to be obtained) and η1, η2, etc., are
independent functions of x that vanish at the end points. The variation of J is given in terms of;
Integrate by parts the integral involved in the second sum
Where the first term vanishes because all curves pass through the fixed end points. Substituting in (2.16), δ J
becomes;
The variation δyi is;
Since the y variables are independent, the variations δyi are independent the condition that δ J is zero requires that the
coefficients of the δyi separately vanish:
for the integral in Hamilton’s
principle,
has just the form stipulated in (2.14) with the transformation;
In deriving Eqs. (2.18), we assumed that the yi variables are independent. The corresponding condition in connection with
Hamilton’s principle is that the generalized coordinates qi be independent, which requires that the constraints be holonomic.
The Euler-Lagrange equations corresponding to the integral I then become the Lagrange equations of motion,
Derivation of Lagrange’s Equations . . . . . . . . . . . .
4. Hamilton’s Principle for Nonholonomic Systems
• Lagrange’s equations from either Hamilton’s or D’Alembert’s principle, the holonomic constraints appear in the last
step when the variations in the qi were considered independent of each other.
• However, the virtual displacements in the δqI ’s may not be consistent with the constraints.
• If there are n variables and m constraint equations, the extra virtual displacements are eliminated by the method of
Lagrange undetermined multipliers.
We modify the integral in Eq. (2.19) to be;
This allow to qα and the λα to vary independently to obtain n +m equations. The variations of the λα’s give the m
constraint equations. The variations of the qi ’s give
However, the δqi ’s are not independent. We choose the λα’s so that m of the equations are satisfied for arbitrary δqi
, and then choose the variations of the δqi in the remaining n − m equations
independently. Thus we obtain m equations of the form
for k = 1, . . . ,m. The equality follows from the choice of the λα’s. We also have the same
expressions as Eq. (2.22)
for k = m + 1, . . . , n, where the equality follows from the
virtual variations of the δqi ’s. Where the Q are generalized forces.
In general, nonholonomic constraints cannot be expressed by a variational principle. One of the exceptions is
semiholonomic constraints where the constraints can be written as a set of functions of the form
4. Hamilton’s Principle for Nonholonomic Systems
where α = 1, 2, . . . ,m. Equation (2.24) commonly appears in the form;
where the fα are a set of nonintegrable differential expressions and the aαk and a0 are functions of the qι and t. In
these cases, since we cannot integrate the constraints. However, we can treat the variations in the same
fashion as before by writing;
where the symbol μ is used to distinguish these multipliers from the holonomic Lagrange multipliers. If we
assume that μα = μα(t), the equations resulting from the virtual displacements are
These two sets (Eq. (2.26) and (2.27)) together constitute n + m equations for the n + m unknowns. Hence
they can be interpreted as equivalent to an n + m holonomic system with generalized forces Qk .

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Variational Principle

  • 1. Lecture # 4 Variational Principles and Lagrange’s Equations 1. Hamilton’s Principle 2. Calculus of Variations 3. Derivation of Lagrange’s Equations from Hamilton’s Principle 4. Hamilton’s Principle for Nonholonomic Systems 5. Conservation Theorems and Symmetry Properties 6. Energy Function and the Conservation of Energy
  • 2. 1. Hamilton’s Principle 1.1 Action Integral and Hamilton’s principle The entire motion of the system between times t1 and t2, and small virtual variations of this motion from the actual motion. This motion is such that the line integral (called the action or the action integral), • where L = T − V, has a stationary value for the actual path of the motion. • The “stationary value means that the integral along the given path has the same value to within first-order infinitesimals as that along all neighboring paths (i.e., those that differ from it by infinitesimal displacements). • We can summarize Hamilton’s principle by saying that the motion is such that the variation of the line integral I for fixed t1 and t2 is zero:
  • 3. 1.2 Motion of a System and Configuration Space • The instantaneous configuration of a system is described by the values of the n generalized coordinates q1, . . . , qn, and corresponds to a particular point in a Cartesian hyperspace. • q’s form the n coordinate axes. This n-dimensional space is therefore known as configuration space. • As time goes on, the state of the system changes and the system point moves in configuration space tracing out a curve, described as “the path of motion of the system.” • The “motion of the system,” refers to the motion of the system point along this path in configuration space. • The configuration space has no necessary connection with the physical three-dimensional space, just as the generalized coordinates are not necessarily position coordinates. • The path of motion in configuration space has no resemblance to the path in space of any actual particle; each point on the path represents the entire system configuration at some given instant of time.
  • 4. 2. Calculus of Variations To find the curve for which some given line integral has a stationary value: Consider the problem in an essentially one-dimensional form, • have a function f (y, 𝑦, x) defined on a path y = y(x) between two values x1 and x2, where 𝑦 is the derivative of y with respect to x. • We wish to find a particular path y(x) such that the line integral J of the function ‘f’ between x1 and x2, has a stationary value relative to paths differing infinitesimally from the correct function y(x). Consider only such varied paths for which y(x1) = y1, y(x2) = y2.
  • 5. • Since J must have a stationary value for the correct path relative to any neighboring path. • The variation must be zero relative to some particular set of neighboring paths labeled by an infinitesimal parameter α. • Such a set of paths might be denoted by y(x, α), with y(x, 0) representing the correct path. • If we select any function η(x) that vanishes at x = x1 and x = x2, then a possible set of varied paths is given by; Calculus of Variations ……………… For simplicity, it is assumed that both the correct path y(x) and the auxiliary function η(x) are well-behaved functions— continuous between x1 and x2. For any such parametric family of curves, J in Eq. (2.3) is also a function of α: The condition for obtaining a stationary point is; and
  • 6. • Since J must have a stationary value for the correct path relative to any neighboring path. • The variation must be zero relative to some particular set of neighboring paths labeled by an infinitesimal parameter α. • Such a set of paths might be denoted by y(x, α), with y(x, 0) representing the correct path. • If we select any function η(x) that vanishes at x = x1 and x = x2, then a possible set of varied paths is given by; Calculus of Variations ……………… For simplicity, it is assumed that both the correct path y(x) and the auxiliary function η(x) are well-behaved functions— continuous and nonsingular between x1 and x2, with continuous first and second derivatives in the same interval. For any such parametric family of curves, J in Eq. (2.3) is also a function of α: The condition for obtaining a stationary point is; and
  • 7. Calculus of Variations ……………… Consider the second term of these integrals: Integrating by parts, the integral becomes; All varied curves pass through the points (x1, y1), (x2, y2), and hence the partial derivative of y with respect to α at x1 and x2 must vanish.
  • 8. Calculus of Variations ……………… Integrating by parts, the integral becomes; All varied curves pass through the points (x1, y1), (x2, y2), and hence the partial derivative of y with respect to α at x1 and x2 must vanish. Consider the second term of these integrals:
  • 9. Calculus of Variations ……………… Integrating by parts, the integral becomes; All varied curves pass through the points (x1, y1), (x2, y2), and hence the partial derivative of y with respect to α at x1 and x2 must vanish. Consider the second term of these integrals:
  • 10. Calculus of Variations ……………… Integrating by parts, the integral becomes; All varied curves pass through the points (x1, y1), (x2, y2), and hence the partial derivative of y with respect to α at x1 and x2 must vanish. Therefore, the first term of (2.8) vanishes and Eq. (2.7) reduces to; The condition for a stationary value, Eq. , is therefore equivalent to the equation • The partial derivative of y with respect to α is a function of x that is arbitrary except for continuity and end point conditions. • For example, for the particular parametric family of varied paths given by Eq. (2.4), it is the arbitrary function η(x). • We can imagine constructing a function η that is positive in the immediate vicinity of any chosen point in the interval and zero everywhere else. From Eq. (2.9), it therefore follows that J can have a stationary value only if; Consider the second term of these integrals:
  • 11. Calculus of Variations ……………… The differential quantity, represents the infinitesimal departure of the varied path from the correct path y(x) at the point x and thus corresponds to the virtual displacement δy; This concluded that J is stationary for the correct path and Eq. 2.9 can be written; The infinitesimal variation of J about the correct path is; ⸫y(x) satisfy the differential equation (2.11). 1. Shortest distance between two points in a plane. 2. Minimum surface of revolution. 3. The brachistochrone problem.
  • 12. 3. Derivation of Lagrange’s Equations from Hamilton’s Principle The fundamental problem of the calculus of variations is easily generalized to the case where f is a function of many independent variables yi , and their derivatives 𝑦i . Then a variation of the integral J , Consider J as a function of parameter α that labels a possible set of curves y1(x, α) . . . . . . where y1(x, 0), y2(x, 0), etc., are the solutions of the extremum problem (to be obtained) and η1, η2, etc., are independent functions of x that vanish at the end points. The variation of J is given in terms of; Integrate by parts the integral involved in the second sum Where the first term vanishes because all curves pass through the fixed end points. Substituting in (2.16), δ J becomes; The variation δyi is;
  • 13. Since the y variables are independent, the variations δyi are independent the condition that δ J is zero requires that the coefficients of the δyi separately vanish: for the integral in Hamilton’s principle, has just the form stipulated in (2.14) with the transformation; In deriving Eqs. (2.18), we assumed that the yi variables are independent. The corresponding condition in connection with Hamilton’s principle is that the generalized coordinates qi be independent, which requires that the constraints be holonomic. The Euler-Lagrange equations corresponding to the integral I then become the Lagrange equations of motion, Derivation of Lagrange’s Equations . . . . . . . . . . . .
  • 14. 4. Hamilton’s Principle for Nonholonomic Systems • Lagrange’s equations from either Hamilton’s or D’Alembert’s principle, the holonomic constraints appear in the last step when the variations in the qi were considered independent of each other. • However, the virtual displacements in the δqI ’s may not be consistent with the constraints. • If there are n variables and m constraint equations, the extra virtual displacements are eliminated by the method of Lagrange undetermined multipliers. We modify the integral in Eq. (2.19) to be; This allow to qα and the λα to vary independently to obtain n +m equations. The variations of the λα’s give the m constraint equations. The variations of the qi ’s give However, the δqi ’s are not independent. We choose the λα’s so that m of the equations are satisfied for arbitrary δqi , and then choose the variations of the δqi in the remaining n − m equations independently. Thus we obtain m equations of the form for k = 1, . . . ,m. The equality follows from the choice of the λα’s. We also have the same expressions as Eq. (2.22) for k = m + 1, . . . , n, where the equality follows from the virtual variations of the δqi ’s. Where the Q are generalized forces.
  • 15. In general, nonholonomic constraints cannot be expressed by a variational principle. One of the exceptions is semiholonomic constraints where the constraints can be written as a set of functions of the form 4. Hamilton’s Principle for Nonholonomic Systems where α = 1, 2, . . . ,m. Equation (2.24) commonly appears in the form; where the fα are a set of nonintegrable differential expressions and the aαk and a0 are functions of the qι and t. In these cases, since we cannot integrate the constraints. However, we can treat the variations in the same fashion as before by writing; where the symbol μ is used to distinguish these multipliers from the holonomic Lagrange multipliers. If we assume that μα = μα(t), the equations resulting from the virtual displacements are These two sets (Eq. (2.26) and (2.27)) together constitute n + m equations for the n + m unknowns. Hence they can be interpreted as equivalent to an n + m holonomic system with generalized forces Qk .