1. STATIC EQUILIBRIUM [4] Calkin, M. G. “Lagrangian and Hamiltonian Mechanics”, World Scientific, Singapore, 1996, ISBN 981-02-2672-1 Consider an object having mass M held in the air at height y > 0 above the ground and released. Energy conservation implies that its height y(t) as a function of time t > 0 since release must satisfy the equation Since this equation holds if and only if either energy conservation doesn’t imply Newtons 2 nd law

2. STATIC EQUILIBRIUM We observe that and, for an unconstrained object having mass M, position r (a vector), and a position dependent potential energy U(r), we observe that energy conservation implies that An object is said to be in static equilibrium if its position r is constant. Therefore, if a system is not in static equilibrium then energy conservation implies Newton’s 2 nd law.

3. ENERGY NOT ENOUGH We made the painful observation, that conservation of energy by itself does not suffice to explain the real world in which things happen: rain falls from the sky and objects move when they are pushed Definition A virtual displacement of an object is any change of its position that can be imagined without violating any specific constraint. The virtual work is the scalar product of the force and the virtual displacement Examples Virtual displacements for a falling object are arbitrary vectors, but those for an object that is constrained to lie on a table must be horizontal

4. STATIC PRINCIPLE OF VIRTUAL WORK Fundamental Principle: An object is in static equilibrium only if the virtual work associated with every virtual displacement is zero Example 1 Consider an unconstrained object that is subjected to a force F. Then, if the object is in static equilibrium, we obtain from the principle above that for every virtual displacementTherefore, since the object is unconstrained we may choose to obtain thathence

5. STATIC PRINCIPLE OF VIRTUAL WORK Example 2 Consider an object that is constrained to lie on a table surface that is flat. Assume that the only force on the object is the gravitational force F. Then if the object is in static equilibrium for every virtual displacementSince the object is constrained to lie on the table surface, the virtual displacements consist of all vectors parallel to the table. Therefore, if the object is in static equilibrium F must be orthogonal to the table surface. (Recall that two vectors are orthogonal if and only if their scalar product equals zero)

6. STATIC PRINCIPLE OF VIRTUAL WORK Example 3 This example shows that the static principle of virtual work applies to systems consisting of one or more objects that may be mutually as well as individually constrained. Consider two weights attached by a flexible rope as illustrated below. Let the configuration of the system be represented by a vector whose coordinates are the heights of each object. If the system is in static equilibrium what is the ratio of the weights ? gravitational force

7. STATIC PRINCIPLE OF VIRTUAL WORK Example 4 Consider two weights on a pulley as illustrated in the figure below If the system is in static equilibrium what is the ratio of the weights ? gravitational force

8. D’ALEMBERT’S PRINCIPLE Definition For a moving object the force of constraint is D’Alemberts Principle: The virtual work done by force of constraint equals zero, this means that for every virtual displacement This is a dynamic principle of virtual work. It can be extended to describe systems with many objects.

9. D’ALEMBERT’S PRINCIPLE Example 1 An object slides on an inclined plane (x, y -horizontal coordinates) having angle and D’Alembert’s principle implies that therefore the equations of motion are

10. GENERALIZED COORDINATES The configuration of a system with N objects having masses whose positions are constrained by 3N-f independent functions can be parameterized by f independent variables Virtual displacements, expressed in GC, are

11. LAGRANGE’S EQUATIONS D’Alembert’s principle expressed in GC has the form Therefore, the chain rule for derivatives implies that

12. LAGRANGE’S EQUATIONS D’Alembert’s Principle implies, but can NOT be derived from Newton’s 2 nd Law. However, it can be derived from conservation of energy together with the static principle of virtual work by defining constraints as the limit of forces orthogonal to the constraint set. If each applied force is conservative then there exists a potential energy function such that This is the reasonare called generalized forces.

13. LAGRANGE’S EQUATIONS Example 1 Consider a single unconstrained object with mass M and position r. We can choose q = r to obtain Therefore, Lagrange’s Equations reduce to Newton’s Second Law This is the reason generalized momenta. are called

14. LAGRANGE’S EQUATIONS Example 2 Compute the trajectory of an object with mass M sliding on a surface whose height = h(x,y) (x and y are horizontal coordinates) Choose generalized coordinates Then the kinetic energy And the generalized momenta

15. LAGRANGE’S EQUATIONS Example 2 For i=1,2 the generalized force and Lagrange’s Equations are a pair of second order differential equations.

16. LAGRANGE’S EQUATIONS Example 2 A tedious, but direct computation yields

17. LAGRANGE’S EQUATIONS Example 2 where A is the 2 x 3 matrix with entries

18. LAGRANGE’S EQUATIONS Example 2

19. GEODESICS AND GENERAL RELATIVITY If we set g = 0 in Example 2 we obtain the equations for a geodesic trajectory on the surface, that is the uniform speed trajectory whose distance between any two points is minimal. Albert Einstein showed that an object in any inertial frame, such as in space moving with constant speed or falling freely under the Earths gravity, follows a geodesic trajectory. This is how he explained the equivalence between gravitational and inertial mass that was measured by experimentalists with such amazing precision.

20. HAMILTON’S EQUATIONS AND ENERGY If, as in Example 2, the kinetic energy is a quadratic function of the generalized velocities and the forces are conservative (arising from a potential energy U) then the total energy is given by the Hamiltonian The system dynamics satisfies Hamilton’s equation the foundation of advanced and quantum mechanics.