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Principles of Mathematical Modelling: Ideas, Methods, Examples

Alexander A. Samarskii (Author); Alexander P. Mikhailov

Chapter 3

MODELS DEDUCED FROM VARIATIONAL PRINCIPLES, HIERARCHIES OF MODELS - all with Video Answers

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Section 1

Equations of Motion, Variational Principles and Conservation Laws in Mechanics

Problem 1

Show that equations (3) describe the motion of balls of masses $m_1, m_2$, connected with a weightless spring of rigidity $k$ ( $l$ is the length of a unloaded spring, $\left.r_1(t) \leq r_2(t)\right)$. Check an invariance of equations at the passage from one inertial system to another.

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Problem 2

Using in (8) the identical transformation of an initial system of coordinates, be convinced that from equations (9) the coordinate form of equations (1) is obtained.

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Problem 3

Using the equality (16), check that for a path along which $\delta Q=0$, the Lagrangian equations (11) are satisfied.

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Problem 4

Establish the equivalence of the Hamiltonian equations (21) and Lagrangian ones (11) in an example of the motion of a single point mass in a stationary potential field, considering it in Cartesian coordinates $x, y, z$.

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Problem 5

In the absence of external forces, the potential energy of a system and hence its Lagrangian function do not vary at the shift of beginning of coordinates. Taking in (22) the transformations of Cartesian coordinates $x_i^*=x_i+\alpha, y^*=y_i, z^*=z_i$, $(i=1, \ldots, N), t^*=t$, prove that (23) has a form $\phi=\sum_{i=1}^N m_i \dot{x}_i=$ const, i.e.for such a system the momentum conservation law in projection on axis $x$ is valid.

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