Conservation Laws In Classical Mechanics Research Articles

On a paradox in the impact dynamics of smooth rigid bodies

Paradoxes in the impact dynamics of rigid bodies are known to arise in the presence of friction. We show here that, on specific occasions, in the absence of friction, the conservation laws of classical mechanics are also incompatible with the collisions of smooth, strictly convex rigid bodies. Under the assumption that the impact impulse is along the normal direction to the surface at the contact point, two convex rigid bodies that are well separated can come into contact, and then interpenetrate each other. This paradox can be demonstrated in both 2D and 3D when the collisions are tangential, in which case no momentum or energy transfer between the two bodies is possible. The postcollisional interpenetration can be realized through the contact points or through neighboring points only. The penetration distance is shown to be . The conclusion is that rigid-body dynamics is not compatible with the conservation laws of classical mechanics.

Symmetries and conservation laws in classical and quantum mechanics

We describe the connection between continuous symmetries and conservation laws in classical mechanics. This is done at successively more sophisticated levels, bringing out important features at each level: the Newtonian, the Euler-Lagrange, and the Hamiltonian phase-space forms of mechanics. The role of the Action Principle is emphasised, and many examples are given.

Lattice gas automata for fluid mechanics

A lattice gas is the representation of a gas by its restriction on the nodes of a regular lattice for discrete time steps. It was recently shown by Frisch, Hasslacher and Pomeau that such very simple models lead to the incompressible Navier-Stokes equation provided the lattice has enough symmetry and the local rules for collisions between particles obey the usual conservation laws of classical mechanics. We present here recent results of numerical simulations to illustrate the power of this new approach to fluid mechanics which may give new tools for numerical studies and build a bridge between cellular automata theory and complex physical problems.

Conservation laws in classical mechanics for quasi-coordinates

Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.

Conservation laws in classical mechanics

Conservation laws in classical mechanics

On the Conservation Laws in Classical Mechanics

It is pointed out that any classical system having n degrees of freedom has 2n − 1 constants of motion which can be found from the dynamical equations. A subsequent analysis is made of the dynamical content of the conservation laws. In this connection, transformation theory is applied by using a complete set of dynamical variables that, in general, are not required to form a canonical conjugate set.

Symmetry and conservation laws in classical mechanics

Ideas of symmetry form an essential part of modern physics. It seems advantageous to introduce these ideas to students at a relatively early stage by using them in the context of classical mechanics. The aim is to demonstrate the results that invariance under time and space translations and space rotations imply conservation of energy, momentum and angular momentum. But how does one teach this and just what is it that has to be invariant?

Invariance and Conservation Laws in Classical Mechanics. II

In this paper, the invariances of the equation of motion of a classical particle, to coordinate and time translations, to scale transformations and inversions, and to Galilean transformations, are considered individually. Resultant conditions on the equation of motion are given, and, for invariance to the one-parameter continuous transformations, it is shown that the equation of motion can be reduced from second to first order. Associated with each such reduction is a conservation law. The implications of the invariance of the system Lagrangian to these transformations are indicated, and the conservation laws, if any, associated with them. Some requirements on the Lagrangian for invariant equations of motion are also presented, and it is shown that the invariance of an equation of motion derived from a Lagrangian does not imply the invariance of that Lagrangian to the transformation. It is also shown that time-translation invariance of the equation of motion does not always require conservation of the Hamiltonian.

Invariance and Conservation Laws in Classical Mechanics

It is commonly supposed that, in classical mechanics, invariance of a physical system to coordinate translation implies conservation of linear momentum. ``Invariance'' may be defined in a number of ways. If it is defined to mean invariance of the equation of motion, it is shown that invariance of this equation with respect to coordinate translation does not imply conservation of linear momentum. The effects of scale transformation and coordinate inversion invariance are also investigated. Both the Lagrangian approach and Newton's (second) law approach are considered. It is shown that each of the above invariances implies a condition on the equation of motion, while a combination of these and time-inversion invariance is needed to obtain ordinary momentum conservation.