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The Lagrangian Method Vs.Other Methods [email protected] COMPARATIVE EXAMPLE

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The Lagrangian Method Vs.Other Methods Jozef.Hanc@Tuke.Sk COMPARATIVE EXAMPLE Jozef HANC 1 The Lagrangian Method vs.other methods [email protected] COMPARATIVE EXAMPLE The Lagrangian Method vs. other methods (COMPARATIVE EXAMPLE) This material written by Jozef HANC, [email protected] Technical University, Kosice, Slovakia For Edwin Taylor‘s website http://www.eftaylor.com/ 17 February 2003 The aim of this material is a demonstration of power of Lagrange’s equations for solving physics problems based on comparing the solution of an example by five different methods: “Vectorial” methods • Using Newton’s laws (inertial frame of reference) • Using Newton’s laws with fictitious forces (accelerated frame of reference) • Using D’Alembert’s principle (The extended principle of virtual work) “Scalar” methods • Using Conservation laws • Using Lagrange’s equations (The Lagrangian method) Here is our problem: MOVING PLANE: A block of mass m is held motionless on a frictionless plane of m mass M and angle of inclination θ. The plane rests on a fric- M tionless horizontal surface. The block is released (see Fig.1). 1 θ What is the horizontal acceleration of the plane? Fig. 1 SOLUTIONS: „VECTORIAL” METHODS: 2 I. METHOD: Newton’s laws (in an inertial frame of reference) : N vertical Fig. 2 Diagram with forces applying on the positive moving block showing the notation: direction N – magnitude of the normal force from mg the plane mg – magnitude of gravity horizontal positive direction 1 The problem was taken from section 2.7 in Morin [Ref.1], ch2.pdf, p.18 2 This solution according to Morin [Ref. 1], ch2.pdf, p. 22 Jozef HANC 2 The Lagrangian Method vs.other methods [email protected] COMPARATIVE EXAMPLE vertical positive F Fig. 3 Diagram of the moving plane shows notation: direction N – magnitude the normal force from the block N θ Mg – magnitude of gravity F –magnitude of the normal force from the base Mg horizontal positive direction Equations of motion of the block θ = horizontally N sin max (1.1) − θ = vertically mg N cos ma y (1.2) Equations of motion of the plane − θ = horizontally N sin MAx (1.3) + θ − = = vertically Mg N cos F MAy 0 (1.4) Kinematical equation resulting from the motion of the block: a t 2 / 2 a y = y = tan θ (1.5) − 2 − (ax Ax )t / 2 ax Ax We have five equations with five unknowns. To find the quantity Ax, it is sufficient to resolve equations (1.1), (1.2), (1.3) and (1.5): a N sin θ = ma mg − N cosθ = ma − N sin θ = MA y = tan θ x y x − a x Ax The solution for Ax has to be done by a more time consuming procedure: θ = − mg tan Ax (1.6) M (1+ tan 2 θ) + m tan 2 θ Jozef HANC 3 The Lagrangian Method vs.other methods [email protected] COMPARATIVE EXAMPLE II. METHOD: Newton’s laws with fictitious forces: FORCES acting on the BLOCK (moving in an accelerated frame of reference) vertical positive N Fig. 4 Diagram of the moving block in accelerated direction frame determined by the plane. N – magnitude of the normal force from the plane Ftrans F – magnitude of the translation fictitious force mg trans equal to mAx and with opposite direction as Ax mg – magnitude of gravity horizontal positive direction Equations of motion of the block: θ + − θ = horizontally mg sin ( mAx ) cos ma (2.1) θ − − − θ = vertically mg cos N ( mAx )sin 0 (2.2) Equations of motion of the plane are identical with those written in Section 1, because we consider the plane in the same inertial frame of reference. So: − θ = horizontally N sin MAx (2.3) + θ − = = vertically Mg N cos F MAy 0 (2.4) We have four equations with four unknowns. To find the quantity Ax, it is sufficient to resolve equations (2.2) and (2.3) θ − + θ = − θ = mg cos N mAx sin 0 N sinMAx The solution for Ax is given by a quick way: θ θ = − mg sin cos Ax (2.5) M + msin 2 θ Using known trigonometric formulas, we get the same result as (1.6). Jozef HANC 4 The Lagrangian Method vs.other methods [email protected] COMPARATIVE EXAMPLE III. METHOD: D’Alembert’s principle (“The extended principle of virtual work”) SHORT SUMMARY: 3 D’Alembert’s Principle : The total virtual work of the effective forces is zero for all reversible variations (virtual displacements), which satisfy the given kinematical conditions. REMARKS: 1) This principle uses the concept of the virtual displacement, which means, not actual small displacement, but it involves a possible, but purely mathematical experiment, so it can be applied in a certain definite time (even if such a displacement would involve physically infinite velocities). At the instant the actual motion of the body does not enter into account. To distinguish between virtual displacement from actual one, we use label δ instead of ∆ or d, e.g. δx, δy, δR. 2) The special name − the effective forces − represents a common name for the given external or applied forces on a mechanical system, augmented by the inertial forces (see fictitious force in Section 2), but without the forces of constraint. 3) Since the forces of constraint do not need to be taken into account, D’Alembert principle immediately gives the consequence: The virtual work of the forces of reaction is always zero for any virtual dis- placement which is in harmony with the given kinematic constrains. Or in other words we restrict our- selves to systems for which the net virtual work of the forces of constraint is zero4. This assertion is eas- ily understandable from the Newtonian mechanics. If e.g. a particle is constrained to move on a surface, the force of constraint arises from the reaction forces (Newton’s third law) and is perpendicular to the surface, while virtual displacement must be tangent to it, and hence the virtual work of the force van- ishes. However, in general case it is not so evident and becomes postulate called POSTULATE A. horizontal positive direction Fig. 5 Diagram of “impressed” forces (mg, vertical positive Mg) − the given external forces acting on the direction m g mechanical system. The forces of constraint have no influence on a value of virtual work θ and can be neglected in a virtual-work calcu- lation. Mg Total virtual work is represented by sum of virtual works of all impressed forces together with the inertial forces on mechanical system: block + plane (the vectorial form): δW = (mg − ma)δr + (Mg − MA)δR = 0 (3.1) where a and A are accelerations of the block and the plane, Mg+(−MA), mg+(−ma) are corre- sponding net effective forces acting on the block and the plane, and δr, δR are virtual dis- placements of the block and plane in harmony with the given kinematical conditions. 3 The formulation of D’Alembert’s Principle according to Lanczos [Ref. 2], page 90, Section IV. See also Ref. 3, Section 1.4 4 According to Goldstein [Ref.3] Jozef HANC 5 The Lagrangian Method vs.other methods [email protected] COMPARATIVE EXAMPLE We can rewrite the equation (3.1) in more appropriate, analytical language. Using Cartesian coordinates: a=(ax, ay), A=(Ax, 0), Mg+(−MA)=(−MAx, Mg), mg+(−ma) = ( − max, mg − may), and δr = (δx, δy), δR = (δX, 0), we get the total virtual work as a sum of virtual works of components of the effective forces in x and y-direction: δ = − δ + − δ + − δ = W (mg ma y ) y ( ma x ) x ( MAx ) X 0 (3.2) All these displacements must be in harmony with the kinematical conditions (the block must 5 slide down the frictionless plane). It gives the condition : δy = (δx − δX ) tan θ (3.3) In other words all the virtual displacements δx, δy, δX are not independent. At the same time the previous equation (3.3) provides the following relations between accelerations ax, ay, Ax: = − θ a y (ax Ax ) tan (3.4) Since one equation (3.3) connects the virtual displacements, one of the displacement, say δx, 6 can by (3.3) be expressed in terms of the other two δy, δX. Substitution of δx in terms of δy, δX into the virtual work (3.2) with an rearrangement leads to: δ = θ − θ − δ − + δ = W (mg tan ma y tan max ) y (max MAx ) X 0 (3.5) This remaining pair δy, δX may be considered as quite independent displacements without further restriction (Think about it!). Then the vanishing of the resultant work requires: θ − θ − = mg tan ma y tan max 0 (3.6a) + = ma x MAx 0 (3.6b) These equations (3.6) together with equation (3.4) provide the system of three equations with three unknowns. Solving the equations, we obtain in short time the same result for Ax as (1.6). ADDITIONAL NOTE (This note can be skipped the first time): The analytical treatment according to D’Alembert’s principle takes only the external and inertial forces into account. Since the inner forces which produce constraints need not be consider, D’Alembert’s method offers a significant simplifica- tion in comparison with Newton’s laws. But it is interested to look at the forces of constraints (see Fig. 6). horizontal positive direction N2 Fig. 6 Diagram of all forces of constraints N1, N2 and F. According to D’Alembert’s princi- vertical positive ple the forces of constraint have no influence N1 F direction on a value of virtual work and their work has zero value θ 5 This condition can be obtained by the same way as in Section IV, Figure 7 of this material.
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