“Uniformly accelerated motion” - Sx = 2t + 3t2; 2) Sx = 1.5t2; 3) Sx = 2t + 1.5t2; 4) Sx = 3t + t2. Equation for the dependence of the projection of the speed of a moving body on time: ?x=2+3t (m/s). How can this be illustrated graphically? uniformly accelerated motion? Speak out the summary. Answer questions. Write down formulas for this lesson topic. How is average speed determined?
“Rectilinear uniformly accelerated motion” - Acceleration. 1. 0. 8. Average speed... Velocity and acceleration do not coincide in direction. Dependency?(t). 2. How can you call this type of movement? Lesson topic: Rectilinear uniformly accelerated motion. The figure shows graphs for 3 bodies. 3. 5. Give examples when the speed of a body changes.
“Inertia in physics” - Test. 1. What is inertia? As friction decreases, the ball rolls further. Without action there is no movement." Report on physics by Guseva Anastasia. A. The stone falls to the bottom of the gorge. Galileo Galileo in inertia. Therefore, the action of a body on another body cannot be one-sided. Inertia translated from Latin means inactivity or inaction.
“Body dynamics” - Reference systems in which Newton’s first law is satisfied are called inertial. Dynamics. Dynamics is a branch of mechanics that examines the causes of the movement of bodies (material points). Newton's first law states: Newton's laws apply only to inertial systems countdown. In what frames of reference do Newton's laws apply?
“Uniform and uneven movement” - Yablunevka. Chistoozernoe. t 2. Uneven movement. L 1. =. Uniform movement. L2. Uniform and t 3. t 1. L3.
“Non-inertial frames of reference” - The principle of relativity. The modulus of the inertial force acting in the rotational reference frame on stationary bodies: Where is the distance from the body to the axis of rotation; - latitude of the area. OY: Example: In a stationary train carriage there is a toy car on a smooth table. Non-inertial reference systems. - Newton's second law.
There are a total of 23 presentations in the topic
Let us consider a closed system with many degrees of freedom, performing finite (in all coordinates) motion. Let us assume that the problem allows complete separation of variables in the Hamilton-Jacobi method. This means that with an appropriate choice of coordinates, the shortened action is the sum
functions, each of which depends on only one of the coordinates.
Since generalized impulses
then each of the functions can be represented in the form
These functions are ambiguous. Due to the finiteness of motion, each of the coordinates can run through values only in a certain finite interval. When changing “forward” and “backward” in this interval, the action receives an increment
where is the integral
taken according to the specified change.
Let's produce now canonical transformation in the same way as was done in § 50 for the case of one degree of freedom. The new variables will be " action variables" and "angular variables"
where the generating function is again the action expressed in terms of coordinates and quantities; equations of motion in these variables
(52,6)
(52,7)
We will also find, similarly to (50.7), that complete change coordinates (“forward” and “backward”) corresponds to changing the corresponding to:
In other words, the quantities are ambiguous functions of the coordinates, which, when changing the latter and returning to the original values, can change by any integer multiple of . This property can also be formulated as a property of a function (expressed in terms of coordinates and momenta) in phase space systems. Since the quantities themselves, if expressed through and q, are single-valued functions of these variables, then by substituting we get a function that, when traversing along any closed curve in phase space, can change to an integer multiple of (or to zero).
It follows that any single-valued function of the state of the system, being expressed through canonical variables, is periodic function angular variables with a period for each of them. It can therefore be expanded into a multiple Fourier series of the form
( - integers). Substituting here the angular variables as functions of time, we find that the time dependence of F is determined by a sum of the form
Each of the terms of this sum is a periodic function of time with frequency
representing the sum of integer multiples of the fundamental frequencies
But since all frequencies (52.10) are not, generally speaking, integral multiples (or rational parts) of any one of them, then the entire sum as a whole is not a strictly periodic function. This applies, in particular, to the coordinates q themselves and the momenta of the system.
Thus, the movement of the system is in general case strictly periodic neither in general nor in any of the coordinates. This means that if the system has passed through any state, then it will not pass through it again through any end time. It can be argued, however, that after sufficient large gap time it will pass as close as desired to this state. This property is meant by calling such a movement conditionally periodic.
In various special cases, two (or more) of the fundamental frequencies may turn out to be commensurate (for arbitrary values of ). In such cases, they speak of the presence of degeneracy, and if all s frequencies are commensurate, then the motion of the system is called completely degenerate.
IN the latter case, obviously, the movement is strictly periodic and thus the trajectories of all particles are closed.
The presence of degeneracy leads, first of all, to a decrease in the number independent quantities, on which the energy of the system depends. Let two frequencies be related by the relation
(52,12)
where are integers. It follows that quantities enter energy only in the form of a sum.
Very important feature degenerate motions is an increase in the number of single-valued integrals of motion compared to their number in the general case of a non-degenerate system (with the same number of degrees of freedom). In the last case of full number of all integrals of action, only s functions of the state of the system are single-valued; their complete set constitute, for example, s values h. The remaining integrals can be represented as differences
(52,13)
The constancy of these quantities follows directly from formula (52.7), but due to the ambiguity of the angular variables, they are not unambiguous functions of the state of the system.
In the presence of degeneracy, the situation changes. Thus, in view of connection (52.12), the integral
(52,14)
although it is ambiguous, its ambiguity is reduced to the addition of any integer multiple. Therefore it is enough to take trigonometric function this quantity in order to obtain a new unambiguous integral of motion.
An example of degenerate motion is motion in a field (see the problem for this section). It is this circumstance that leads to the emergence of a new, specific single-valued integral of motion (15.17), in addition to the two (we consider the motion immediately as flat) ordinary single-valued integrals - moment M and energy E - characteristic of motion in any central field.
We also note that the appearance of additional single-valued integrals leads in turn to another property of degenerate motions - they allow complete separation of variables for different, rather than for one specific choice of coordinates.
Indeed, the quantities in the coordinates that separate the variables are single-valued integrals of motion. But in the presence of degeneracy, the number of single-valued integrals exceeds s, and therefore the choice of those of them that we want to obtain as quantities becomes ambiguous
As an example, let us again mention Keplerian motion, which allows separation of variables in both spherical and parabolic coordinates.
In the previous paragraph it was shown that in one-dimensional finite motion the action variable is an adiabatic invariant. This statement remains valid for systems with many degrees of freedom. It is proved in the general case by a direct generalization of the method outlined at the beginning of § 51.
For a multidimensional system with a variable parameter, the equations of motion in canonical variables give for the rate of change of each of the action variables an expression similar to (50.10):
where, as before, averaging of this equality must be carried out over a period of time that is large compared to the main periods of the system, but small compared to the time of change in the parameter. In this case, it is again taken out from under the averaging sign, and the averaging of the derivatives is carried out as if the movement occurred at a constant k and therefore was conditionally periodic. Then A will be a single-valued periodic function of angular variables and the average values of its derivatives vanish.
In conclusion, let us make some comments about the properties of finite motion closed systems with many degrees of freedom in the most general case, which does not assume separability of variables in the corresponding Hamilton-Jacobi equation.
The main property of systems with separable variables is the uniqueness of the integrals of motion, the number of which is equal to the number of degrees of freedom. In the general case of systems with nonseparable variables, the set of unambiguous integrals of motion is limited to those whose constancy is an expression of the properties of homogeneity and isotropy of space and time, i.e., the laws of conservation of energy, momentum and momentum.
The phase trajectory of the system passes through those regions of the phase space that are determined by the given constant values single-valued integrals of motion. For a system with separable variables with its s single-valued integrals, these conditions define an -dimensional manifold in the phase space. Over a sufficiently long time, the trajectory of the system will cover this variety as densely as desired.
For a system with nonseparable variables, with its smaller (for the same s) number of single-valued integrals phase trajectory can fill domains (manifolds) in phase space more measurements.
Finally, we point out that if the Hamiltonian function of a system differs from a function that allows separation of variables only by small terms, then the properties of the motion are close to the properties of conditionally periodic motions, and the degree of this closeness is much higher than the degree of smallness of the additional terms in the Hamilton function.
Task
Calculate action variables for elliptical motion in a field.
Solution. IN polar coordinates in the plane of motion we have:
PERIODIC MOTION
In mechanics: uneven movement, in which, after a certain period of time, the previous circumstances are resumed.
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Above, we discussed the main cycles of movement in the geographic shell, which differ primarily in the nature of the material carrier. Cycles also differ in the nature of dynamic modes, which are understood as the types of changes in system parameters over time. One of these dynamic modes is periodic. In this case, the system reaches the same state at regular intervals. In physical-geographical phenomena, strict periodicity does not exist, therefore it is more correct to speak of “quasi-periodicity” (quasi - almost).
Periodicity in the geographic envelope manifests itself in many processes: tectonic, magmatic, sedimentation, climatic, hydrological and many others.
Numerous facts indicate climate fluctuations that are caused by periodic changes parameters earth's orbit, solar activity, tidal force and many other factors. This is evidenced quite reliably by geological, glaciological, archaeological data, as well as observations of historical period. Well, for example, climatic fluctuations can be traced with a duration of 35 years (this cycle of fluctuations was first established by the famous climatologist Brickner) and 1800 years. The latter is recorded in the development of the nature of the Sahara, where eras of humid and arid climates repeatedly alternated.
Periodicity is characteristic of tectono-magmatic processes: uplifts and subsidences, earthquakes, folding movements, intrusive and effusive volcanism. Tectono-magmatic epochs are separated by periods of relative tectonic peace of 50-150 million years. There is a reduction in the duration of periods between epochs of tectonic activity - the rate of geotectonic movements increases during the development of the Earth.
Periodicity can be traced in sections of geological deposits. It is clearly visible in terrigenous-carbonate and glaciolacustrine formations. In terrigenous-carbonate deposits (mainly Carboniferous and Permian in age), there is an alternation along the section of limestones, dolomites, clays, marls, sandstones, siltstones and other sediments. The rhythmicity of these deposits is associated with periodic oscillatory movements earth's crust and changes in sea level, as well as climate fluctuations.
In periglacial lakes, ribbon layering is formed. In summer, when the glacier melts, coarser material is brought into the lake; in winter, a fine clay sediment is deposited. A pair of such layers thus corresponds to one year.
Numerous evidence of the recurrence of phenomena has been found in the biosphere, glaciers, and relief.
Forced vibrations. The frequency of phenomena is associated with the influence of external factors (forced fluctuations) and internal laws development geographic envelope(autonomous oscillations, self-oscillations).
External factors that cause periodic phenomena include the position of the Solar system in orbit in our Galaxy, fluctuations in the eccentricity of the Earth’s orbit, changes in the inclination of its axis, etc. During the galactic year solar system passes through spaces with different densities substances (dust matter). The magnitude changes throughout the galactic year. gravitational field due to a change in the position of the masses relative to each other. A change in the density of dust matter leads to a change in the value of the solar constant, and the value gravitational forces- to fluctuations in the atmospheric and oceanic circulation system, changes in the compression of the ellipsoid of revolution, the position of the geoid surface, which, in turn, affects the configuration of land and sea, and sedimentation processes, etc. Classic example forced oscillations annual and daily rhythms may serve. They are associated with the intensity change mode solar radiation, which depends on planetary-astronomical factors - the rotation of the Earth around the Sun and around its axis and inclination earth's axis to the orbital plane. Since solar radiation is one of the most powerful factors influencing natural processes, daily and annual rhythms are characteristic of almost all physical and geographical phenomena. Due to their clear repeatability, the day and year serve as natural units of time in physical geography.
Changes in the timing of the equinoxes, the inclination of the rotation axis to the ecliptic and the eccentricity of the earth's orbit correspond to periods of about 21 thousand years, 40 thousand years and about 92 thousand years. These periods were studied by the Yugoslav scientist Milanković from the point of view of the impact on the distribution of solar radiation on the earth's surface. Changes in the listed characteristics are very weak, but their combined influence, observed during periods of coinciding phases of oscillations, is quite large and can serve as a cause of climate fluctuations.
Forced oscillations are also created under the influence of such planetary and astronomical factors as tidal forces. Rhythms with a duration of 1.2; 8.9; 18.9; about 111 and 1800-1900 years (Kalesnik S.V., 1970).
The occurrence of periodicity in many cases is a reflection of a change in the spatial position of the system. For example, seasonal and daily periodicity in the flow of solar radiation is associated with changes in the position of the Earth relative to the Sun. Fluctuations in the magnitude of the tidal force with a period of 1800 years, causing climate fluctuations, are associated with changes in the location of the Sun, Earth and Moon relative to each other. IN in this case the inextricable unity of space and time is manifested: temporal characteristics - rhythms, periods - arise as a reflection of the movements of objects in space.
Autonomous oscillations. In addition to the vibrations caused external factors, the geographic envelope is characterized by autonomous fluctuations. The latter are generally characteristic of systems that have at least two inertial links. Inertial objects are those that, with an instantaneous change in influences external to each of them, change their parameters not instantly, but gradually, as a result of the transition process. The longer the transition process, the more inertial the object. Strictly speaking, everything is inertial geographical features. However, the inertia of many of them is small, it is measured in minutes, hours, days. At the same time, such geographic shell systems as the ocean and continental ice, when exposed to external forces rebuild much more slowly. For example, the ocean cools slowly and warms up just as slowly. It still retains the cold that accumulated during the Pleistocene ice age. The advance and retreat of continental glaciers takes place over tens of thousands of years.
In the theory of automatic control (one of the branches of cybernetics), it is proven that in a system containing two or more inertial subsystems interacting according to a negative feedback scheme (see section III.4), self-oscillatory phenomena can occur. Moreover, fluctuations occur even under constant external influences. That is why they are called autonomous, that is, arising independently of external forces.
Climate changes and glaciations in the Pleistocene (about Pleistocene glaciation and its role in the development of the nature of the earth's surface, see section IV. 6). V. Ya. Sergin and S. Ya. Sergin ( System analysis problems of large climate fluctuations and glaciation of the Earth. L., 1978) built mathematical models systems "glaciers - ocean - atmosphere". In Fig. III.26 presents a graphical display of a system of equations connecting all elementary processes of heat and moisture exchange on the earth's surface. Such circuits are called functional. They allow us to imagine the system of interaction between the elements of the object under study and are the basis for constructing a mathematical model.
A study of computer models showed that the glacier-ocean-atmosphere system is characterized by self-oscillations that arise as a result of the transfer of mass and energy between two large inertial systems: the ocean and continental ice. The inertial properties of the ocean are associated with the high heat capacity of its waters, and of glaciers - with the low rate of accumulation and melting ice sheets. These inertial systems are united by nonlinear straight lines and feedback. Oscillations occur with a constant influx of solar radiation to the Earth. Setting external disturbances, including changes in latitudinal and annual distribution solar radiation and tectonically determined changes in land area, the authors obtained theoretical curves of glacial fluctuations (Fig. III. 27). The period of oscillation varies from 20 to 80 thousand years. Range of average fluctuations long-term temperature northern hemisphere is approximately 15°, southern - about 7 °C. The volume of continental glaciation changes by 20 million km3 in the northern and by 18-28 million km3 in southern hemispheres. The model study also made it possible to establish asymmetry in changes in the mass of glaciers, temperature and humidity of the earth's surface. Changes in the temperature of the earth's surface in relation to changes in ice mass lag behind. In the late Pleistocene era, this lag could be 1-3 thousand years. Thus, it cannot be said that glaciation is controlled by temperature.
There is an asymmetry of glacial cycles in relation to humidity: interglacial periods and the beginning of glaciations are characterized by a relatively humid climate, and the glaciations themselves and the beginning of interglacial periods are characterized by a relatively dry climate.
Apparently, weather changes also have a self-oscillating nature. They are not associated with intensity fluctuations electromagnetic radiation The sun, and are caused by the interaction of the atmosphere with the ocean, continents and glaciers. Factors such as cloudiness and differences in the thermodynamic characteristics of the atmosphere and ocean play a significant role. Cloudiness is an effective converter of a constant flow of solar radiation into a heat flow, the distribution of which is uneven in space and time. At the same time, cloudiness depends on the heat flux.
The inertia of the ocean, i.e. its slower (compared to the atmosphere) response to external influences (for example, to changes in the influx of solar radiation), causes a shift in all its thermodynamic characteristics over time. The ocean turns out to be a kind of “memory device” that stores information about states and processes for an earlier period of time. Thus, the existence and interaction of objects such as the atmosphere, ocean, glaciers, which have different characteristic times, regardless of external influences, inevitably leads to the occurrence of oscillatory movements.
The combination of vibrations associated with external influences, and self-oscillations leads to a complication of periodicity. However, it is most often impossible to strictly separate forced and autonomous oscillations. The superposition of oscillations of different frequencies and durations leads to the emergence of complex rhythms.
After the end of the full phase of the rhythm earth's surface and its individual subsystems do not return to their original state. Each phase of the rhythm brings something new. As a result, the system changes and evolves. The development of the system is carried out on the basis of those irreversible changes that accumulate over a long period of time.
Periodicity natural phenomena and their forecasts. Revealing the rhythm of natural phenomena has important to predict them. Rhythm is the repetition of phenomena in time, and if sufficiently stable repetitions of phenomena in the past are identified, then there is a high probability that they will be repeated in the future. Development Forecast Basis natural environment- knowledge of its previous states. The past is the key to knowing the future. Analysis of the past allows us to establish sustainable development trends natural processes and in many cases, extrapolate—carry established trends forward into the future.
There are many examples of forecasts based on knowledge of the rhythms of natural phenomena: forecasting general the annual course of weather conditions, and with them the nature of intra-annual changes river flow, development of vegetation cover and other phenomena. They also confidently predict the daily dynamics of phenomena. Particularly successful is forecasting the movement of planets, the Sun, solar and lunar eclipses. Clear rhythm of movements celestial bodies makes it possible to predict their relative position tens and even hundreds of years in advance.
However, the movements of celestial bodies are mechanical, not physical-geographical phenomena, the patterns of movement of which are more complex, and the rhythm is not so clearly expressed. Even in the daily and annual rhythm of physical-geographical phenomena, which has a planetary-astronomical nature, significant distortions are found. For example, it may be warmer at night than during the day. There may be frosts in the summer and thaws in the winter. These features arise due to the superposition of the daily and annual rhythms associated with radiation factors with atmospheric circulation, which has a complex and not yet sufficiently studied nature.
